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Chapter 4 Resource Masters
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such materials be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with the Glencoe Geometry program. Any other reproduction, for sale or other use, is expressly prohibited.
Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240 - 4027
ISBN: 978-0-07-890513-1
MHID: 0-07-890513-3
Printed in the United States of America.
1 2 3 4 5 6 7 8 9 10 024 14 13 12 11 10 09 08
Consumable Workbooks Many of the worksheets contained in the Chapter Resource Masters are available as consumable workbooks in both English and Spanish.
ISBN10 ISBN13
Study Guide and Intervention Workbook 0-07-890848-5 978-0-07-890848-4
Homework Pratice Workbook 0-07-890849-3 978-0-07-890849-1
Spanish Version
Homework Practice Workbook 0-07-890853-1 978-0-07-890853-8
Answers for Workbooks The answers for Chapter 4 of these workbooks can be found in the back of this Chapter Resource Masters booklet.
StudentWorks PlusTM This CD-ROM includes the entire Student Edition test along with the English workbooks listed above.
TeacherWorks PlusTM All of the materials found in this booklet are included for viewing, printing, and editing in this CD-ROM.
Spanish Assessment Masters (ISBN10: 0-07-890856-6, ISBN13: 978-0-07-890856-9) These masters contain a Spanish version of Chapter 4 Test Form 2A and Form 2C.
iii
Contents
Teacher’s Guide to Using the Chapter 4
Resource Masters .........................................iv
Chapter ResourcesStudent-Built Glossary ....................................... 1
Anticipation Guide (English) .............................. 3
Anticipation Guide (Spanish) ............................. 4
Lesson 4-1Classifying Triangles
Study Guide and Intervention ............................ 5
Skills Practice .................................................... 7
Practice .............................................................. 8
Word Problem Practice ..................................... 9
Enrichment ...................................................... 10
Lesson 4-2Angles of Triangles
Study Guide and Intervention .......................... 11
Skills Practice .................................................. 13
Practice ............................................................ 14
Word Problem Practice ................................... 15
Enrichment ...................................................... 16
Cabri Junior. ................................................... 17
Geometer’s Sketchpad Activity ....................... 18
Lesson 4-3Congruent Triangles
Study Guide and Intervention .......................... 19
Skills Practice .................................................. 21
Practice ............................................................ 22
Word Problem Practice ................................... 23
Enrichment ...................................................... 24
Lesson 4-4Proving Triangles Congruent—SSS, SAS
Study Guide and Intervention .......................... 25
Skills Practice .................................................. 27
Practice ............................................................ 28
Word Problem Practice ................................... 29
Enrichment ...................................................... 30
Lesson 4-5Proving Triangles Congruent—ASA, AAS
Study Guide and Intervention .......................... 31
Skills Practice .................................................. 33
Practice ............................................................ 34
Word Problem Practice ................................... 35
Enrichment ...................................................... 36
Lesson 4-6Isosceles and Equilateral Triangles
Study Guide and Intervention .......................... 37
Skills Practice .................................................. 39
Practice ............................................................ 40
Word Problem Practice ................................... 41
Enrichment ...................................................... 42
Lesson 4-7Congruence Transformations
Study Guide and Intervention .......................... 43
Skills Practice .................................................. 45
Practice ............................................................ 46
Word Problem Practice ................................... 47
Enrichment ...................................................... 48
Lesson 4-8Triangles and Coordinate Proof
Study Guide and Intervention .......................... 49
Skills Practice .................................................. 51
Practice ............................................................ 52
Word Problem Practice ................................... 53
Enrichment ...................................................... 54
AssessmentStudent Recording Sheet ................................ 55
Rubric for Extended Response ....................... 56
Chapter 4 Quizzes 1 and 2 ............................. 57
Chapter 4 Quizzes 3 and 4 ............................. 58
Chapter 4 Mid-Chapter Test ............................ 59
Chapter 4 Vocabulary Test ............................. 60
Chapter 4 Test, Form 1 ................................... 61
Chapter 4 Test, Form 2A ................................. 63
Chapter 4 Test, Form 2B ................................. 65
Chapter 4 Test, Form 2C ................................ 67
Chapter 4 Test, Form 2D ................................ 69
Chapter 4 Test, Form 3 ................................... 71
Chapter 4 Extended-Response Test ............... 73
Standardized Test Practice ............................. 74
Answers ........................................... A1–A40
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Teacher’s Guide to Using the
Chapter 4 Resource MastersThe Chapter 4 Resource Masters includes the core materials needed for Chapter 4. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing on
the TeacherWorks PlusTM CD-ROM.
Chapter ResourcesStudent-Built Glossary (pages 1–2) These masters are a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar. Give this to students before beginning Lesson 4-1. Encourage them to add these pages to their mathematics study notebooks. Remind them to complete the appropriate words as they study each lesson.
Anticipation Guide (pages 3–4) This master, presented in both English and Spanish, is a survey used before beginning the chapter to pinpoint what students may or may not know about the concepts in the chapter. Students will revisit this survey after they complete the chapter to see if their perceptions have changed.
Lesson ResourcesStudy Guide and Intervention These masters provide vocabulary, key concepts, additional worked-out examples and Check Your Progress exercises to use as a reteaching activity. It can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent.
Skills Practice This master focuses more on the computational nature of the lesson. Use as an additional practice option or as homework for second-day teaching of the lesson.
Practice This master closely follows the types of problems found in the Exercises section of the Student Edition and includes word problems. Use as an additional practice option or as homework for second-day teaching of the lesson.
Word Problem Practice This master includes additional practice in solving word problems that apply the concepts of the lesson. Use as an additional practice or as homework for second-day teaching of the lesson.
Enrichment These activities may extend the concepts of the lesson, offer a historical or multicultural look at the concepts, or widen students’ perspectives on the mathematics they are learning. They are written for use with all levels of students.
Graphing Calculator or Spreadsheet
Activities These activities present ways in which technology can be used with the con-cepts in some lessons of this chapter. Use as an alternative approach to some con-cepts or as an integral part of your lesson presentation.
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Assessment OptionsThe assessment masters in the
Chapter 4 Resource Masters offer a wide range of assessment tools for formative (monitoring) assessment and summative (final) assessment.
Student Recording Sheet This master corresponds with the standardized test practice at the end of the chapter.
Extended Response Rubric This master provides information for teachers and stu-dents on how to assess performance on open-ended questions.
Quizzes Four free-response quizzes offer assessment at appropriate intervals in the chapter.
Mid-Chapter Test This 1-page test provides an option to assess the first half of the chapter. It parallels the timing of the Mid-Chapter Quiz in the Student Edition and includes both multiple-choice and free-response questions.
Vocabulary Test This test is suitable for all students. It includes a list of vocabulary words and 10 questions to assess students’ knowledge of those words. This can also be used in conjunction with one of the leveled chapter tests.
Leveled Chapter Tests
• Form 1 contains multiple-choice questions and is intended for use with below grade level students.
• Forms 2A and 2B contain multiple-choice questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.
• Forms 2C and 2D contain free-response questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.
• Form 3 is a free-response test for use with above grade level students.
All of the above mentioned tests include a free-response Bonus question.
Extended-Response Test Performance assessment tasks are suitable for all students. Sample answers and a scoring rubric are included for evaluation.
Standardized Test Practice These three pages are cumulative in nature. It includes three parts: multiple-choice questions with bubble-in answer format, griddable questions with answer grids, and short-answer free-response questions.
Answers• The answers for the Anticipation Guide
and Lesson Resources are provided as reduced pages.
• Full-size answer keys are provided for the assessment masters.
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NAME DATE PERIOD
Chapter 4 1 Glencoe Geometry
Student-Built Glossary
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 4.
As you study the chapter, complete each term’s definition or description.
Remember to add the page number where you found the term. Add these pages to
your Geometry Study Notebook to review vocabulary at the end of the chapter.
4
Vocabulary TermFound
on PageDefinition/Description/Example
auxiliary line
base angle
congruent
corresponding parts
congruent polygons
equiangular triangle
equilateral triangle
image
included angle
(continued on the next page)
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Chapter 4 2 Glencoe Geometry
Student-Built Glossary (continued)
Vocabulary TermFound
on PageDefinition/Description/Example
included side
isosceles triangle
preimage
reflection
rotation
scalene (SKAY·leen) triangle
transformation
translation
4
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Chapter 4 3 Glencoe Geometry
Anticipation Guide
Congruent Triangles
Before you begin Chapter 4
• Read each statement.
• Decide whether you Agree (A) or Disagree (D) with the statement.
• Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure).
After you complete Chapter 4
• Reread each statement and complete the last column by entering an A or a D.
• Did any of your opinions about the statements change from the first column?
• For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.
4
Step 1
STEP 1A, D, or NS
StatementSTEP 2A or D
1. For a triangle to be acute, all three angles must be acute.
2. All sides of an equilateral triangle are congruent.
3. An exterior angle is any angle outside of the given triangle.
4. The sum of the measures of the angles in a triangle is 360°.
5. An obtuse triangle is a triangle with three obtuse angles.
6. Triangles that are the same shape and size are congruent.
7. If the angles of one triangle are congruent to the angles of another triangle, then the two triangles are congruent.
8. An isosceles triangle is a triangle with two congruent sides.
9. A triangle can be equiangular and not equilateral.
10. To make the calculations in a coordinate proof easier, place either a vertex or the center of the polygon at the origin.
Step 2
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NOMBRE FECHA PERÍODO
Ejercicios preparatorios
Triá ngulos congruentes
Antes de comenzar el Capítulo 4
• Lee cada enunciado.
• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.
• Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta, escribe NS (No estoy seguro(a)).
Después de completar el Capítulo 4
• Vuelve a leer cada enunciado y completa la última columna con una A o una D.
• ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna?
• En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con los enunciados que marcaste con una D.
Paso 2
Paso 1
PASO 1A, D, o NS
EnunciadoPASO 2A o D
1. Para que un triángulo sea acutángulo, todos sus tres ángulos deben ser agudos.
2. Todos los lados de un triángulo equilátero son iguales.
3. Un ángulo exterior es cualquier ángulo fuera del triángulo dado.
4. La suma de las medidas de los ángulos de un triángulo es 360°.
5. Un triángulo obtusángulo es un triángulo con tres ángulos obtusos.
6. Los triángulos que son iguales en forma y tamaño son congruentes.
7. Si los ángulos de dos triángulos son congruentes, entonces los dos triángulos son congruentes.
8. Un triángulo isósceles es un triángulo con dos lados congruentes.
9. Un triángulo puede ser equiángulo y no equilátero.
10. Para facilitar los cálculos en una demostración por coordenadas, ubica ya sea un vértice o el centro del polígono en el origen.
4
Capítulo 4 4 Geometría de Glencoe
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Less
on
4-1
Chapter 4 5 Glencoe Geometry
Study Guide and Intervention
Classifying Triangles
Classify Triangles by Angles One way to classify a triangle is by the measures of its angles.
• If all three of the angles of a triangle are acute angles, then the triangle is an acute triangle.
• If all three angles of an acute triangle are congruent, then the triangle is an equiangular triangle.
• If one of the angles of a triangle is an obtuse angle, then the triangle is an obtuse triangle.
• If one of the angles of a triangle is a right angle, then the triangle is a right triangle.
Classify each triangle.
a.
60°
A
B C
All three angles are congruent, so all three angles have measure 60°.
The triangle is an equiangular triangle.
b.
25°35°
120°
D F
E
The triangle has one angle that is obtuse. It is an obtuse triangle.
c.
90°
60° 30°
G
H J
The triangle has one right angle. It is a right triangle.
Exercises
Classify each triangle as acute, equiangular, obtuse, or right.
1. 67°
90° 23°
K
L M
2.
120°
30° 30°N O
P
3.
60° 60°
60°
Q
R S
4.
65° 65°
50°
U V
T 5.
45°
45°90°X Y
W 6. 60°
28° 92°F D
B
4-1
Example
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Chapter 4 6 Glencoe Geometry
Study Guide and Intervention (continued)
Classifying Triangles
Classify Triangles by Sides You can classify a triangle by the number of congruent sides. Equal numbers of hash marks indicate congruent sides.
• If all three sides of a triangle are congruent, then the triangle is an equilateral triangle.
• If at least two sides of a triangle are congruent, then the triangle is an isosceles triangle.
Equilateral triangles can also be considered isosceles.
• If no two sides of a triangle are congruent, then the triangle is a scalene triangle.
Classify each triangle.
a.
L J
H b. N
R P
c.
2312
15X V
T
Two sides are congruent. All three sides are The triangle has no pair
The triangle is an congruent. The triangle of congruent sides. It is
isosceles triangle. is an equilateral triangle. a scalene triangle.
Exercises
Classify each triangle as equilateral, isosceles, or scalene.
1. 2
1
3√G C
A 2. G
K I18
18 18
3.
12 17
19Q O
M
4.
UW
S 5. 8x
32x
32x
B
C
A 6. D E
F
x
x x
Example
4-1
7. ALGEBRA Find x and the length of each
side if △RST is an equilateral triangle.
8. ALGEBRA Find x and the length of
each side if △ABC is isosceles with
AB = BC.
5x - 42x + 2
3x
3y + 24y
3y
Lesso
n 4
-1
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Chapter 4 7 Glencoe Geometry
Skills Practice
Classifying Triangles Classify each triangle as acute, equiangular, obtuse, or right.
1.
60° 60°
60° 2. 40°
95° 45°
3.
50° 40°
90°
4. 80°
50°
50°
5.
100°
30°
50° 6. 70°
55° 55°
Classify each triangle as equilateral, isosceles, or scalene.
E CD
AB9
8
8√2 7. △ABE 8. △EDB
9. △EBC 10. △DBC
11. ALGEBRA Find x and the length of
each side if △ABC is an isosceles
triangle with −−
AB # −−
BC .
12. ALGEBRA Find x and the length of each
side if △FGH is an equilateral triangle.
5x - 83x + 10
4x + 1
13. ALGEBRA Find x and the length of
each side if △RST is an isosceles
triangle with −−
RS # −−
TS .
3x - 1
9x + 2
6x + 8
14. ALGEBRA Find x and the length of each
side if △DEF is an equilateral triangle.
5x - 212x + 6
7x - 39
4-1
4x - 3 2x + 5
x + 2
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Chapter 4 8 Glencoe Geometry
Practice
Classifying Triangles Classify each triangle as acute, equiangular, obtuse, or right.
1. 2. 3.
Classify each triangle in the figure at the right by its angles and sides.
4. △ABD 5. △ABC
6. △EDC 7. △BDC
ALGEBRA For each triangle, find x and the measure of each side.
8. △FGH is an equilateral triangle with FG = x + 5, GH = 3x - 9, and FH = 2x - 2.
9. △LMN is an isosceles triangle, with LM = LN, LM = 3x - 2, LN = 2x + 1, and MN = 5x - 2.
Find the measures of the sides of △KPL and classify each triangle by its sides.
10. K(-3, 2), P(2, 1), L(-2, -3)
11. K(5, -3), P(3, 4), L(-1, 1)
12. K(-2, -6), P(-4, 0), L(3, -1)
13. DESIGN Diana entered the design at the right in a logo contest sponsored by a wildlife environmental group. Use a protractor. How many right angles are there?
A CD
E
B
60°
30°
90°
65°30°
85°
40°
100°
40°
4-1
Lesso
n 4
-1
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Chapter 4 9 Glencoe Geometry
Word Problem Practice
Classifying Triangles
1. MUSEUMS Paul is standing in front of
a museum exhibition. When he turns his
head 60° to the left, he can see a statue
by Donatello. When he turns his head
60° to the right, he can see a statue by
Della Robbia. The two statues and Paul
form the vertices of a triangle. Classify
this triangle as acute, right, or obtuse.
2. PAPER Marsha cuts a rectangular piece
of paper in half along a diagonal. The
result is two triangles. Classify these
triangles as acute, right, or obtuse.
3. WATERSKIING Kim and Cassandra
are waterskiing. They are holding on to
ropes that are the same length and tied
to the same point on the back of a speed
boat. The boat is going full speed ahead
and the ropes are fully taut.
Kim, Cassandra, and the point where
the ropes are tied on the boat form the
vertices of a triangle. The distance
between Kim and Cassandra is never
equal to the length of the ropes. Classify
the triangle as equilateral, isosceles, or
scalene.
4. BOOKENDS Two bookends are shaped
like right triangles.
The bottom side of each triangle is
exactly half as long as the slanted side
of the triangle. If all the books between
the bookends are removed and they are
pushed together, they will form a single
triangle. Classify the triangle that can
be formed as equilateral, isosceles, or
scalene.
5. DESIGNS Suzanne saw this pattern on
a pentagonal floor tile. She noticed many
different kinds of triangles were created
by the lines on the tile.
a. Identify five triangles that appear to
be acute isosceles triangles.
b. Identify five triangles that appear to
be obtuse isosceles triangles.
4-1
Kim
Cassandra
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IF
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Chapter 4 10 Glencoe Geometry
Enrichment
Reading MathematicsWhen you read geometry, you may need to draw a diagram to make the text
easier to understand.
Consider three points, A, B, and C on a coordinate grid. The
y-coordinates of A and B are the same. The x-coordinate of B is greater than the
x-coordinate of A. Both coordinates of C are greater than the corresponding
coordinates of B. Is △ABC acute, right, or obtuse?
To answer this question, first draw a sample triangle
that fits the description.
Side −−
AB must be a horizontal segment because the
y-coordinates are the same. Point C must be located
to the right and up from point B.
From the diagram you can see that triangle ABC
must be obtuse.
Exercises
Draw a simple triangle on grid paper to help you.
1. Consider three points, R, S, and 2. Consider three noncollinear points,
T on a coordinate grid. The J, K, and L on a coordinate grid. The
x-coordinates of R and S are the y-coordinates of J and K are the
same. The y-coordinate of T is same. The x-coordinates of K and L
between the y-coordinates of R are the same. Is △JKL acute,
and S. The x-coordinate of T is less right, or obtuse?
than the x-coordinate of R. Is angle
R of △RST acute, right, or
obtuse?
3. Consider three noncollinear points, 4. Consider three points, G, H, and I
D, E, and F on a coordinate grid. on a coordinate grid. Points G and
The x-coordinates of D and E are H are on the positive y-axis, and
opposites. The y-coordinates of D and the y-coordinate of G is twice the
E are the same. The x-coordinate of y-coordinate of H. Point I is on the
F is 0. What kind of triangle must positive x-axis, and the x-coordinate
△DEF be: scalene, isosceles, or of I is greater than the y-coordinate
equilateral? of G. Is △GHI scalene, isosceles, or
equilateral?
BA
C
x
y
O
4-1
Example
Less
on
4-2
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Chapter 4 11 Glencoe Geometry
Study Guide and Intervention
Angles of Triangles
Triangle Angle-Sum Theorem If the measures of two angles of a triangle are known, the measure of the third angle can always be found.
Triangle Angle Sum
Theorem
The sum of the measures of the angles of a triangle is 180.
In the figure at the right, m∠A + m∠B + m∠C = 180.
CA
B
Find m∠T.
m∠R + m∠S + m∠T = 180 Triangle Angle -
Sum Theorem
25 + 35 + m∠T = 180 Substitution
60 + m∠T = 180 Simplify.
m∠T = 120 Subtract 60 from
each side.
Find the missing
angle measures.
m∠1 + m∠A + m∠B = 180 Triangle Angle - Sum
Theorem
m∠1 + 58 + 90 = 180 Substitution
m∠1 + 148 = 180 Simplify.
m∠1 = 32 Subtract 148 from
each side.
m∠2 = 32 Vertical angles are
congruent.
m∠3 + m∠2 + m∠E = 180 Triangle Angle - Sum
Theorem
m∠3 + 32 + 108 = 180 Substitution
m∠3 + 140 = 180 Simplify.
m∠3 = 40 Subtract 140 from
each side.
58°
90°
108°
1
2 3
E
DAC
B
Exercises
Find the measures of each numbered angle.
1. 2.
3. 4.
5. 6. 20°
152°
DG
A
130°60°
1 2
S
R
TW
V
W T
U
30°
60°
2
1
S
Q R30°
1
90°
62°
1N
M
P
4-2
Example 1
35°
25°R T
S
Example 2
Q
O
NM
P58°
66°
50°
3
21
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Chapter 4 12 Glencoe Geometry
Study Guide and Intervention (continued)
Angles of Triangles
Exterior Angle Theorem At each vertex of a triangle, the angle formed by one side and an extension of the other side is called an exterior angle of the triangle. For each exterior angle of a triangle, the remote interior angles are the interior angles that are not adjacent to that exterior angle. In the diagram below, ∠B and ∠A are the remote interior angles for exterior ∠DCB.
Exterior Angle
Theorem
The measure of an exterior angle of a triangle is equal to
the sum of the measures of the two remote interior angles.
AC
B
D
1m∠1 = m∠A + m∠B
Find m∠1.
m∠1 = m∠R + m∠S Exterior Angle Theorem
= 60 + 80 Substitution
= 140 Simplify.
RT
S
60°
80°
1
Find x.
m∠PQS = m∠R + m∠S Exterior Angle Theorem
78 = 55 + x Substitution
23 = x Subtract 55 from each
side.
S R
Q
P
55°
78°
x°
Exercises
Find the measures of each numbered angle.
1. 2.
3. 4.
Find each measure.
5. m∠ABC 6. m∠F
E
FGH
58°
x°
x°B
A
DC
95°
2x° 145°
U T
SRV
35° 36°
80°
13
2
POQ
N M60°
60°
3 2
1
BC D
A
25°
35°
12
YZ W
X
65°
50°
1
4-2
Example 1 Example 2
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Chapter 4 13 Glencoe Geometry
Skills Practice
Angles of Triangles
Find the measure of each numbered angle.
1. 2.
Find each measure.
3. m∠1
4. m∠2
5. m∠3
Find each measure.
6. m∠1
7. m∠2
8. m∠3
Find each measure.
9. m∠1
10. m∠2
11. m∠3
12. m∠4
13. m∠5
Find each measure.
14. m∠1
15. m∠2 63°
1
2
DA C
B
80°
60°
40°
105°
1 4 5
2
3
150°55°
70°
1 2
3
85°55°
40°
1 2
3
146°1 2
TIGERS80°
73°
1
4-2
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Chapter 4 14 Glencoe Geometry
Practice
Angles of Triangles
Find the measure of each numbered angle.
1. 2.
Find each measure.
3. m∠1
4. m∠2
5. m∠3
Find each measure.
6. m∠1
7. m∠4
8. m∠3
9. m∠2
10. m∠5
11. m∠6
Find each measure.
12. m∠1
13. m∠2
14. CONSTRUCTION The diagram shows an
example of the Pratt Truss used in bridge
construction. Use the diagram to find m∠1.
145°1
64°58°1
2A
B
C
D
118°
36°
68°
70°
65°
82°
1
2
3 4
5
6
58°
39°
35°
12
3
40°
1
55°
72°
1
2
4-2
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Chapter 4 15 Glencoe Geometry
Word Problem Practice
Angles of Triangles
62˚136˚
1. PATHS Eric walks around a triangular
path. At each corner, he records the
measure of the angle he creates.
He makes one complete circuit around
the path. What is the sum of the
three angle measures that he wrote
down?
2. STANDING Sam, Kendra, and Tony are
standing in such a way that if lines were
drawn to connect the friends they would
form a triangle.
If Sam is looking at Kendra he needs to
turn his head 40° to look at Tony. If
Tony is looking at Sam he needs to turn
his head 50° to look at Kendra. How
many degrees would Kendra have to
turn her head to look at Tony if she is
looking at Sam?
3. TOWERS A lookout tower sits on a
network of struts and posts. Leslie
measured two angles on the tower.
What is the measure of angle 1?
4. ZOOS The zoo lights up the chimpanzee
pen with an overhead light at night. The
cross section of the light beam makes an
isosceles triangle.
The top angle of the triangle is 52 and
the exterior angle is 116. What is the
measure of angle 1?
5. DRAFTING Chloe bought a drafting
table and set it up so that she can draw
comfortably from her stool. Chloe
measured the two angles created by the
legs and the tabletop in case she had to
dismantle the table.
a. Which of the four numbered angles
can Chloe determine by knowing the
two angles formed with the tabletop?
What are their measures?
b. What conclusion can Chloe make
about the unknown angles before she
measures them to find their exact
measurements?
4-2
52˚
116˚1
Kendra
Tony
Sam
29˚
68˚
12
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Chapter 4 16 Glencoe Geometry
Enrichment
Finding Angle Measures in TrianglesYou can use algebra to solve problems involving triangles.
In triangle ABC, m∠A is twice m∠B, and m∠C
is 8 more than m∠B. What is the measure of each angle?
Write and solve an equation. Let x = m∠B.
m∠A + m∠B + m∠C = 180
2x + x + (x + 8) = 180
4x + 8 = 180
4x = 172
x = 43
So, m∠A = 2(43) or 86, m∠B = 43, and m∠C = 43 + 8 or 51.
Solve each problem.
1. In triangle DEF, m∠E is three times 2. In triangle RST, m∠T is 5 more than m∠D, and m∠F is 9 less than m∠E. m∠R, and m∠S is 10 less than m∠T.What is the measure of each angle? What is the measure of each angle?
3. In triangle JKL, m∠K is four times 4. In triangle XYZ, m∠Z is 2 more than twice m∠J, and m∠L is five times m∠J. m∠X, and m∠Y is 7 less than twice m∠X. What is the measure of each angle? What is the measure of each angle?
5. In triangle GHI, m∠H is 20 more than 6. In triangle MNO, m∠M is equal to m∠N, m∠G, and m∠G is 8 more than m∠I. and m∠O is 5 more than three times What is the measure of each angle? m∠N. What is the measure of each angle?
7. In triangle STU, m∠U is half m∠T, 8. In triangle PQR, m∠P is equal to and m∠S is 30 more than m∠T. What m∠Q, and m∠R is 24 less than m∠P.is the measure of each angle? What is the measure of each angle?
9. Write your own problems about measures of triangles.
4-2
Example
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Chapter 4 17 Glencoe Geometry
Graphing Calculator Activity
Cabri Junior: Angles of Triangles
Cabri Junior can be used to investigate the relationships between the angles in a triangle.
Use Cabri Junior to investigate the sum of the measures of the
angles of a triangle and to investigate the measure of the exterior angle in a
triangle in relation to its remote interior angles.
Step 1 Use Cabri Junior to draw a triangle. • Select F2 Triangle to draw a triangle. • Move the cursor to where you want the first vertex. Then press ENTER . • Repeat this procedure to determine the next two vertices of the triangle.
Step 2 Label each vertex of the triangle. • Select F5 Alph-num.
• Move the cursor to a vertex and press ENTER . Enter A and press ENTER again. • Repeat this procedure to label vertex B and vertex C.
Step 3 Draw an exterior angle of △ABC.
• Select F2 Line to draw a line through −−−
BC .
• Select F2 Point, Point on to draw a point on # $% BC so that C is between B and the new point.
• Select F5 Alph-num to label the point D.
Step 4 Find the measure of the three interior angles of △ABC and the exterior angle, ∠ACD.
• Select F5 Measure, Angle. • To find the measure of ∠ABC, select points A, B, and C
(with the vertex B as the second point selected). • Use this method to find the remaining angle measures.
Exercises
Analyze your drawing.
1. Use F5 Calculate to find the sum of the measures of the three interior angles.
2. Make a conjecture about the sum of the interior angles of a triangle.
3. Change the dimensions of the triangle by moving point A. (Press CLEAR so the pointer becomes a black arrow. Move the pointer close to point A until the arrow becomes transparent and point A is blinking. Press ALPHA to change the arrow to a hand. Then move the point.) Is your conjecture still true?
4. What is the measure of ∠ACD? What is the sum of the measures of the two remote interior angles (∠ABC and ∠BAC)?
5. Make a conjecture about the relationship between the measure of an exterior angle of a triangle and its two remote interior angles.
6. Change the dimensions of the triangle by moving point A. Is your conjecture still true?
4-2
Example
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Chapter 4 18 Glencoe Geometry
Geometer’s Sketchpad Activity
Angles of Triangles
The Geometer’s Sketchpad can be used to investigate the relationships between the angles in a triangle.
Use The Geometer’s Sketchpad to investigate the sum of the
measures of the angles of a triangle and to investigate the measure of the
exterior angle in a triangle in relation to its remote interior angles.
Step 1 Use The Geometer’s Sketchpad to draw a triangle. • Construct a ray by selecting the Ray tool from the toolbar. First, click where
you want the first point. Then click on a second point to draw the ray. • Next, select the Segment tool from the toolbar. Use the endpoint of the ray as
the first point for the segment and click on a second point to construct the segment.
• Construct another segment joining the second point of the new segment to a point on the ray.
Step 2 Display the labels for each point. Use the pointer tool to select all four points. Display the labels by selecting Show Labels from the Display menu.
Step 3 Find the measures of the three interior angles and the one exterior angle. To find the measure of angle ABC, use the Selection Arrow tool to select points A, B, and C (with the vertex B as the second point selected). Then, under the Measure menu, select Angle. Use this method to find the remaining angle measures.
Exercises
Analyze your drawing.
1. What is the sum of the measures of the three interior angles?
2. Make a conjecture about the sum of the interior angles of a triangle.
3. Change the dimensions of the triangle by selecting point A with the pointer tool and moving it. Is your conjecture still true?
4. What is the measure of ∠ACD? What is the sum of the measures of the two remote interior angles (∠ABC and ∠BAC)?
5. Make a conjecture about the relationship between the measure of an exterior angle of a triangle and its two remote interior angles.
6. Change the dimensions of the triangle by selecting point A with the pointer tool and moving it. Is your conjecture still true?
m/ABC = 60.00°
m/BAC = 58.19°
m/ACB = 61.81°
m/ACD = 118.19°
m/ABC + m/BAC + m/ACB = 180.00°
m/ABC + m/BAC = 118.19°
B C D
A
4-2
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Less
on
4-3
Chapter 4 19 Glencoe Geometry
Congruence and Corresponding Parts Triangles that have the same size and same shape are congruent triangles. Two triangles are congruent if and only if all three pairs of corresponding angles are congruent and all three pairs of corresponding sides are congruent. In the figure, △ABC ! △RST.
If △XYZ ! △RST, name the pairs of
congruent angles and congruent sides.
∠X ! ∠R, ∠Y ! ∠S, ∠Z ! ∠T
−−
XY ! −−
RS , −−
XZ ! −−
RT , −−
YZ ! −−
ST
Exercises
Show that the polygons are congruent by identifying all congruent corresponding
parts. Then write a congruence statement.
1.
CA
B
LJ
K 2.
C
D
A
B 3. K
J
L
M
4. F G L K
JE
5. B D
CA
6. R
T
U S
7. Find the value of x.
8. Find the value of y.
Y
XZ
T
SR
AC
B
R
T
S
4-3 Study Guide and Intervention
Congruent Triangles
Third Angles
Theorem
If two angles of one triangle are congruent to two angles
of a second triangle, then the third angles of the triangles are congruent.
Example
2x+ y(2y- 5)°75°
65° 40°
96.6
90.664.3
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Chapter 4 20 Glencoe Geometry
Study Guide and Intervention (continued)
Congruent Triangles
4-3
Example
Prove Triangles Congruent Two triangles are congruent if and only if their corresponding parts are congruent. Corresponding parts include corresponding angles and corresponding sides. The phrase “if and only if” means that both the conditional and its converse are true. For triangles, we say, “Corresponding parts of congruent triangles are congruent,” or CPCTC.
Write a two-column proof.
Given: −−
AB " −−−
CB , −−−
AD " −−−
CD , ∠BAD " ∠BCD
−−−
BD bisects ∠ABC
Prove: △ABD " △CBD
Proof:
Statement Reason
1. −−
AB " −−−
CB , −−−
AD " −−−
CD
2. −−−
BD " −−−
BD
3. ∠BAD " ∠BCD
4. ∠ABD " ∠CBD
5. ∠BDA " ∠BDC
6. △ABD " △CBD
1. Given
2. Reflexive Property of congruence
3. Given
4. Definition of angle bisector
5. Third Angles Theorem
6. CPCTC
ExercisesWrite a two-column proof.
1. Given: ∠A " ∠C, ∠D " ∠B, −−−
AD " −−−
CB , −−
AE " −−−
CE ,
−−
AC bisects −−−
BD
Prove: △AED " △CEB
Write a paragraph proof.
2. Given: −−−
BD bisects ∠ABC and ∠ADC,
−−
AB " −−−
CB , −−
AB " −−−
AD , −−−
CB " −−−
DC
Prove: △ABD " △CBD
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Lesso
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-3
Chapter 4 21 Glencoe Geometry
4-3
Show that polygons are congruent by identifying all congruent corresponding
parts. Then write a congruence statement.
1.
L
P
J
S
V
T
2.
In the figure, △ABC " △FDE.
3. Find the value of x.
4. Find the value of y.
5. PROOF Write a two-column proof.
Given: −−−
AB " −−−
CB , −−−
AD " −−−
CD , ∠ABD " ∠CBD,
∠ADB " ∠CDB
Prove: △ABD " △CBD
Skills Practice
Congruent Triangles
D
E
G
F
108° 48° y°
(2x - y)°
A F
BC ED
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Chapter 4 22 Glencoe Geometry
Practice
Congruent Triangles
Show that the polygons are congruent by indentifying all congruent
corresponding parts. Then write a congruence statement.
1.
D
R
S
CA
B 2. M
N
L
P
Q
Polygon ABCD polygon PQRS.
3. Find the value of x.
4. Find the value of y.
5. PROOF Write a two-column proof.
Given: ∠P ! ∠R, ∠PSQ ! ∠RSQ, −−−
PQ ! −−−
RQ ,
−−
PS ! −−
RS
Prove: △PQS △RQS
6. QUILTING
a. Indicate the triangles that appear to be congruent.
b. Name the congruent angles and congruent sides of a pair of
congruent triangles.
B
A
I
E
FH G
C D
4-3
(2x+ 4)°
(3y- 3)
80°
100°
10
12
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Lesso
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-3
Chapter 4 23 Glencoe Geometry
Word Problem Practice
Congruent Triangles
1. PICTURE HANGING Candice hung a picture that was in a triangular frame on her bedroom wall.
Before
A B
C
After
One day, it fell to the floor, but did not break or bend. The figure shows the object before and after the fall. Label the vertices on the frame after the fall according to the “before” frame.
2. SIERPINSKI’S TRIANGLE The figure below is a portion of Sierpinski’s Triangle. The triangle has the property that any triangle made from any combination of edges is equilateral. How many triangles in this portion are congruent to the black triangle at the bottom corner?
3. QUILTING Stefan drew this pattern for a piece of his quilt. It is made up of congruent isosceles right triangles. He drew one triangle and then repeatedly drew it all the way around.
45˚
What are the missing measures of the angles of the triangle?
4. MODELS Dana bought a model airplane kit. When he opened the box, these two congruent triangular pieces of wood fell out of it.
A
B
CQ
R
P
Identify the triangle that is congruent to △ABC.
5. GEOGRAPHY Igor noticed on a map that the triangle whose vertices are the supermarket, the library, and the post office (△SLP) is congruent to the triangle whose vertices are Igor’s home, Jacob’s home, and Ben’s home (△IJB). That is, △SLP ! △IJB.
a. The distance between the supermarket and the post office is 1 mile. Which path along the triangle △IJB is congruent to this?
b. The measure of ∠LPS is 40. Identify the angle that is congruent to this angle in △IJB.
4-3
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Chapter 4 24 Glencoe Geometry
Enrichment4-3
Overlapping TrianglesSometimes when triangles overlap each other, they share corresponding parts. When trying to picture the corresponding parts, try redrawing the triangles separately.
In the figure △ABD ! △DCA and m∠BDA = 30°.
Find m∠CAD and m∠CDA.
30°
First, redraw △ABD and △DCA like this.
30°
Now you can see that ∠BDA ! ∠CAD because they are corresponding parts of congruent triangles, so m∠CAD = 30. Using the Triangle Sum Theorem, 180 - (90 + 30) = 60, so m∠CDA = 60.
Exercises
1. In the figure △ABC ! △CDB, m∠ABC = 110, and m∠D = 20. Find m∠ACB and m∠DCB.
2. Write a two column proof.
Given: ∠A and ∠B are right angles, ∠ACD ! ∠BDC,
−−−
AD ! −−−
BC , −−
AC ! −−−
BD
Prove: △ADC ! △CBD
Example
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Lesso
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-4
Chapter 4 25 Glencoe Geometry
Study Guide and Intervention
Proving Triangles Congruent—SSS, SAS
SSS Postulate You know that two triangles are congruent if corresponding sides are congruent and corresponding angles are congruent. The Side-Side-Side (SSS) Postulate lets you show that two triangles are congruent if you know only that the sides of one triangle are congruent to the sides of the second triangle.
Write a two-column proof.
Given: −−
AB " −−−
DB and C is the midpoint of −−−
AD .
Prove: △ABC " △DBC B
CDA
Statements Reasons
1. −−
AB " −−−
DB 1. Given
2. C is the midpoint of −−−
AD . 2. Given
3. −−
AC " −−−
DC 3. Midpoint Theorem
4. −−−
BC " −−−
BC 4. Reflexive Property of "
5. △ABC " △DBC 5. SSS Postulate
Exercises
Write a two-column proof.
1. B Y
CA XZ
Given: AB "−−XY ,
−−AC "
−−XZ ,
−−−BC "
−−YZ
Prove: △ABC " △XYZ
2. T U
R S
Given: −−
RS " −−−
UT , −−
RT " −−−
US
Prove: △RST " △UTS
4-4
SSS PostulateIf three sides of one triangle are congruent to three sides of a second
triangle, then the triangles are congruent.
Example
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Chapter 4 26 Glencoe Geometry
Study Guide and Intervention (continued)
Proving Triangles Congruent—SSS, SAS
SAS Postulate Another way to show that two triangles are congruent is to use the Side-Angle-Side (SAS) Postulate.
For each diagram, determine which pairs of triangles can be
proved congruent by the SAS Postulate.
a. A
B C
X
Y Z
b. D H
F E
G
J
c. P Q
S
1
2R
In △ABC, the angle is not The right angles are The included angles, “included” by the sides
−− AB congruent and they are the ∠1 and ∠2, are congruent
and −−
AC . So the triangles included angles for the because they are cannot be proved congruent congruent sides. alternate interior angles by the SAS Postulate. △DEF $ △JGH by the for two parallel lines. SAS Postulate. △PSR $ △RQP by the SAS Postulate.
Exercises
4-4
SAS Postulate
If two sides and the included angle of one triangle are congruent to two
sides and the included angle of another triangle, then the triangles are
congruent.
Example
Write the specified type of proof.
1. Write a two column proof. 2. Write a two-column proof. Given: NP = PM,
−−− NP ⊥
−− PL Given: AB = CD,
−− AB ‖
−−− CD
Prove: △NPL $ △MPL Prove: △ACD $ △CAB
3. Write a paragraph proof.
Given: V is the midpoint of −−
YZ .
V is the midpoint of −−−
WX .
Prove: △XVZ $ △WVY
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Lesso
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-4
Chapter 4 27 Glencoe Geometry
Determine whether △ABC ! △KLM. Explain.
1. A(-3, 3), B(-1, 3), C(-3, 1), K(1, 4), L(3, 4), M(1, 6)
2. A(-4, -2), B(-4, 1), C(-1, -1), K(0, -2), L(0, 1), M(4, 1)
PROOF Write the specified type of proof.
3. Write a flow proof.
Given: −−
PR " −−−
DE , −−
PT " −−−
DF
∠R " ∠E, ∠T " ∠F
Prove: △PRT " △DEF
4. Write a two-column proof.
Given: −−
AB " −−−
CB , D is the midpoint of −−
AC .
Prove: △ABD " △CBD
T
R
P
F
E
D
4-4 Skills Practice
Proving Triangles Congruent—SSS, SAS
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Chapter 4 28 Glencoe Geometry
Practice
Proving Triangles Congruent—SSS, SASDetermine whether △DEF ! △PQR given the coordinates of the vertices. Explain.
1. D(-6, 1), E(1, 2), F(-1, -4), P(0, 5), Q(7, 6), R(5, 0)
2. D(-7, -3), E(-4, -1), F(-2, -5), P(2, -2), Q(5, -4), R(0, -5)
3. Write a flow proof.
Given: −−
RS " −−
TS
V is the midpoint of −−
RT .
Prove: △RSV " △TSV
Determine which postulate can be used to prove that the triangles are congruent.
If it is not possible to prove congruence, write not possible.
4. 5. 6.
7. INDIRECT MEASUREMENT To measure the width of a sinkhole on his property, Harmon marked off congruent triangles as shown in the diagram. How does he know that the lengths A'B' and AB are equal?
A'B'
A B
C
S
R
V
T
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Lesso
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-4
Chapter 4 29 Glencoe Geometry
Word Problem Practice
Proving Triangles Congruent—SSS, SAS
1. STICKS Tyson had three sticks of
lengths 24 inches, 28 inches, and
30 inches. Is it possible to make two
noncongruent triangles using the same
three sticks? Explain.
2. BAKERY Sonia made a sheet of
baklava. She has markings on her pan
so that she can cut them into large
squares. After she cuts the pastry in
squares, she cuts them diagonally to
form two congruent triangles. Which
postulate could you use to prove the
two triangles congruent?
3. CAKE Carl had a piece of cake in the
shape of an isosceles triangle with
angles 26, 77, and 77. He wanted to
divide it into two equal parts, so he cut
it through the middle of the 26 angle to
the midpoint of the opposite side. He
says that because he is dividing it at the
midpoint of a side, the two pieces are
congruent. Is this enough information?
Explain.
4. TILES Tammy installs bathroom tiles.
Her current job requires tiles that are
equilateral triangles and all the tiles
have to be congruent to each other. She
has a big sack of tiles all in the shape
of equilateral triangles. Although she
knows that all the tiles are equilateral,
she is not sure they are all the same
size. What must she measure on each
tile to be sure they are congruent?
Explain.
5. INVESTIGATION An investigator at a
crime scene found a triangular piece of
torn fabric. The investigator
remembered that one of the suspects
had a triangular hole in their coat.
Perhaps it was a match. Unfortunately,
to avoid tampering with evidence, the
investigator did not want to touch the
fabric and could not fit it to the coat
directly.
a. If the investigator measures all three
side lengths of the fabric and the
hole, can the investigator make a
conclusion about whether or not the
hole could have been filled by the
fabric?
b. If the investigator measures two sides
of the fabric and the included angle
and then measures two sides of the
hole and the included angle can he
determine if it is a match? Explain.
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Chapter 4 30 Glencoe Geometry
Enrichment4-4
Congruent Triangles in the Coordinate Plane
If you know the coordinates of the vertices of two triangles in the coordinate plane, you can often decide whether the two triangles are congruent. There may be more than one way to do this.
1. Consider △ABD and △CDB whose vertices have coordinates A(0, 0), B(2, 5), C(9, 5), and D(7, 0). Briefly describe how you can use what you know about congruent triangles and the coordinate plane to show that △ABD ! △CDB. You may wish to make a sketch to help get you started.
2. Consider △PQR and △KLM whose vertices are the following points.
P(1, 2) Q(3, 6) R(6, 5)
K(-2, 1) L(-6, 3) M(-5, 6)
Briefly describe how you can show that △PQR ! △KLM.
3. If you know the coordinates of all the vertices of two triangles, is it always possible to tell whether the triangles are congruent? Explain.
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Lesso
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-5
Chapter 4 31 Glencoe Geometry
Study Guide and Intervention
Proving Triangles Congruent—ASA, AAS
ASA Postulate The Angle-Side-Angle (ASA) Postulate lets you show that two triangles are congruent.
Write a two column proof.
Given: −−
AB ‖ −−−
CD
∠CBD $ ∠ADB
Prove: △ABD $ △CDB
Statements Reasons
1. −−
AB ‖ −−−
CD
2. ∠CBD $ ∠ADB
3. ∠ABD $ ∠BDC
4. BD = BD
5. △ABD $ △CDB
1. Given
2. Given
3. Alternate Interior Angles Theorem
4. Reflexive Property of congruence
5. ASA
4-5
ASA PostulateIf two angles and the included side of one triangle are congruent to two angles and
the included side of another triangle, then the triangles are congruent.
Example
Exercises
PROOF Write the specified type of proof.
1. Write a two column proof. 2. Write a paragraph proof.
S
V
U
R T
A C
B
E
D
Given: ∠S $ ∠V, Given: −−−
CD bisects −−
AE , −−
AB ‖ −−−
CD
T is the midpoint of −−
SV ∠E $ ∠BCA
Prove: △RTS $ △UTV Prove: △ABC $ △CDE
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Chapter 4 32 Glencoe Geometry
Study Guide and Intervention (continued)
Proving Triangles Congruent—ASA, AAS
AAS Theorem Another way to show that two triangles are congruent is the Angle-Angle-Side (AAS) Theorem.
You now have five ways to show that two triangles are congruent.
• definition of triangle congruence • ASA Postulate
• SSS Postulate • AAS Theorem
• SAS Postulate
In the diagram, ∠BCA ! ∠DCA. Which sides
are congruent? Which additional pair of corresponding parts
needs to be congruent for the triangles to be congruent by
the AAS Theorem?
−−
AC " −−
AC by the Reflexive Property of congruence. The congruent angles cannot be ∠1 and ∠2, because
−− AC would be the included side.
If ∠B " ∠D, then △ABC " △ADC by the AAS Theorem.
Exercises
PROOF Write the specified type of proof.
1. Write a two column proof.
Given: −−−
BC ‖ −−
EF
AB = ED
∠C " ∠F
Prove: △ABD " △DEF
2. Write a flow proof.
Given: ∠S " ∠U; −−
TR bisects ∠STU.
Prove: ∠SRT " ∠URT
S
R T
U
4-5
AAS TheoremIf two angles and a nonincluded side of one triangle are congruent to the corresponding
two angles and side of a second triangle, then the two triangles are congruent.
Example
D
C1
2A
B
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Lesso
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-5
Chapter 4 33 Glencoe Geometry
PROOF Write a flow proof.
1. Given: ∠N ! ∠L
−−
JK ! −−−
MK
Prove: △JKN ! △MKL
2. Given: −−
AB ! −−−
CB
∠A ! ∠C
−−−
DB bisects ∠ABC.
Prove: −−−
AD ! −−−
CD
3. Write a paragraph proof.
Given: −−−
DE ‖ −−−
FG
∠E ! ∠G
Prove: △DFG ! △FDE
FG
D E
A C
B
D
N
J
M
K L
4-5 Skills Practice
Proving Triangles Congruent—ASA, AAS
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Chapter 4 34 Glencoe Geometry
PROOF Write the specified type of proof.
1. Write a flow proof.
Given: S is the midpoint of −−−QT .
−−−
QR ‖ −−−
TU
Prove: △QSR $ △TSU
2. Write a paragraph proof.
Given: ∠D $ ∠F
−−−
GE bisects ∠DEF.
Prove: −−−
DG $ −−−
FG
ARCHITECTURE For Exercises 3 and 4, use the following information.
An architect used the window design in the diagram when remodeling
an art studio. −−
AB and −−−
CB each measure 3 feet.
3. Suppose D is the midpoint of −−
AC . Determine whether △ABD $ △CBD.
Justify your answer.
4. Suppose ∠A $ ∠C. Determine whether △ABD $ △CBD. Justify your answer.
D
B
A C
D
G
F
E
U
Q S
R
T
4-5 Practice
Proving Triangles Congruent—ASA, AAS
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Lesso
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-5
Chapter 4 35 Glencoe Geometry
1. DOOR STOPS Two door stops have
cross-sections that are right triangles.
They both have a 20 angle and the
length of the side between the 90 and
20 angles are equal. Are the cross-
sections congruent? Explain.
2. MAPPING Two people decide to take a
walk. One person is in Bombay and the
other is in Milwaukee. They start by
walking straight for 1 kilometer. Then
they both turn right at an angle of 110,
and continue to walk straight again.
After a while, they both turn right
again, but this time at an angle of 120.
They each walk straight for a while in
this new direction until they end up
where they started. Each person walked
in a triangular path at their location.
Are these two triangles congruent?
Explain.
3. CONSTRUCTION The rooftop of
Angelo’s house creates an equilateral
triangle with the attic floor. Angelo
wants to divide his attic into 2 equal
parts. He thinks he should divide it by
placing a wall from the center of the roof
to the floor at a 90 angle. If Angelo does
this, then each section will share a side
and have corresponding 90 angles. What
else must be explained to prove that the
two triangular sections are congruent?
90˚ 90˚
4. LOGIC When Carolyn finished her
musical triangle class, her teacher gave
each student in the class a certificate in
the shape of a golden triangle. Each
student received a different shaped
triangle. Carolyn lost her triangle on her
way home. Later she saw part of a
golden triangle under a grate.
Is enough of the triangle visible to allow
Carolyn to determine that the triangle is
indeed hers? Explain.
5. PARK MAINTENANCE Park officials
need a triangular tarp to cover a field
shaped like an equilateral triangle 200
feet on a side.
a. Suppose you know that a triangular
tarp has two 60 angles and one side
of length 200 feet. Will this tarp cover
the field? Explain.
b. Suppose you know that a triangular
tarp has three 60 angles. Will this
tarp necessarily cover the field?
Explain.
4-5 World Problem Pratice
Proving Triangles Congruent—ASA, AAS
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Chapter 4 36 Glencoe Geometry
Enrichment4-5
What About AAA to Prove Triangle Congruence?You have learned about the SSS and ASA postulates to prove that two triangles are
congruent. Do you think that triangles can be proven congruent by AAA?
Investigate the AAS theorem. AAS represents the conjecture
that two triangles are congruent if they have two pairs of
corresponding angles congruent and corresponding
non-included sides congruent. For example, in the
triangles at the right, ∠A ! ∠Y, ∠B ! ∠X, and −−
AC ! −−
YZ .
1. Refer to the drawing above. What is the measure of the missing angle in the first
triangle? What is the measure of the missing angle in the second triangle?
2. With the information from Exercise 1, what postulate can you now use to prove that the
two triangles are congruent?
Now investigate the AAA conjecture. AAA represents the conjecture that two triangles are
congruent if they have three pairs of corresponding angles congruent. The triangles below
demonstrate AAA.
25˚
25˚
70˚
70˚85˚
85˚
3. Are the triangles above congruent? Explain.
4. Can you draw two triangles that have all three angles congruent that are not congruent?
Can AAA be used to prove that two triangles are congruent?
5. What characteristics do these two triangles drawn by AAA seem to possess?
89˚
89˚
66˚
66˚A
B C
5 in
5 in
X
Y
Z
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Chapter 4 37 Glencoe Geometry
Exercises
ALGEBRA Find the value of each variable.
1. R
P
Q2x°
40°
2. S
V
T 3x - 6
2x + 6 3. W
Y Z3x°
4. 2x°
(6x + 6)°
5.
5x°
6.
R S
T
3x°
x°
7. PROOF Write a two-column proof.
Given: ∠1 ! ∠2
Prove: −−
AB ! −−−
CB
Study Guide and Intervention
Isosceles and Equilateral Triangles
Properties of Isosceles Triangles An isosceles triangle has two congruent sides called the legs. The angle formed by the legs is called the vertex angle. The other two angles are called base angles. You can prove a theorem and its converse about isosceles triangles.
• If two sides of a triangle are congruent, then the angles opposite
those sides are congruent. (Isosceles Triangle Theorem)
• If two angles of a triangle are congruent, then the sides opposite
those angles are congruent. (Converse of Isosceles Triangle
Theorem)
If −−
AB ! −−−
CB , then ∠A ! ∠C.
If ∠A ! ∠C, then −−
AB ! −−−
CB .
A
B
C
Find x, given −−
BC # −−
BA .
B
A
C (4x + 5)°
(5x - 10)°
BC = BA, so
m∠A = m∠C Isos. Triangle Theorem
5x - 10 = 4x + 5 Substitution
x - 10 = 5 subtract 4x from each side.
x = 15 Add 10 to each side.
Example 1 Example 2 Find x.
RT
S
3x - 13
2x
m∠S = m∠T, so
SR = TR Converse of Isos. △ Thm.
3x - 13 = 2x Substitution
3x = 2x + 13 Add 13 to each side.
x = 13 Subtract 2x from each side.
B
AC
D
E
1 32
4-6
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Chapter 4 38 Glencoe Geometry
Study Guide and Intervention (continued)
Isosceles and Equilateral Triangles
Properties of Equilateral Triangles An equilateral triangle has three congruent sides. The Isosceles Triangle Theorem leads to two corollaries about equilateral triangles.
1. A triangle is equilateral if and only if it is equiangular.
2. Each angle of an equilateral triangle measures 60°.
Prove that if a line is parallel to one side of an
equilateral triangle, then it forms another equilateral triangle.
Proof:
Statements Reasons
1. △ABC is equilateral; −−−
PQ ‖ −−−
BC . 1. Given
2. m∠A = m∠B = m∠C = 60 2. Each ∠ of an equilateral △ measures 60°.
3. ∠1 & ∠B, ∠2 & ∠C 3. If ‖ lines, then corres. ' are &.
4. m∠1 = 60, m∠2 = 60 4. Substitution
5. △APQ is equilateral. 5. If a △ is equiangular, then it is equilateral.
Exercises
ALGEBRA Find the value of each variable.
1. D
F E6x°
2. G
J H
6x - 5 5x
3.
3x°
4.
4x 40
60°
5. X
Z Y4x - 4
3x + 8 60°
6.
4x°
60°
7. PROOF Write a two-column proof.
Given: △ABC is equilateral; ∠1 & ∠2.
Prove: ∠ADB & ∠CDB
A
D
C
B1
2
A
B
P Q
C
1 2
4-6
Example
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Lesso
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-6
Chapter 4 39 Glencoe Geometry
Skills Practice
Isosceles and Equilateral Triangles
D
C
B
A E
Refer to the figure at the right.
1. If −−
AC " −−−
AD , name two congruent angles.
2. If −−−
BE " −−−
BC , name two congruent angles.
3. If ∠EBA " ∠EAB, name two congruent segments.
4. If ∠CED " ∠CDE, name two congruent segments.
Find each measure.
5. m∠ABC 6. m∠EDF
60°
40°
ALGEBRA Find the value of each variable.
7.
2x + 4 3x - 10
8.
(2x + 3)°
9. PROOF Write a two-column proof.
Given: −−−
CD " −−−
CG
−−−
DE " −−−
GF
Prove: −−−
CE " −−
CF
D
E
F
G
C
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Chapter 4 40 Glencoe Geometry
Practice
Isosceles and Equilateral Triangles
Lincoln H
awks
E
D B
C
A1
23
4
U
R
TV
S
Refer to the figure at the right.
1. If −−−
RV " −−
RT , name two congruent angles.
2. If −−
RS " −−
SV , name two congruent angles.
3. If ∠SRT " ∠STR, name two congruent segments.
4. If ∠STV " ∠SVT, name two congruent segments.
Find each measure.
5. m∠KML 6. m∠HMG 7. m∠GHM
8. If m∠HJM = 145, find m∠MHJ.
9. If m∠G = 67, find m∠GHM.
10. PROOF Write a two-column proof.
Given: −−−
DE ‖ −−−
BC
∠1 " ∠2
Prove: −−
AB " −−
AC
11. SPORTS A pennant for the sports teams at Lincoln High
School is in the shape of an isosceles triangle. If the measure
of the vertex angle is 18, find the measure of each base angle.
4-6
50°
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Lesso
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-6
Chapter 4 41 Glencoe Geometry
Word Problem Practice
Isosceles and Equilateral Triangles
1
1. TRIANGLES At an art supply store, two
different triangular rulers are available.
One has angles 45°, 45°, and 90°. The
other has angles 30°, 60°, and 90°.
Which triangle is isosceles?
2. RULERS A foldable ruler has two hinges
that divide the ruler into thirds. If the
ends are folded up until they touch,
what kind of triangle results?
3. HEXAGONS Juanita placed one end of
each of 6 black sticks at a common point
and then spaced the other ends evenly
around that point. She connected the
free ends of the sticks with lines. The
result was a regular hexagon. This
construction shows that a regular
hexagon can be made from six congruent
triangles. Classify these triangles.
Explain.
4. PATHS A marble path is constructed
out of several congruent isosceles
triangles. The vertex angles are all 20.
What is the measure of angle 1 in the
figure?
5. BRIDGES Every day, cars drive through
isosceles triangles when they go over the
Leonard Zakim Bridge in Boston. The
ten-lane roadway forms the bases of the
triangles.
a. The angle labeled A in the picture
has a measure of 67. What is the
measure of ∠B?
b. What is the measure of ∠C?
c. Name the two congruent sides.
A
B
C
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Chapter 4 42 Glencoe Geometry
Enrichment
Triangle ChallengesSome problems include diagrams. If you are not sure how to solve the problem, begin by using the given information. Find the measures of as many angles as you can, writing each measure on the diagram. This may give you more clues to the solution.
1. Given: BE = BF, ∠BFG ! ∠BEF ! ∠BED, m∠BFE = 82 and ABFG and BCDE each have opposite sides parallel and congruent.
Find m∠ABC.
A
G D
F E
CB
3. Given: m∠UZY = 90, m∠ZWX = 45, △YZU ! △VWX, UVXY is a square (all sides congruent, all angles right angles).
Find m∠WZY.
2. Given: AC = AD, and −−
AB ⊥ −−−
BD , m∠DAC = 44 and −−−
CE bisects ∠ACD.
Find m∠DEC.
A
D C
B
E
4. Given: m∠N = 120, −−−
JN ! −−−
MN , △JNM ! △KLM.
Find m∠JKM.
J
K
L
MN
UV
W
XYZ
4-6
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Chapter 4 43 Glencoe Geometry
Identify Congruence Transformations A congruence transformation is a transformation where the original figure, or preimage, and the transformed figure, or image, figure are still congruent. The three types of congruence transformations are reflection (or flip), translation (or slide), and rotation (or turn).
Identify the type of congruence transformation shown as a
reflection, translation, or rotation.
x
y
Each vertex and its image are the
same distance from the y-axis.
This is a reflection.
x
y
Each vertex and its image are in
the same position, just two units
down. This is a translation.
x
y
Each vertex and its image are the
same distance from the origin.
The angles formed by each pair of
corresponding points and the
origin are congruent. This is a
rotation.
Exercises
Identify the type of congruence transformation shown as a reflection, translation,
or rotation.
1.
x
y 2.
x
y 3.
x
y
4.
x
y 5.
x
y 6.
x
y
Study Guide and Intervention
Congruence Transformations
Example
4-7
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Chapter 4 44 Glencoe Geometry
Verify Congruence You can verify that reflections, translations, and rotations of triangles produce congruent triangles using SSS.
Verify congruence after a transformation.
△WXY with vertices W(3, -7), X(6, -7), Y(6, -2) is a transformation of △RST with vertices R(2, 0), S(5, 0), and T(5, 5). Graph the original figure and its image. Identify the transformation and verify that it is a congruence transformation.
Graph each figure. Use the Distance Formula to show the sides are congruent and the triangles are congruent by SSS.
RS = 3, ST = 5, TR = √ ####### (5 - 2) 2 + (5 - 0) 2 = √ ## 34
WX = 3, XY = 5, YW= √ ######### (6 - 3) 2 + (-2- (-7)) 2 = √ ## 34
−−
RS % −−−
WX , −−
ST , % −−
XY , −−
TR % −−−
YW
By SSS, △RST % △WXY.
Exercises
COORDINATE GEOMETRY Graph each pair of triangles with the given vertices.
Then identify the transformation, and verify that it is a congruence
transformation.
1. A(−3, 1), B(−1, 1), C(−1, 4) 2. Q(−3, 0), R(−2, 2), S(−1, 0)
D(3, 1), E(1, 1), F(1, 4) T(2, −4), U(3, −2), V(4, −4)
x
y
x
y
4-7 Study Guide and Intervention (continued)
Congruence Transformations
Example
x
y
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Chapter 4 45 Glencoe Geometry
4-7
Identify the type of congruence transformation shown as a reflection, translation,
or rotation.
1.
x
y 2.
x
y
COORDINATE GEOMETRY Identify each transformation and verify that it is a
congruence transformation.
3.
x
y 4.
x
y
COORDINATE GEOMETRY Graph each pair of triangles with the given vertices.
Then, identify the transformation, and verify that it is a congruence
transformation.
5. A(1, 3), B(1, 1), C(4, 1); 6. J(−3, 0), K(−2, 4), L(−1, 0);
D(3, -1), E(1, -1), F(1, -4) Q(2, −4), R(3, 0), S(4, −4)
x
y
x
y
4-7 Skills Practice
Congruence Transformations
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Chapter 4 46 Glencoe Geometry
Identify the type of congruence transformation shown as a reflection, translation,
or rotation.
1.
x
y 2.
x
y
3. Identify the type of congruence transformation shown as a reflection, translation, or rotation, and verify that it
is a congruence transformation.
4. ∆ABC has vertices A(−4, 2), B(−2, 0), C(−4, -2). ∆DEF has vertices D(4, 2), E(2, 0), F(4, −2). Graph the original figure and its image. Then identify the transformation and verify that it is a congruence transformation.
5. STENCILS Carly is planning on stenciling a pattern of flowers along the ceiling in her bedroom. She wants all of the flowers to look exactly the same. What type of congruence transformation should she use? Why?
4-7 Practice
Congruence Transformations
x
y
x
y
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Lesso
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Chapter 4 47 Glencoe Geometry
4-7 Word Problem Practice
Congruence Transformations
1. QUILTING You and your two friends
visit a craft fair and notice a quilt, part
of whose pattern is shown below. Your
first friend says the pattern is repeated
by translation. Your second friend says
the pattern is repeated by rotation. Who
is correct?
2. RECYCLING The international symbol
for recycling is shown below. What type
of congruence transformation does this
illustrate?
3. ANATOMY Carl notices that when he
holds his hands palm up in front of him,
they look the same. What type of
congruence transformation does this
illustrate?
4. STAMPS A sheet of postage stamps
contains 20 stamps in 4 rows of 5
identical stamps each. What type of
congruence transformation does this
illustrate?
5. MOSAICS José cut rectangles out of
tissue paper to create a pattern. He cut
out four congruent blue rectangles and
then cut out four congruent red
rectangles that were slightly smaller to
create the pattern shown below.
a. Which congruence transformation
does this pattern illustrate?
b. If the whole square were rotated 90,
would the transformation still be the
same?
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Chapter 4 48 Glencoe Geometry
4-7 Enrichment
Transformations in The Coordinate Plane
The following statement tells one way to relate points to corresponding translated points in the coordinate plane.
(x, y) → (x + 6, y - 3)
This can be read, “The point with coordinates (x, y) is mapped to the point with coordinates (x + 6, y - 3).” With this transformation, for example, (3, 5) is mapped or moved to (3 + 6, 5 - 3) or (9, 2). The figure shows how the triangle ABC is mapped to triangle XYZ.
1. Does the transformation above appear to be a congruence transformation? Explain your answer.
Draw the transformation image for each figure. Then tell whether the
transformation is or is not a congruence transformation.
2. (x, y) → (x - 4, y) 3. (x, y) → (x + 5, y + 4)
4. (x, y) → (-x , -y) 5. (x, y) → (- 1 −
2 x, y)
x
y
Ox
y
O
x
y
Ox
y
O
x
y
B
A
C
X
Z
Y
O
(x, y) # (x + 6, y - 3)
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Chapter 4 49 Glencoe Geometry
Study Guide and Intervention
Triangles and Coordinate Proof
Position and Label Triangles A coordinate proof uses points, distances, and slopes to prove geometric properties. The first step in writing a coordinate proof is to place a figure on the coordinate plane and label the vertices. Use the following guidelines.
1. Use the origin as a vertex or center of the figure.
2. Place at least one side of the polygon on an axis.
3. Keep the figure in the first quadrant if possible.
4. Use coordinates that make computations as simple as possible.
Position an isosceles triangle on the
coordinate plane so that its sides are a units long and
one side is on the positive x-axis.
Start with R(0, 0). If RT is a, then another vertex is T(a, 0).
For vertex S, the x-coordinate is a −
2 . Use b for the y-coordinate,
so the vertex is S ( a −
2 , b) .
Exercises
Name the missing coordinates of each triangle.
1.
x
y
B(2p, 0)
C(?, q)
A(0, 0)
2.
x
y
S(2a, 0)
T(?, ?)
R(0, 0)
3.
x
y
G(2g, 0)
F(?, b)
E(?, ?)
Position and label each triangle on the coordinate plane.
4. isosceles triangle 5. isosceles right △DEF 6. equilateral triangle △RST with base
−− RS with legs e units long △EQI with vertex
4a units long Q(0, √ $ 3 b) and sides 2b units long
x
y
x
y
x
y
x
y
T(a, 0)R(0, 0)
S(a–2, b)Example
4-8
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Chapter 4 50 Glencoe Geometry
Write Coordinate Proofs Coordinate proofs can be used to prove theorems and to verify properties. Many coordinate proofs use the Distance Formula, Slope Formula, or Midpoint Theorem.
Prove that a segment from the vertex
angle of an isosceles triangle to the midpoint of the base
is perpendicular to the base.
First, position and label an isosceles triangle on the coordinate plane. One way is to use T(a, 0), R(-a, 0), and S(0, c). Then U(0, 0) is the midpoint of
−− RT .
Given: Isosceles △RST; U is the midpoint of base −−
RT .
Prove: −−−
SU ⊥ −−
RT
Proof:
U is the midpoint of −−
RT so the coordinates of U are ( -a + a
− 2 ,
0 + 0 −
2 ) = (0, 0). Thus
−−− SU lies on
the y-axis, and △RST was placed so −−
RT lies on the x-axis. The axes are perpendicular, so
−−−
SU ⊥ −−
RT .
Exercise
PROOF Write a coordinate proof for the statement.
Prove that the segments joining the midpoints of the sides of a right triangle form a right triangle.
x
y
T(a, 0)U(0, 0)R(–a, 0)
S(0, c)
4-8 Study Guide and Intervention (continued)
Triangles and Coordinate Proof
Example
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Chapter 4 51 Glencoe Geometry
Position and label each triangle on the coordinate plane.
1. right △FGH with legs 2. isosceles △KLP with 3. isosceles △AND with
a units and b units long base −−
KP 6b units long base −−−
AD 5a units long
x
y
x
y
x
y
Name the missing coordinates of each triangle.
4.
x
y
B(2a, 0)
A(0, ?)
C(0, 0)
5.
x
y
Y(2b, 0)
Z(?, ?)
X(0, 0)
6.
x
y
N(3b, 0)
M(?, ?)
O(0, 0)
7.
x
y
Q(?, ?)
R(2a, b)
P(0, 0)
8.
x
y
P(7b, 0)
R(?, ?)
N(0, 0)
9.
x
y
U(a, 0)
T(?, ?)
S(–a, 0)
10. PROOF Write a coordinate proof to prove that in an isosceles right triangle, the segment
from the vertex of the right angle to the midpoint of the hypotenuse is perpendicular to
the hypotenuse.
Given: isosceles right △ABC with ∠ABC the right angle and M the midpoint of −−
AC
Prove: −−−
BM ⊥ −−
AC
4-8 Skills Practice
Triangles and Coordinate Proof
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Chapter 4 52 Glencoe Geometry
Position and label each triangle on the coordinate plane.
1. equilateral △SWY with 2. isosceles △BLP with 3. isosceles right △DGJ
sides 1 − 4 a units long base
−− BL 3b units long with hypotenuse
−− DJ and
legs 2a units long
x
y
x
y
x
y
Name the missing coordinates of each triangle.
4.
x
yS(?, ?)
J(0, 0) R( b, 0)1
–3
5.
x
y
C(?, 0)
E(0, ?)
B(–3a, 0)
6.
x
y
P(2b, 0)
M(0, ?)
N(?, 0)
NEIGHBORHOODS For Exercises 7 and 8, use the following information.
Karina lives 6 miles east and 4 miles north of her high school. After school she works part
time at the mall in a music store. The mall is 2 miles west and 3 miles north of the school.
7. Proof Write a coordinate proof to prove that Karina’s high school, her home, and the
mall are at the vertices of a right triangle.
Given: △SKM
Prove: △SKM is a right triangle.
8. Find the distance between the mall and Karina’s home.
x
y
S(0, 0)
K(6, 4)
M(–2, 3)
4-8 Practice
Triangles and Coordinate Proof
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Chapter 4 53 Glencoe Geometry
1. SHELVES Martha has a shelf bracket shaped like a right isosceles triangle. She wants to know the length of the hypotenuse relative to the sides. She does not have a ruler, but remembers the Distance Formula. She places the bracket on a coordinate grid with the right angle at the origin. The length of each leg is a. What are the coordinates of the vertices making the two acute angles?
2. FLAGS A flag is shaped like an isosceles triangle. A designer would like to make a drawing of the flag on a coordinate plane. She positions it so that the base of the triangle is on the y-axis with one endpoint located at (0, 0). She locates
the tip of the flag at (a, b−2
).
What are the coordinates of the third vertex?
3. BILLIARDS The figure shows a situation on a billiard table.
(–b, ?) y
x
O
(0, 0)
What are the coordinates of the cue ball before it is hit and the point where the cue ball hits the edge of the table?
4. TENTS The entrance to Matt’s tent makes an isosceles triangle. If placed on a coordinate grid with the base on the x–axis and the left corner at the origin, the right corner would be at (6, 0) and the vertex angle would be at (3, 4). Prove that it is an isosceles triangle.
5. DRAFTING An engineer is designing a roadway. Three roads intersect to form a triangle. The engineer marks two points of the triangle at (-5, 0) and (5, 0) on a coordinate plane.
a. Describe the set of points in the coordinate plane that could not be used as the third vertex of the triangle.
b. Describe the set of points in the coordinate plane that would make the vertex of an isosceles triangle together with the two congruent sides.
c. Describe the set of points in the coordinate plane that would make a right triangle with the other two points if the right angle is located at (-5, 0).
Word Problem Practice
Triangles and Coordinate Proof
4-8
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Chapter 4 54 Glencoe Geometry
Rectangle Paradox
Use grid paper to draw the two rectangles below. Cut them into pieces along the
dashed lines. Then rearrange the pieces to form a new rectangle.
1. Make a sketch of the new rectangle. Label the whole number lengths
2. What is the combined area of the two original rectangles?
3. What is the area of the new rectangle?
4. Make a conjecture as to why there is a discrepancy between the answers to
Exercises 2 and 3.
5. Use a straight edge to precisely draw the four shapes that
make up the larger rectangle. What does you drawing show?
6. How could you use slope to show that you drawing
is accurate?
3
5
3
35
5
4-8 Enrichment
Assessment
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Chapter 4 55 Glencoe Geometry
SCORE 4 Student Recording Sheet
Use this recording sheet with pages 316–317 of the Student Edition.
Read each question. Then fill in the correct answer.
1. A B C D
2. F G H J
3. A B C D
4. F G H J
5. A B C D
6. F G H J
7. A B C D
Multiple Choice
Record your answer in the blank.
For gridded response questions, also enter your answer in the grid by writing
each number or symbol in a box. Then fill in the corresponding circle for that
number or symbol.
8. ———————— (grid in)
9. ———————— (grid in)
10. ———————— (grid in)
11. ————————
12. ————————
13. ———————— (grid in)
14. ————————
Extended Response
Short Response/Gridded Response
Record your answers for Question 15 on the back of this paper.
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
. . . . .
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
. . . . .
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
8.
10.
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
. . . . .
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9.
13.
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
. . . . .
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
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NAME DATE PERIOD
Chapter 4 56 Glencoe Geometry
General Scoring Guidelines
• If a student gives only a correct numerical answer to a problem but does not show how
he or she arrived at the answer, the student will be awarded only 1 credit. All extended-
response questions require the student to show work.
• A fully correct answer for a multiple-part question requires correct responses for all
parts of the question. For example, if a question has three parts, the correct response to
one or two parts of the question that required work to be shown is not considered a fully
correct response.
• Students who use trial and error to solve a problem must show their method. Merely
showing that the answer checks or is correct is not considered a complete response for
full credit.
Exercise 15 Rubric
Score Specific Criteria
4
The points for the verties of △ABC are plotted properly and the
distance formula is appropritely used to find AB = BC = √ #### 4a2+b2 .
Students draw the appropriate conclusion that △ABC is isosceles.
3A generally correct solution, but may contain minor flaws in reasoning or computation.
2 A partially correct interpretation and/or solution to the problem.
1 A correct solution with no evidence or explanation.
0An incorrect solution indicating no mathematical understanding of the concept or task, or no solution is given.
4 Rubric for Scoring Extended Response
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Assessm
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NAME DATE PERIOD
Chapter 4 57 Glencoe Geometry
SCORE
SCORE
1. Use a protractor to classify the triangle by its angles and sides.
2. MULTIPLE CHOICE What is the best classification of this triangle by its angles and sides?
A acute isosceles C right isosceles
B obtuse isosceles D obtuse equilateral
3. If △ABC is an isosceles triangle, ∠B is the vertex angle, AB = 6x + 3, BC = 8x - 1, and AC = 10x - 10, find the value of x and the length of each side of the triangle.
4. Find the measures of the sides of △ABC with vertices A(1, 5), B(3, -2), and C(-3, 0). Classify the triangle by sides.
Find the measure of each angle in
the figure.
5. m∠1 6. m∠2
7. m∠3 8. m∠4
9. m∠5 10. m∠6
1. Identify the congruent triangles in the figure. K
MLN
2. If △JGO " △RWI, which angle corresponds to ∠I?
3. In the diagram, △ABC " △DEF. Find the values of x and y.
4. In the figure −−−
QR " −−
SR and −−−
PQ " −−
TS . Point R is the midpoint of
−− PT . Determine
which theorem or postulate can be used to show that △QRP " △SRT.
5. What is the relationship between ∠Q and ∠S?
1.
2.
3.
4.
5.
4
4
Chapter 4 Quiz 1(Lessons 4-1 and 4-2)
70°65°
1
2 3
4
5
6 107°
43°
Chapter 4 Quiz 2(Lessons 4-3 and 4-4)
Q
P T
R
S
3x - y
(6y - 4)°
80°
70° 30°
19.7
18.810
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
SCORE
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NAME DATE PERIOD
Chapter 4 58 Glencoe Geometry
SCORE
A
K
Z C
1.
2.
3.
4.
5.
For Questions 1 and 2, complete the two-column proof by supplying the missing information for each corresponding location.
Given: ∠Z ! ∠C −−−
AK bisects ∠ZKC.
Prove: △AKZ ! △AKC
Proof:
Statements Reasons
1. ∠Z ! ∠C −−−
AK bisects ∠ZKC. 1. Given
2. ∠ZKA ! ∠CKA 2. (Question 1)
3. −−−
AK ! −−−
AK 3. Reflexive Property
4. △AKZ ! △AKC 4. (Question 2)
For Questions 3 and 4 refer to the figure.
3. Find m∠1 4. Find m∠2
5. The base angle of an isosceles triangle is 30. What is the measure of the vertex angle?
1. Identify the type of congruence transformation shown as a reflection, translation, or rotation.
Position and label the following triangle on a coordinate plane.
2. Right △DJL with hypotenuse −−
DJ ; LJ = 1 − 2 DL and
−−− DL is a
units long.
3. Two of the vertices of an equilateral triangle are (0, -a) and (0, a). The third vertex is on the x-axis to the right of the origin. Name the coordinates of the third vertex.
For Questions 4 and 5, complete the coordinate proof by supplying the missing information for each corresponding location.
Given: △ABC with A(-1, 1), B(5, 1), and C(2, 6).
Prove: △ABC is isosceles.
By the Distance Formula the lengths of the three sides are as follows: (Question 4). Since (Question 5), △ABC is isosceles.
1.
2.
3.
4.
5.
4
4
Chapter 4 Quiz 3(Lessons 4-5 and 4-6)
Chapter 4 Quiz 4(Lessons 4-7 and 4-8)
x
y
SCORE
21
Assessm
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Chapter 4 59 Glencoe Geometry
SCORE
1. Use a ruler to measure each side and a protractorto measure each angle. Which best describes the type of triangle?
A acute scalene
B obtuse equilateral
C acute isosceles
D obtuse isosceles
Find the missing angle measures.
2. What is m∠1?
F 50 H 100
G 60 J 105
3. What is m∠2?
A 40 C 60
B 50 D 100
4. If △SJL " △DMT, which segment corresponds to −−
LS ?
F −−−
DT H −−−
MD
G −−−
TD J −−−
MT
50°
50°
60°
2
1
5.
6.
7.
8.
1.
2.
3.
4.
Part II
5. Find the measures of the sides of △ABC with vertices A(1, 3), B(5, -2), and C(0, -4). Classify the triangle by its sides.
6. Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove congruence, write not possible.
7. What is the maximum number of acute angles that a triangle can have?
8. Determine whether △DEF " △JKL, given that D(2, 0), E(5, 0), F(5, 5), J(3, -7), K(6, -7), and L(6, -2).
Part I Write the letter for the correct answer in the blank at the right of each question.
4 Chapter 4 Mid-Chapter Test(Lessons 4-1 through 4-4)
SCORE SCORE
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Chapter 4 60 Glencoe Geometry
Write whether each sentence is true or false. If false,
replace the underlined word or number to make a true
sentence.
1. A triangle that is equilateral is also called a(n) acute triangle.
2. A(n) obtuse triangle has at least one obtuse angle.
Choose the correct term to complete the sentence.
3. If you slide, flip, or turn a triangle you are performing a (coordinate proof or congruence transformation).
4. An (interior or exterior) angle is formed by one side of a triangle and the extension of another side.
5. The SAS Postulate involves two corresponding sides and the (exterior angle or included angle) they form.
Choose from the terms above to complete each sentence.
6. A(n) organizes a series of statements in logical order, starting with the given statements.
7. A triangle that has one 90° angle is called a(n) .
8. The ASA postulate involves two corresponding angles and
their corresponding .
9. A(n) uses figures in the coordinate plane and algebra to prove geometric concepts.
10. The is formed by the congruent legs of an isosceles triangle.
Define each term in your own words.
11. corollary
12. congruent triangles
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
acute triangle
auxiliary line
base angles
congruence
transformation
congruent triangles
coordinate proof
corollary
corresponding parts
equiangular triangle
equilateral triangle
exterior angle
flow proof
included angle
included side
isosceles triangle
obtuse triangle
reflection
remote interior angles
right triangle
rotation
scalene triangle
translation
vertex angle
4
?
?
?
?
?
Chapter 4 Vocabulary Test SCORE
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Chapter 4 61 Glencoe Geometry
SCORE
80˚
A
E
CB
D
A C
B
10x - 5
6x + 37.5x
1.
2.
3.
4.
5.
6.
7.
8.
1. Which best describes the type of triangle?
A acute C obtuse
B equiangular D right
2. What is the value of x if △ABC is equilateral?
F -8 H 1 − 2
G - 1 −
8 J 2
Use the figure for Questions 3 and 4.
3. What is m∠2?
A 50 B 70 C 110 D 120
4. What is m∠4?
F 10 G 60 H 100 J 120
5. What are the congruent triangles in the diagram?
A △ABC $ △EBD
B △ABE $ △CBD
C △AEB $ △CBD
D △ABE $ △CDB
6. If △CJW $ △AGS, m∠A = 50, m∠J = 45, and m∠S = 16x + 5, what is the value of x?
F 17.5 H 6
G 11.875 J 5
7. Two triangles are graphed with the following vertices: A(1, 1), B(0, 1), C(0, 0)
and Q(−2, 1), R(−1, 1), S(−1, 0). When a transformation is applied to △ABC,
the result is △QRS. Which best describes the transformation?
A △QRS is a translation of △ABC
B △QRS is a rotation of △ABC
C △QRS is a reflection of △ABC
D △QRS and △ABC are not congruent
8. A triangular-shaped roof of a house has congruent legs. What is the measure of each of the two base angles?
F 25 H 100
G 50 J 120
70°
2 1
3 460°
40°
4 Chapter 4 Test, Form 1
Write the letter for the correct answer in the blank at the right of each question.
C W
J
(16x + 5)°
45°
A S
G
50°
SCORE
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Chapter 4 62 Glencoe Geometry
N
K
LMJ
x
yM(a, c)
N(?, ?)L(0, 0)
1 2
M
TL N
Use the proof for Questions 9 and 10.
Given: L is the midpoint of −−−
JM
−−
JK || −−−
NM
Prove: △JKL # △MNL
Proof:
Statements Reasons
1. L is the midpoint of −−−
JM . 1. Given
2. −−
JL # −−−
ML 2. Definition of midpoint
3. −−
JK ‖ −−−
MN 3. Given
4. ∠JKL # ∠MNL 4. Alt. int. & are #.
5. ∠JLK # ∠MLN 5. (Question 9)
6. △JKL # △MNL 6. (Question 10)
9. What is the reason for ∠JLK # ∠MLN?
A definition of midpoint
B corresponding angles
C vertical angles
D alternate interior angles
10. What is the reason for △JKL # △MNL?
F AAS G ASA H SAS J SSS
Use the figure for Questions 11 and 12.
11. If △LMN is isosceles and T is the midpoint of −−−
LN , which postulate can be used to prove △MLT # △MNT?
A AAA B AAS C SAS D ABC
12. If △MLT # △MNT, what is used to prove ∠1 # ∠2?
F CPCTC
G definition of isosceles triangle
H definition of perpendicular
J definition of angle bisector
13. What are the missing coordinates of this triangle?
A (2a, 2c) C (0, 2a)
B (2a, 0) D (a, 2c)
Bonus Classify the triangle with coordinates A(5, 0), B(0, 5), and C(-5, 0).
4 Chapter 4 Test, Form 1 (continued)
9.
10.
11.
12.
13.
B:
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Chapter 4 63 Glencoe Geometry
SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. What is the length of the sides of this equilateral triangle?
A 42 C 15
B 30 D 12
2. How would △ABC with vertices A(4, 1), B(2, -1), and C(-2, -1) be classified based on the length of its sides?
F equilateral G isosceles H scalene J right
Use the figure for Questions 3 and 4.
3. What is m∠1?
A 40 B 50 C 70 D 90
4. What is m∠3?
F 40 G 70 H 90 J 110
5. If △DJL " △EGS, which segment in △EGS corresponds to −−−
DL ?
A −−−
EG B −−
ES C −−−
GS D −−−
GE
6. Which triangles are congruent in the figure?
F △KLJ " △MNL
G △JLK " △NLM
H △JKL " △LMN
J △JKL " △MNL
7. Quadrilateral MNQP is made of two congruent triangles. −−−
NP bisects ∠N and ∠P. In the quadrilateral, m∠N = 50 and m∠P = 100. What is the measure of ∠M?
A 25 C 60
B 50 D 105
8. The coordinates of the vertices of △CDE are C(-3, 1), D(-1, 4), and E(-6, 4). A transformation applied to △CDE creates a congruent triangle △SQR. The new coordinates of two vertices are Q(-1, 6) and R(-6, 6). What are the coordinates of S?
F (-3, 3) G (1, 3) H (-1, 1) J (-1, 3)
M
N
Q
P
L
K
NJ
M
70°
50°1
2
3
6x - 3
9x - 123x + 6
1.
2.
3.
4.
5.
6.
7.
8.
4 Chapter 4 Test, Form 2A SCORE
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Chapter 4 64 Glencoe Geometry
D A
NY
A C
B
E F
x
y
B(0, a)
A(?, ?)
x
y
(0, c)
(?, ?) (2b, 0)
9.
10.
11.
12.
13.
B:
9. If △ABC is isosceles with vertex angle ∠B, and
−−AE $
−−FC , which theorem or postulate can be
used to prove △AEB $ △CFB?
A SSS C ASA
B SAS D AAS
Use the proof for Questions 10 and 11.
Given: −−−
DA || −−−
YN
−−−
DA $ −−−
YN
Prove: ∠NDY $ ∠DNA
Proof:
Statements Reasons
1. −−−
DA ‖ −−−
YN 1. Given
2. ∠ADN $ ∠YND 2. Alt. int. & are $.
3. −−−
DA $ −−−
YN 3. Given
4. −−−
DN $ −−−
DN 4. Reflexive Property
5. △NDY $ △DNA 5. (Question 10)
6. ∠NDY $ ∠DNA 6. (Question 11)
10. What is the reason for statement 5?
F ASA H SAS
G AAS J SSS
11. What is the reason for statement 6?
A Alt. int. & are $. C Corr. angles are $.
B CPCTC D Isosceles Triangle Theorem
12. What is the classification of a triangle with vertices A(3, 3), B(6, -2), C(0, -2) by the length of its sides?
F isosceles H equilateral
G scalene J right
13. What are the missing coordinates of the triangle?
A (-2b, 0) C (-c, 0)
B (0, 2b) D (0, -c)
Bonus Name the coordinates of points A and C in isosceles right △ABC if point C is in the second quadrant.
4 Chapter 4 Test, Form 2A (continued)
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Chapter 4 65 Glencoe Geometry
SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. What are the lengths of the sides of this equilateral triangle?
A 2.5 C 15
B 5 D 20
2. What is the classification of △ABC with vertices A(0, 0), B(4, 3), and C(4, -3) by its sides?
F equilateral G isosceles H scalene J right
Use the figure for Questions 3 and 4.
3. What is m∠1?
A 120 B 90 C 60 D 30
4. What is m∠2?
F 120 G 90 H 60 J 30
5. If △TGS " △KEL, which angle in △KEL corresponds to ∠T ?
A ∠L B ∠E C ∠K D ∠A
6. Which triangles are congruent in the figure?
F △HMN " △HGN
G △HMN " △NGH
H △NMH " △NGH
J △MNH " △HGN
7. The rhombus QRST is made of two congruent isosceles triangles. Given m∠QRS = 34, what is the measure of ∠S?
A 56 C 112
B 73 D 146
8. The vertices of △ABC are A(2, 0), B(5, 0), C(2, 6). The vertices of △DEF
are D(-6, -7), E(-3, -7), F(-6, -1) so that △ABC " △DEF. Which best describes the congruence transformation?
F flip G rotation H reflection J translation
H G
NM
120°1
2
7x - 15
3x + 54x
1.
2.
3.
4.
5.
6.
7.
8.
4 Chapter 4 Test, Form 2B
Q
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R
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Chapter 4 66 Glencoe Geometry
G A
N
E I
A DF E
CB
(5x + 60)° (2x + 51)°
(30 - 10x)°(43 - 2x)°
x
y
(?, ?)
(a, 0)(-a, 0)
9.
10.
11.
12.
13.
B:
9. If −−AF !
−−−DE ,
−−AB !
−−FC and
−−AB ||
−−FC , which theorem
or postulate can be used to prove △ABE ! △FCD?
A AAS C SAS
B ASA D SSS
Use the proof for Questions 10 and 11.
Given: −−−
EG ! −−
IA ∠EGA ! ∠IAG
Prove: ∠GEN ! ∠AIN
Proof:
Statements Reasons
1. −−−
EG ! −−
IA 1. Given
2. ∠EGA ! ∠IAG 2. Given
3. −−−
GA ! −−−
GA 3. Reflexive Property
4. △EGA ! △IAG 4. (Question 10)
5. ∠GEN ! ∠AIN 5. (Question 11)
10. What is the reason for statement 4?
F SSS G ASA H SAS J AAS
11. What is the reason for statement 5?
A Alt. int. % are !. C Corr. angles are !.
B Same side interior angles D CPCTC
12. What is the classification of a triangle with vertices A(-3, -1), B(-2, 2), C(3, 1) by its sides?
F scalene H equilateral
G isosceles J right
13. What are the missing coordinates of the triangle?
A (a, 0) C (c, 0)
B (b, 0) D (∅, a −
2 )
Bonus Find the value of x.
4 Chapter 4 Test, Form 2B (continued)
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Chapter 4 67 Glencoe Geometry
SCORE
1. Use a protractor and ruler to classify the triangle by its angles and sides.
2. Find x, AB, BC, and AC if △ABC is equilateral.
3. Find the measure of the sides of the triangle if the vertices of △EFG are E(-3, 3), F(1, -1), and G(-3, -5). Then classify the triangle by its sides.
Use the figure for Question 4-6. Find the measure of
each angle.
4. m∠1
5. m∠2
6. m∠3
7. Identify the congruent triangles and name their corresponding congruent angles.
8. Identify the transformation and verify that it is a congruence transformation.
x
y
A
D F
G
B
C
110° 2
1
3
A C
B
8x
7x + 310x - 6
1.
2.
3.
4.
5.
6.
7.
8.
4 Chapter 4 Test, Form 2C
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Chapter 4 68 Glencoe Geometry
9. A rectangular piece of paper measuring 8 inches by 16 inches
is folded in half to form a square. It is then folded again along
its diagonal to form a triangle. Find the measures of the
angles of the triangle.
10. △KLM is an isosceles triangle
and ∠1 " ∠2. Name the theorem
that could be used to determine
∠LKP " ∠LMN. Then name the
postulate that could be used to
prove △LKP " △LMN. Choose
from SSS, SAS, ASA, and AAS.
11. Find m∠1.
12. Find the value of x.
13. Position and label isosceles △ABC with base −−
AB , b units long,
on the coordinate plane.
14. −−
CP joins point C in isosceles right
△ABC to the midpoint P, of −−
AB .
Name the coordinates of P. Then
determine the relationship
between −−
AB and −−
CP .
Bonus Without finding any other angles or sides congruent,
determine which pair of triangles can be proved to be
congruent by the HL Theorem.
A
B
C D
E
F X
Y
Z M
N
O
x
y
A(0, b)
B(b, 0)C(0, 0)
P
(10x + 20)°15x°
(18x - 12)°
40°
1
P
1 2
NMK
L
4 Chapter 4 Test, Form 2C (continued)
9.
10.
11.
12.
13.
14.
B:
Assessment
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Chapter 4 69 Glencoe Geometry
SCORE
1. Use a protractor and ruler to classify the triangle by its angles and sides.
2. Find x, AB, BC, AC if △ABC is isosceles with AB = BC.
3. Find the measure of the sides of the triangle if the vertices of △EFG are E(1, 4), F(5, 1), and G(2, -3). Then classify the triangle by its sides.
Use the figure for Questions 4-6. Find the measure of
each angle.
4. m∠1
5. m∠2
6. m∠3
7. Identify the congruent triangles and name their corresponding congruent angles.
8. Identify the transformation and verify that it is a congruence transformation.
9. A square piece of paper measuring 8 inches on each side is folded in half forming a triangle. It is then folded in half again to form another triangle. Find the angles of the resulting triangle.
DB
EC
FA
70°1 32
80°
A C
B
2x + 20
9x - 5
5x + 5
4 Chapter 4 Test, Form 2D
x
y
1.
2.
3.
4.
5.
6.
7.
8.
9.
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Chapter 4 70 Glencoe Geometry
10. △ABC is an isosceles triangle with −−−
BD ⊥ −−
AC . Name the theorem that
could be used to determine
∠A % ∠C. Then name the
postulate that could be used to
prove △BDA % △BDC. Choose
from SSS, SAS, ASA, and AAS.
11. Find m∠1.
12. Find the value of x.
13. Position and label equilateral △KLM with side lengths
3a units long on the coordinate plane.
14. −−−
MN joins the midpoint of −−
AB and the
midpoint of −−
AC in △ABC. Find the
coordinates of M and N.
Bonus Without finding any other angles or sides congruent,
determine which pair of triangles can be proved to be
congruent by the LL Theorem.
A
B
C D
E
F X
Y
Z M
N
O
x
y
B(a, 0)
C(0, b)
M(?, ?)
N(?, ?)
A(0, 0)
(6x + 4)°
(18x - 8)°
80°1
B D
C
A
10.
11.
12.
13.
14.
B:
4 Chapter 4 Test, Form 2D (continued)
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Chapter 4 71 Glencoe Geometry
SCORE
1. If △ABC is isosceles, ∠B is the vertex angle, AB = 20x - 2, BC = 12x + 30, and AC = 25x, find x and the length of each side of the triangle.
2. Find the length of the sides of the triangle with vertices A(0, 4), B(5, 4), and C(-3, -2). Classify the triangle by its sides and angles.
Use the figure to answer Questions 3–5.
3. Find the value of x.
4. Find m∠1, if m∠1 = 4x + 10.
5. Find m∠2.
6. The vertices of △ABC are A(-1, 1), B(4, 2), and C(1, 5). The vertices of △DEF are D(-1, -1), E(4, -2), and F(1, -5) so that △ABC " △DEF. Identify the congruence transformation.
7. Determine whether △GHI " △JKL, given G(1, 2), H(5, 4), I(3, 6) and J(-4, -5), K(0, -3), L(-2, -1). Explain.
8. In the figure, −−
AC " −−−
FD , −−
AB || −−−
DE ,
and −−
AC || −−−
FD . Determine which postulate can be used to prove △ABC " △DEF. Choose from SSS, SAS, ASA, and AAS.
D
E
A
B
F
C
(8x - 30)°(3x - 10)°2
1
1.
2.
3.
4.
5.
6.
7.
8.
4 Chapter 4 Test, Form 3
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Chapter 4 72 Glencoe Geometry
For Questions 9 and 10, complete this two-column proof.
Given: △ABC is an isosceles triangle with base −−
AC . D is the midpoint of
−− AC .
Prove: −−−
BD bisects ∠ABC
Proof:
Statements Reasons
1. △ABC is isosceles with base
−− AC .
1. Given
2. −−
AB $ −−−
CB 2. Def. of isosceles triangle.
3. ∠A $ ∠C 3. (Question 9)
4. D is the midpoint of −−
AC . 4. Given
5. −−−
AD $ −−−
CD 5. Midpoint Theorem
6. △ABD $ △CBD 6. (Question 10)
7. ∠1 $ ∠2 7. CPCTC
8. −−−
BD bisects ∠ABC 8. Def. of angle bisector
11. Find x.
12. Position and label isosceles △ABC with base −−
AB ,
(a + b) units long, on a coordinate plane
13. A wood column is supported by four cablesthat form four triangles. Determine which postulate can be used to show that all the triangles are congruent.
Bonus In the figure, △ABC is isosceles, △ADC is equilateral, △AEC is isosceles, and the measures of ∠9, ∠1, and ∠3 are all equal. Find the measures of the nine numbered angles.
987
1
65
23
4
B
C
DE
A
(15x + 15)°(21x - 3)°
(17x + 9)°
21
A C
B
D
9.
10.
11.
12.
13.
B:
4 Chapter 4 Test, Form 3 (continued)
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Chapter 4 73 Glencoe Geometry
SCORE
Demonstrate your knowledge by giving a clear, concise solution to
each problem. Be sure to include all relevant drawings and justify
your answers. You may show your solution in more than one way
or investigate beyond the requirements of the problem.
1.
a. Classify the triangle by its angles and sides.
b. Show the steps needed to solve for x.
2. a. Describe how to determine whether a triangle with coordinates A(1, 4), B(1, -1), and C(-4, 4) is an equilateral triangle.
b. Is the triangle equilateral? Explain.
3. Explain how to find m∠1 and m∠2 in the figure.
4.
a. Determine which theorem or postulate can be used to prove that the triangles are congruent.
b. List their corresponding congruent angles and sides.
5. Write a two-column proof.
Given: −−
AB || −−−
DE , −−−
AD bisects −−−
BE . Prove: △ABC $ △DEC using the ASA postulate.
A
C
B
E D
J
DL
E
G
S
C
BD E
A 58°
40° 62°
1 2
(9x + 4)°
(20x - 10)°
4 Chapter 4 Extended-Response Test
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Chapter 4 74 Glencoe Geometry
4 Standardized Test Practice(Chapters 1– 4)
A CB
D
?
U V
XW
1. If m∠1 = 5x - 4 and m∠2 = 52 - 9y, which values for x and y would make ∠1 and ∠2 complementary? (Lesson 1-5)
A x = 2, y = 12 C x = 12, y = 2
B x = 27, y = 1 − 3 D x = 1 −
3 , y = 27
2. Which is not a polygon? (Lesson 1-6)
F G H J
3. Complete the statement so that its conditional and its converse are true.
If ∠1 # ∠2, then ∠1 and ∠2 . (Lesson 2-3)
A are supplementary. C are complementary.
B have the same measure. D are consecutive interior angles.
4. Complete this proof. (Lesson 2-7)
Given: −−−
UV # −−−
VW −−−
VW # −−−
WX Prove: UV = WX
Proof:
Statements Reasons
1. −−−
UV # −−−
VW ; −−−
VW # −−−
WX 1. Given
2. UV = VW; VW = WX 2.
3. UV = WX 3. Transitive Property
F Definition of congruent segments
G Substitution Property
H Segment Addition Postulate
J Symmetric Property
5. Which equation has a slope of 1 − 3 and a y-intercept of -2? (Lesson 3-4)
A y = 1 − 3 x + 2 C y = 1 −
3 x - 2
B y = 2x - 1 − 3 D y = -2x + 1 −
3
6. Classify △DEF with vertices D(2, 3), E(5, 7) and F(9, 4). (Lesson 4-1)
F acute G equiangular H obtuse J right
7. Which postulate or theorem can be used to prove △ABD # △CBD? (Lesson 4-4)
A SAS C SSS
B ASA D AAS
?
SCORE
Part 1: Multiple Choice
Instructions: Fill in the appropriate circle for the best answer.
1.
2. F G H J
3. A B C D
4. F G H J
5.
6. F G H J
7. A B C D
A B C D
A B C D
Assessment
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pyri
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t ©
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Mc
Gra
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.
NAME DATE PERIOD
Chapter 4 75 Glencoe Geometry
SCORE 4 Standardized Test Practice (continued)
8. If the volume of a cylinder with a height of 3 feet is 75π cubic feet, find the surface area of the cylinder in square feet. (Lesson 1-7)
F 25π G 50π H 80π J 30π
9. Let C be the midpoint of the line segment AB having endpoints A(2, -7) and B(6, -3). Find the length of
−−− CB . (Lesson 1-3)
A 2 √ $ 2 units B √ $$ 53 units C √ $$ 20 units D √ $$ 10 units
10. Find x so that p || q. (Lesson 3-5)
F 11.50 H 20.5
G 20 J 25
11. If ∠1 is supplementary to ∠2 and ∠3 is complementary to ∠2, find m∠3 if m∠1 is 145. (Lesson 2-8)
A 35 C 55
B 45 D 90
12. Let △ABC be an isosceles triangle with △ABC ' △PQR. If m∠B = 154, find m∠R. (Lesson 4-3)
F 154 G 126 H 26 J 13
13. In proving the congruence of two triangles, which postulate involves an included angle? (Lessons 4-4 and 4-5)
A SSS B SAS C ASA D AAS
14. △PQR is an isosceles triangle with base −−−
QR . If m∠P = 6x + 40 and m∠Q = x - 10, find x. (Lesson 4-6)
F 20 G 25 H 30 J 100
qp
(2x -
55)°
(8x + 30)°
8.
9.
10.
11.
12.
13.
14. F G H J
A B C D
F G H J
A B C D
F G H J
A B C D
F G H J
15. If −−
JK || −−− LM , then ∠4 must
be supplementary to
∠ . (Lesson 3-5)
16. Find PR if △PQR is isosceles, ∠Q is the vertex angle, PQ = 4x - 8, QR = x + 7, and PR = 6x - 12. (Lesson 4-1)
?
6
7
4
5H
L
M
J
K
15.
16. 9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
Part 2: Gridded Response
Instructions: Enter your answer by writing each digit of the
answer in a column box and then shading in the appropriate circle
that corresponds to that entry.
Co
pyrig
ht ©
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Gra
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ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
NAME DATE PERIOD
Chapter 4 76 Glencoe Geometry
17. The perimeter of a regular pentagon is 14.5 feet. Find the new
perimeter if the length of each side of the pentagon is doubled.
(Lesson 1-6)
18. Make a conjecture about the next number in the sequence
5, 7, 11, 17, 25. . .. (Lesson 2-1)
19. Find m∠PQR. (Lesson 4-2)
20. If PQ = QS, QS = SR, and
m∠R = 20, find m∠PSQ.
(Lesson 4-6)
21. Name the corresponding congruent angles and sides for
△PQR " △HGB. (Lesson 4-3)
22. Refer to the figure at the right to
answer the questions below.
a. Name the segment that represents
the distance from F to # %& AD . (Lesson 3-6)
b. Classify △ADC. (Lesson 4-1)
c. Find m∠ACD. (Lesson 4-2)
50°30°
85°ED
A CB
F
P RS
Q
63° 125° 10°P S
Q
R
T
17.
18.
19.
20.
21.
22a.
22b.
22c.
4 Standardized Test Practice (continued)
Part 3: Short Response
Instructions: Write your answers in the space provided.
Answers
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t ©
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Mc
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nie
s,
Inc
.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 4-1
Ch
ap
ter
4
5
Gle
nc
oe
Ge
om
etr
y
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Cla
ssif
yin
g T
rian
gle
s
Cla
ssif
y T
ria
ng
les
by
An
gle
s O
ne w
ay t
o c
lass
ify a
tri
an
gle
is
by t
he m
easu
res
of
its
an
gle
s.
• If
all
thre
e o
f th
e a
ng
les o
f a
tria
ng
le a
re a
cu
te a
ng
les,
the
n t
he
tria
ng
le is a
n a
cu
te t
ria
ng
le.
• If
all
thre
e a
ng
les o
f a
n a
cu
te t
ria
ng
le a
re c
on
gru
en
t, t
he
n t
he
tria
ng
le is a
n e
qu
ian
gu
lar
tria
ng
le.
• If
on
e o
f th
e a
ng
les o
f a
tria
ng
le is a
n o
btu
se
an
gle
, th
en
th
e t
ria
ng
le is a
n o
btu
se
tri
an
gle
.
• If
on
e o
f th
e a
ng
les o
f a
tria
ng
le is a
rig
ht
an
gle
, th
en
th
e t
ria
ng
le is a
rig
ht
tria
ng
le.
C
lassif
y e
ach
tria
ng
le.
a
.
60°A
BC
All
th
ree a
ngle
s are
con
gru
en
t, s
o a
ll t
hre
e a
ngle
s h
ave m
easu
re 6
0°.
Th
e t
rian
gle
is
an
equ
ian
gu
lar
tria
ngle
.
b.
25°
35°
120°
DF
E
Th
e t
rian
gle
has
on
e a
ngle
th
at
is o
btu
se.
It i
s an
obtu
se t
rian
gle
.
c.
90°
60°
30°
G
HJ
Th
e t
rian
gle
has
on
e r
igh
t an
gle
. It
is
a r
igh
t tr
ian
gle
.
Exerc
ises
Cla
ssif
y e
ach
tria
ng
le a
s a
cu
te,
eq
uia
ng
ula
r,
ob
tuse,
or r
igh
t.
1
. 67°
90°
23°
K LM
2.
120°
30°
30°
NO
P
3
.
60°
60°
60°
Q
RS
r
igh
t o
btu
se
eq
uia
ng
ula
r
4
.
65°
65°
50°
UV
T
5.
45°
45°
90°
XY
W
6.
60°
28°
92°
FDB
a
cu
te
rig
ht
ob
tuse
4-1 Exam
ple
Chapter Resources
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
3
Gle
nc
oe
Ge
om
etr
y
An
tici
pati
on
Gu
ide
Co
ng
ruen
t Tri
an
gle
s
B
efo
re y
ou
beg
in C
ha
pte
r 4
•
Read
each
sta
tem
en
t.
•
Deci
de w
heth
er
you
Agre
e (
A)
or
Dis
agre
e (
D)
wit
h t
he s
tate
men
t.
•
Wri
te A
or
D i
n t
he f
irst
colu
mn
OR
if
you
are
not
sure
wh
eth
er
you
agre
e o
r d
isagre
e,
wri
te N
S (
Not
Su
re).
A
fter y
ou
co
mp
lete
Ch
ap
ter 4
•
Rere
ad
each
sta
tem
en
t an
d c
om
ple
te t
he l
ast
colu
mn
by e
nte
rin
g a
n A
or
a D
.
•
Did
an
y o
f you
r op
inio
ns
abou
t th
e s
tate
men
ts c
han
ge f
rom
th
e f
irst
colu
mn
?
•
For
those
sta
tem
en
ts t
hat
you
mark
wit
h a
D,
use
a p
iece
of
pap
er
to w
rite
an
exam
ple
of
wh
y y
ou
dis
agre
e.
4 Ste
p 1
ST
EP
1A
, D
, o
r N
SS
tate
men
tS
TE
P 2
A o
r D
1
. F
or
a t
rian
gle
to b
e a
cute
, all
th
ree a
ngle
s m
ust
be
acu
te.
A
2
. A
ll s
ides
of
an
equ
ilate
ral
tria
ngle
are
con
gru
en
t.A
3
. A
n e
xte
rior
an
gle
is
an
y a
ngle
ou
tsid
e o
f th
e g
iven
tr
ian
gle
.D
4
. T
he s
um
of
the m
easu
res
of
the a
ngle
s in
a t
rian
gle
is
360°.
D
5
. A
n o
btu
se t
rian
gle
is
a t
rian
gle
wit
h t
hre
e o
btu
se
an
gle
s.D
6
. T
rian
gle
s th
at
are
th
e s
am
e s
hap
e a
nd
siz
e a
re
con
gru
en
t.A
7
. If
th
e a
ngle
s of
on
e t
rian
gle
are
con
gru
en
t to
th
e
an
gle
s of
an
oth
er
tria
ngle
, th
en
th
e t
wo t
rian
gle
s are
co
ngru
en
t.D
8
. A
n i
sosc
ele
s tr
ian
gle
is
a t
rian
gle
wit
h t
wo c
on
gru
en
t si
des.
A
9
. A
tri
an
gle
can
be e
qu
ian
gu
lar
an
d n
ot
equ
ilate
ral.
D
10
. T
o m
ak
e t
he c
alc
ula
tion
s in
a c
oord
inate
pro
of
easi
er,
p
lace
eit
her
a v
ert
ex o
r th
e c
en
ter
of
the p
oly
gon
at
the o
rigin
.A
Ste
p 2
Answers (Anticipation Guide and Lesson 4-1)
Chapter 4 A1 Glencoe Geometry
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Lesson 4-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
7
Gle
nc
oe
Ge
om
etr
y
Sk
ills
Pra
ctic
e
Cla
ssif
yin
g T
rian
gle
s
Cla
ssif
y e
ach
tria
ng
le a
s a
cu
te,
eq
uia
ng
ula
r,
ob
tuse,
or r
igh
t.
1
.
60°
60°
60°
2
. 40
° 95°
45°
3.
50°
40°
90°
e
qu
ian
gu
lar
ob
tuse
rig
ht
4
. 80
°50
°
50°
5
.
100°
30°
50°
6.
70°
55°
55°
a
cu
te
ob
tuse
acu
te
Cla
ssif
y e
ach
tria
ng
le a
s e
qu
ila
tera
l, i
so
scele
s,
or s
ca
len
e.
EC
D
AB
9
88√2
7
. △
AB
E
8.
△E
DB
s
cale
ne
iso
scele
s
9
. △
EB
C
10
. △
DB
C
s
cale
ne
eq
uala
tera
l
11
. A
LG
EB
RA
F
ind
x a
nd
th
e l
en
gth
of
each
sid
e i
f △
AB
C i
s an
iso
scele
s
tria
ngle
wit
h −−
A
B #
−−
BC
.
12
. A
LG
EB
RA
F
ind
x a
nd
th
e l
en
gth
of
each
sid
e i
f △
FG
H i
s an
equ
ilate
ral
tria
ngle
.
5x-8
3x+10
4x+1
13
. A
LG
EB
RA
F
ind
x a
nd
th
e l
en
gth
of
each
sid
e i
f △
RS
T i
s an
iso
scele
s
tria
ngle
wit
h −
−
RS
# −
−
TS .
3x-1
9x+2
6x+8
14
. A
LG
EB
RA
F
ind
x a
nd
th
e l
en
gth
of
each
sid
e i
f △
DE
F i
s an
equ
ilate
ral
tria
ngle
.
5x-21
2x+6
7x-39
4-1 x
= 4
; A
B =
BC
= 1
3;
AC
= 6
x =
9;
FG
= G
H =
FH
= 3
7
x =
2;
RS =
ST =
20;
RT
= 5
x =
9;
DE =
EF =
DF =
24
4x-3
2x+5
x+2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
6
Gle
nc
oe
Ge
om
etr
y
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Cla
ssif
yin
g T
rian
gle
s
Cla
ssif
y T
ria
ng
les
by
Sid
es
You
can
cla
ssif
y a
tri
an
gle
by t
he n
um
ber
of
con
gru
en
t si
des.
Equ
al
nu
mbers
of
hash
mark
s in
dic
ate
con
gru
en
t si
des.
• If
all
thre
e s
ide
s o
f a
tria
ng
le a
re c
on
gru
en
t, t
he
n t
he
tria
ng
le is a
n e
qu
ila
tera
l tr
ian
gle
.
• If
at
lea
st
two s
ide
s o
f a
tria
ng
le a
re c
on
gru
en
t, t
he
n t
he
tria
ng
le is a
n i
so
sc
ele
s t
ria
ng
le.
Eq
uila
tera
l tr
ian
gle
s c
an
als
o b
e c
on
sid
ere
d iso
sce
les.
• If
no
tw
o s
ide
s o
f a
tria
ng
le a
re c
on
gru
en
t, t
he
n t
he
tria
ng
le is a
sc
ale
ne
tri
an
gle
.
C
lassif
y e
ach
tria
ng
le.
a
.
LJ
H
b.
N
RP
c.
2312
15X
V
T
T
wo s
ides
are
con
gru
en
t.
A
ll t
hre
e s
ides
are
T
he t
rian
gle
has
no p
air
Th
e t
rian
gle
is
an
co
ngru
en
t. T
he t
rian
gle
of
con
gru
en
t si
des.
It
is
isosc
ele
s tr
ian
gle
. is
an
equ
ilate
ral
tria
ngle
. a s
cale
ne t
rian
gle
.
Exerc
ises
Cla
ssif
y e
ach
tria
ng
le a
s e
qu
ila
tera
l, i
so
scele
s,
or s
ca
len
e.
1
. 2
1
3√
G
CA
2.
G
KI
18
1818
3
.
1217
19Q
O
M
s
cale
ne
eq
uilate
ral
scale
ne
4
.
UW
S
5.
8x
32x
32x
B C
A
6.
DE
Fx
xx
i
so
scele
s
iso
scele
s
eq
uilate
ral
Exam
ple
4-1
7
. A
LG
EB
RA
F
ind
x a
nd
th
e l
en
gth
of
each
sid
e i
f △
RS
T i
s an
equ
ilate
ral
tria
ngle
.
8
. A
LG
EB
RA
F
ind
x a
nd
th
e l
en
gth
of
each
sid
e i
f △
AB
C i
s is
osc
ele
s w
ith
AB
= B
C.
5x-4
2x+2
3x
3y+2
4y
3y
2;
62;
AB
= 8
; B
C =
8,
AC
= 6
Answers (Lesson 4-1)
Chapter 4 A2 Glencoe Geometry
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
8
Gle
nc
oe
Ge
om
etr
y
Practi
ce
Cla
ssif
yin
g T
rian
gle
s
Cla
ssif
y e
ach
tria
ng
le a
s acute
, equiangular, obtuse,
or right.
1
.
2.
3.
Cla
ssif
y e
ach
tria
ng
le i
n t
he
fig
ure
at
the
rig
ht
by
its
an
gle
s a
nd
sid
es.
4
. △
AB
D
5.
△A
BC
eq
uia
ng
ula
r; e
qu
ilate
ral
rig
ht;
scale
ne
6
. △
ED
C
7.
△B
DC
ri
gh
t; s
cale
ne
ob
tuse;
iso
scele
s
ALG
EB
RA
F
or e
ach
tria
ng
le,
fin
d x
an
d t
he
me
asu
re
of
ea
ch
sid
e.
8
. △
FG
H i
s an
equ
ilate
ral
tria
ngle
wit
h F
G =
x +
5,
GH
= 3
x -
9,
an
d F
H =
2x -
2.
9.
△L
MN
is
an
iso
scele
s tr
ian
gle
, w
ith
LM
= L
N,
LM
= 3
x -
2,
LN
= 2
x +
1,
an
d
MN
= 5
x -
2.
Fin
d t
he
me
asu
re
s o
f th
e s
ide
s o
f △KPL
an
d c
lassif
y e
ach
tria
ng
le b
y i
ts s
ide
s.
10
. K
(-3,
2),
P(2
, 1),
L(-
2,
-3)
11
. K
(5,
-3),
P(3
, 4),
L(-
1,
1)
12
. K
(-2,
-6),
P(-
4,
0),
L(3
, -
1)
13
. D
ES
IGN
D
ian
a e
nte
red
th
e d
esi
gn
at
the r
igh
t in
a l
ogo c
on
test
sp
on
sore
d b
y a
wil
dli
fe e
nvir
on
men
tal
gro
up
. U
se a
pro
tract
or.
H
ow
man
y r
igh
t an
gle
s are
th
ere
?
AC
DE
B
60°
30°
90°
65°
30°
85°
40°
100°
40°
4-1 x
= 7
; FG
= 1
2,
GH
= 1
2,
FH
= 1
2
x =
3;
LM
= 7
, LN
= 7
, M
N =
13
KP
= √
##
26 ,
PL
= 4
√ #
2 ,
LK
= √
##
26 ;
iso
scele
s
KP
= √
##
53 ,
PL
= 5
, LK
= 2
√ #
#
13 ; s
cale
ne
KP
= 2
√ #
#
10 ,
PL
= 5
√ #
2 ,
LK
= 5
√ #
2 ;
iso
scele
s
5
o
btu
se
acu
te
rig
ht
Lesson 4-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
9
Gle
nc
oe
Ge
om
etr
y
Wo
rd
Pro
ble
m P
racti
ce
Cla
ssif
yin
g T
rian
gle
s
1
. M
US
EU
MS
Pau
l is
sta
nd
ing i
n f
ron
t of
a m
use
um
exh
ibit
ion
. W
hen
he t
urn
s h
is
head
60
° to
th
e l
eft
, h
e c
an
see a
sta
tue
by D
on
ate
llo.
Wh
en
he t
urn
s h
is h
ead
60
° to
th
e r
igh
t, h
e c
an
see a
sta
tue b
y
Dell
a R
obbia
. T
he t
wo s
tatu
es
an
d P
au
l fo
rm t
he v
ert
ices
of
a t
rian
gle
. C
lass
ify
this
tri
an
gle
as
acu
te,
righ
t, o
r obtu
se.
2
. P
AP
ER
M
ars
ha c
uts
a r
ect
an
gu
lar
pie
ce
of
pap
er
in h
alf
alo
ng a
dia
gon
al.
Th
e
resu
lt i
s tw
o t
rian
gle
s. C
lass
ify t
hese
tr
ian
gle
s as
acu
te,
righ
t, o
r obtu
se.
3. W
AT
ER
SK
IIN
G K
im a
nd
Cass
an
dra
are
wate
rsk
iin
g.
Th
ey a
re h
old
ing o
n t
o
rop
es
that
are
th
e s
am
e l
en
gth
an
d t
ied
to
th
e s
am
e p
oin
t on
th
e b
ack
of
a s
peed
boat.
Th
e b
oat
is g
oin
g f
ull
sp
eed
ah
ead
an
d t
he r
op
es
are
fu
lly t
au
t.
K
im,
Cass
an
dra
, an
d t
he p
oin
t w
here
th
e r
op
es
are
tie
d o
n t
he b
oat
form
th
e
vert
ices
of
a t
rian
gle
. T
he d
ista
nce
betw
een
Kim
an
d C
ass
an
dra
is
never
equ
al
to t
he l
en
gth
of
the r
op
es.
Cla
ssif
y
the t
rian
gle
as
equ
ilate
ral,
iso
scele
s, o
r sc
ale
ne.
4
. B
OO
KE
ND
S
Tw
o b
ook
en
ds
are
sh
ap
ed
li
ke r
igh
t tr
ian
gle
s.
Th
e b
ott
om
sid
e o
f each
tri
an
gle
is
exact
ly h
alf
as
lon
g a
s th
e s
lan
ted
sid
e
of
the t
rian
gle
. If
all
th
e b
ook
s betw
een
th
e b
ook
en
ds
are
rem
oved
an
d t
hey a
re
pu
shed
togeth
er,
th
ey w
ill
form
a s
ingle
tr
ian
gle
. C
lass
ify t
he t
rian
gle
th
at
can
be f
orm
ed
as
equ
ilate
ral,
iso
scele
s, o
r sc
ale
ne.
5
. D
ES
IGN
S
Su
zan
ne s
aw
th
is p
att
ern
on
a p
en
tagon
al
floor
tile
. S
he n
oti
ced
man
y
dif
fere
nt
kin
ds
of
tria
ngle
s w
ere
cre
ate
d
by t
he l
ines
on
th
e t
ile.
a.
Iden
tify
fiv
e t
rian
gle
s th
at
ap
pear
to
be a
cute
iso
scele
s tr
ian
gle
s.
b.
Iden
tify
fiv
e t
rian
gle
s th
at
ap
pear
to
be o
btu
se i
sosc
ele
s tr
ian
gle
s.
4-1
Kim
Cassandra
CA
GH
IF
JB
ED
Th
ey a
re b
oth
rig
ht
tria
ng
les.
ob
tuse
eq
uilate
ral
Sam
ple
an
sw
ers
: △
EB
D,
△C
AE
, △
AD
C,
△B
GH
, △
CH
I, △
DIJ
, △
EJF,
△A
GF
Sam
ple
an
sw
ers
: △
AFE
, △
EJD
, △
CID
, △
BH
C,
△B
GA
, △
AH
D,
△C
GE
, △
AC
J
iso
scele
s
Answers (Lesson 4-1)
Chapter 4 A3 Glencoe Geometry
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
10
G
len
co
e G
eo
me
try
En
rich
men
t
Read
ing
Math
em
ati
cs
Wh
en
you
read
geom
etr
y,
you
may n
eed
to d
raw
a d
iagra
m t
o m
ak
e t
he t
ext
easi
er
to u
nd
ers
tan
d.
C
on
sid
er t
hre
e p
oin
ts, A
, B
, a
nd
C o
n a
co
ord
ina
te g
rid
. T
he
y-c
oo
rd
ina
tes o
f A
an
d B
are
th
e s
am
e.
Th
e x
-co
ord
ina
te o
f B
is g
re
ate
r t
ha
n t
he
x-c
oo
rd
ina
te o
f A
. B
oth
co
ord
ina
tes o
f C
are
gre
ate
r t
ha
n t
he
co
rre
sp
on
din
g
co
ord
ina
tes o
f B
. Is
△ABC
acu
te,
rig
ht,
or o
btu
se
?
To a
nsw
er
this
qu
est
ion
, fi
rst
dra
w a
sam
ple
tri
an
gle
that
fits
th
e d
esc
rip
tion
.
Sid
e −
−
AB
mu
st b
e a
hori
zon
tal
segm
en
t beca
use
th
e
y-c
oord
inate
s are
th
e s
am
e.
Poin
t C
mu
st b
e l
oca
ted
to t
he r
igh
t an
d u
p f
rom
poin
t B
.
Fro
m t
he d
iagra
m y
ou
can
see t
hat
tria
ngle
AB
C
mu
st b
e o
btu
se.
Exerc
ises
Dra
w a
sim
ple
tria
ng
le o
n g
rid
pa
pe
r t
o h
elp
yo
u.
1
. C
on
sid
er
thre
e p
oin
ts,
R,
S,
an
d
2. C
on
sid
er
thre
e n
on
coll
inear
poin
ts,
T o
n a
coord
inate
gri
d.
Th
e
J
, K
, an
d L
on
a c
oord
inate
gri
d.
Th
e
x-c
oord
inate
s of
R a
nd
S a
re t
he
y-c
oord
inate
s of
J a
nd
K a
re t
he
sam
e.
Th
e y
-coord
inate
of
T i
s sa
me.
Th
e x
-coord
inate
s of
K a
nd
L
betw
een
th
e y
-coord
inate
s of
R
are
th
e s
am
e.
Is △
JK
L a
cute
,
an
d S
. T
he x
-coord
inate
of
T i
s le
ss
ri
gh
t, o
r obtu
se?
than
th
e x
-coord
inate
of
R.
Is a
ngle
R o
f △
RS
T a
cute
, ri
gh
t, o
r
obtu
se?
3
. C
on
sid
er
thre
e n
on
coll
inear
poin
ts,
4. C
on
sid
er
thre
e p
oin
ts,
G,
H,
an
d I
D,
E,
an
d F
on
a c
oord
inate
gri
d.
on
a c
oord
inate
gri
d.
Poin
ts G
an
d
Th
e x
-coord
inate
s of
D a
nd
E a
re
H
are
on
th
e p
osi
tive y
-axis
, an
d
op
posi
tes.
Th
e y
-coord
inate
s of
D a
nd
th
e y
-coord
inate
of
G i
s tw
ice t
he
E a
re t
he s
am
e.
Th
e x
-coord
inate
of
y-c
oord
inate
of
H.
Poin
t I
is o
n t
he
F i
s 0.
Wh
at
kin
d o
f tr
ian
gle
mu
st
p
osi
tive x
-axis
, an
d t
he x
-coord
inate
△D
EF
be:
scale
ne,
isosc
ele
s, o
r of
I is
gre
ate
r th
an
th
e y
-coord
inate
equ
ilate
ral?
of
G.
Is △
GH
I sc
ale
ne,
isosc
ele
s, o
r
equ
ilate
ral?
BA
C
x
y
O
4-1 Exam
ple
iso
scele
s
rig
ht
scale
ne
acu
te
Lesson 4-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
11
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
An
gle
s o
f Tri
an
gle
s
Tri
an
gle
An
gle
-Su
m T
he
ore
m
If t
he m
easu
res
of
two a
ngle
s of
a t
rian
gle
are
k
now
n,
the m
easu
re o
f th
e t
hir
d a
ngle
can
alw
ays
be f
ou
nd
.
Tri
an
gle
An
gle
Su
m
Th
eo
rem
Th
e s
um
of
the
me
asu
res o
f th
e a
ng
les o
f a
tria
ng
le is 1
80
.
In t
he
fig
ure
at
the
rig
ht,
m∠
A +
m∠
B +
m∠
C =
18
0.
CA
B
F
ind
m∠T
.
m∠
R +
m∠
S +
m∠
T =
180
Triangle
Angle
-
Sum
Theore
m
25 +
35 +
m∠
T =
180
Substitu
tion
60 +
m∠
T =
180
Sim
plif
y.
m
∠T
= 1
20
Subtr
act
60 f
rom
each s
ide.
F
ind
th
e m
issin
g
an
gle
me
asu
re
s.
m∠
1 +
m∠
A +
m∠
B =
180
Triangle
Angle
- S
um
T
heore
m
m∠
1 +
58 +
90 =
180
Substitu
tion
m∠
1 +
148 =
180
Sim
plif
y.
m∠
1 =
32
Subtr
act
148 f
rom
each s
ide.
m∠
2 =
32
Vert
ical angle
s a
re
congru
ent.
m∠
3 +
m∠
2 +
m∠
E =
180
Triangle
Angle
- S
um
T
heore
m
m∠
3 +
32 +
108 =
180
Substitu
tion
m∠
3 +
140 =
180
Sim
plif
y.
m∠
3 =
40
Subtr
act
140 f
rom
each s
ide.
58°90°
108°
1
23
E
DA
C
B
Exerc
ises
Fin
d t
he
me
asu
re
s o
f e
ach
nu
mb
ere
d a
ng
le.
1
.
2.
3
.
4.
5
.
6.
20°
152°
DG
A
130°
60°1
2
S
R
TW
V WT
U
30°
60°
2
1
S
QR
30°
1
90°
62°
1NM
P
4-2
Exam
ple
1
35°
25°
RT
S
Exam
ple
2
m∠
1 =
28
m∠
1 =
120
m∠
1 =
30,
m∠
2 =
60
m∠
1 =
56,
m∠
2 =
56,
m∠
3 =
74
m∠
1 =
30,
m∠
2 =
60
m∠
1 =
8
Q
O
NM P
58°
66°
50°
3
21
Answers (Lesson 4-1 and Lesson 4-2)
Chapter 4 A4 Glencoe Geometry
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
12
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
An
gle
s o
f Tri
an
gle
s
Ex
teri
or
An
gle
Th
eo
rem
A
t each
vert
ex o
f a t
rian
gle
, th
e a
ngle
form
ed
by o
ne s
ide
an
d a
n e
xte
nsi
on
of
the o
ther
sid
e i
s ca
lled
an
ex
terio
r a
ng
le o
f th
e t
rian
gle
. F
or
each
exte
rior
an
gle
of
a t
rian
gle
, th
e r
em
ote
in
terio
r a
ng
les a
re t
he i
nte
rior
an
gle
s th
at
are
not
ad
jace
nt
to t
hat
exte
rior
an
gle
. In
th
e d
iagra
m b
elo
w, ∠
B a
nd
∠A
are
th
e r
em
ote
in
teri
or
an
gle
s fo
r exte
rior ∠
DC
B.
Ex
teri
or
An
gle
Th
eo
rem
Th
e m
ea
su
re o
f a
n e
xte
rio
r a
ng
le o
f a
tria
ng
le is e
qu
al to
the
su
m o
f th
e m
ea
su
res o
f th
e t
wo
re
mo
te in
terio
r a
ng
les.
AC
B
D
1m∠
1 =
m∠
A +
m∠
B
F
ind
m∠
1.
m∠
1 =
m∠
R +
m∠
S
Exte
rior
Angle
Theore
m
= 6
0 +
80
Substitu
tion
= 1
40
Sim
plif
y.
RT
S
60°80°
1
F
ind
x.
m∠
PQ
S =
m∠
R +
m∠
S
Exte
rior
Angle
Theore
m
78 =
55 +
x
Substitu
tion
23 =
x
Subtr
act
55 f
rom
each
sid
e.
SR
Q
P
55°
78°
x°
Exerc
ises
Fin
d t
he
me
asu
re
s o
f e
ach
nu
mb
ere
d a
ng
le.
1
.
2.
3
.
4.
Fin
d e
ach
me
asu
re
.
5
. m∠
AB
C
6. m∠
F
E
FG
H58°
x°
x°
B
A
DC
95°
2x°
145°
UT
SR
V
35°
36°
80°
13
2
PO
Q
NM
60°
60°
32
1
BC
D
A
25°
35°
12
YZ
W
X
65°50°
1
4-2
Exam
ple
1Exam
ple
2
m∠
1 =
115
m∠
1 =
60,
m∠
2 =
120
m∠
1 =
60,
m∠
2 =
60,
m∠
3 =
12
0m∠
1 =
109,
m∠
2 =
29,
m∠
3 =
71
50
29
Lesson 4-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
13
G
len
co
e G
eo
me
try
Sk
ills
Pra
ctic
e
An
gle
s o
f Tri
an
gle
s
Fin
d t
he
me
asu
re
of
ea
ch
nu
mb
ere
d a
ng
le.
1
.
2.
Fin
d e
ach
me
asu
re
.
3
. m∠
1
4
. m∠
2
5
. m∠
3
Fin
d e
ach
me
asu
re
.
6
. m∠
1
7
. m∠
2
8
. m∠
3
Fin
d e
ach
me
asu
re
.
9
. m∠
1
10
. m∠
2
11
. m∠
3
12
. m∠
4
13
. m∠
5
Fin
d e
ach
me
asu
re
.
14
. m∠
1
15
. m∠
263°
1
2
DA
C
B
80°
60°
40°
105°
14
5
2
3
150°
55°
70°
12
3
85°
55°
40°
12
3
146°
12
TIG
ER
S80°
73°
1
4-2
m∠
1 =
27
m∠
1 =
m∠
2 = 1
7
55
55
70
125
55
95
140
40
65
75
115
27
27
Answers (Lesson 4-2)
Chapter 4 A5 Glencoe Geometry
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
14
G
len
co
e G
eo
me
try
Practi
ce
An
gle
s o
f Tri
an
gle
s
Fin
d t
he
me
asu
re
of
ea
ch
nu
mb
ere
d a
ng
le.
1
.
2.
Fin
d e
ach
me
asu
re
.
3
. m∠
1
97
4
. m∠
2
83
5
. m∠
3
62
Fin
d e
ach
me
asu
re
.
6
. m∠
1
104
7
. m∠
4
45
8
. m∠
3
65
9
. m∠
2
79
10
. m∠
5
73
11
. m∠
6
147
Fin
d e
ach
me
asu
re
.
12
. m∠
1
26
13
. m∠
2
32
14
. C
ON
ST
RU
CT
ION
T
he d
iagra
m s
how
s an
exam
ple
of
the P
ratt
Tru
ss u
sed
in
bri
dge
con
stru
ctio
n.
Use
th
e d
iagra
m t
o f
ind
m∠
1.
145°
1
64°
58°
1
2AB
C
D
118°
36°
68°
70° 65° 82°
1
2
34
5
6
58°
39°
35°
12
3
40°
1
55°
72°
1
2
4-2 5
5
m
∠1 =
18,
m∠
1 =
90
m∠
1 =
85
Lesson 4-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
15
G
len
co
e G
eo
me
try
Wo
rd
Pro
ble
m P
racti
ce
An
gle
s o
f Tri
an
gle
s
62˚
136˚
1
. P
AT
HS
E
ric
walk
s aro
un
d a
tri
an
gu
lar
path
. A
t each
corn
er,
he r
eco
rds
the
measu
re o
f th
e a
ngle
he c
reate
s.
He m
ak
es
on
e c
om
ple
te c
ircu
it a
rou
nd
the p
ath
. W
hat
is t
he s
um
of
the
thre
e a
ngle
measu
res
that
he w
rote
dow
n?
2
. S
TA
ND
ING
S
am
, K
en
dra
, an
d T
on
y a
re
stan
din
g i
n s
uch
a w
ay t
hat
if l
ines
were
dra
wn
to c
on
nect
th
e f
rien
ds
they w
ou
ld
form
a t
rian
gle
.
If S
am
is
look
ing a
t K
en
dra
he n
eed
s to
turn
his
head
40°
to l
ook
at
Ton
y.
If
Ton
y i
s lo
ok
ing a
t S
am
he n
eed
s to
tu
rn
his
head
50°
to l
ook
at
Ken
dra
. H
ow
man
y d
egre
es
wou
ld K
en
dra
have t
o
turn
her
head
to l
ook
at
Ton
y i
f sh
e i
s
look
ing a
t S
am
?
3
. T
OW
ER
S
A l
ook
ou
t to
wer
sits
on
a
netw
ork
of
stru
ts a
nd
post
s. L
esl
ie
measu
red
tw
o a
ngle
s on
th
e t
ow
er.
W
hat
is t
he m
easu
re o
f an
gle
1?
4
. Z
OO
S T
he z
oo l
igh
ts u
p t
he c
him
pan
zee
pen
wit
h a
n o
verh
ead
lig
ht
at
nig
ht.
Th
e
cross
sect
ion
of
the l
igh
t beam
mak
es
an
isosc
ele
s tr
ian
gle
.
Th
e t
op
an
gle
of
the t
rian
gle
is
52 a
nd
the e
xte
rior
an
gle
is
116.
Wh
at
is t
he
measu
re o
f an
gle
1?
5
. D
RA
FT
ING
C
hlo
e b
ou
gh
t a d
raft
ing
table
an
d s
et
it u
p s
o t
hat
she c
an
dra
w
com
fort
ably
fro
m h
er
stool.
Ch
loe
measu
red
th
e t
wo a
ngle
s cr
eate
d b
y t
he
legs
an
d t
he t
able
top
in
case
sh
e h
ad
to
dis
man
tle t
he t
able
.
a.
Wh
ich
of
the f
ou
r n
um
bere
d a
ngle
s
can
Ch
loe d
ete
rmin
e b
y k
now
ing t
he
two a
ngle
s fo
rmed
wit
h t
he t
able
top
?
Wh
at
are
th
eir
measu
res?
b.
Wh
at
con
clu
sion
can
Ch
loe m
ak
e
abou
t th
e u
nk
now
n a
ngle
s befo
re s
he
measu
res
them
to f
ind
th
eir
exact
measu
rem
en
ts?
4-2
52˚
116˚
1
Kendra
Tony
Sam
29˚
68˚
12
34
180
90
98
64
∠1 a
nd
∠2;
m∠
1 =
97,
m∠
2 =
83
Th
e s
um
of
the
me
as
ure
s o
f a
ng
le
3 a
nd
4 i
s e
qu
al
to t
he
me
as
ure
of
an
gle
1;
m∠
3 +
m∠
4 =
97
Answers (Lesson 4-2)
Chapter 4 A6 Glencoe Geometry
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
16
G
len
co
e G
eo
me
try
En
rich
men
t
Fin
din
g A
ng
le M
easu
res i
n T
rian
gle
sY
ou
can
use
alg
ebra
to s
olv
e p
roble
ms
involv
ing t
rian
gle
s.
In
tria
ng
le ABC
, m∠A
is t
wic
e m∠B
, a
nd
m∠C
is 8
mo
re
th
an
m∠B
. W
ha
t is
th
e m
ea
su
re
of
ea
ch
an
gle
?
Wri
te a
nd
solv
e a
n e
qu
ati
on
. L
et
x =
m∠
B.
m∠
A +
m∠
B +
m∠
C =
180
2
x +
x +
(x +
8)
= 1
80
4x +
8 =
180
4x =
172
x =
43
So,
m∠
A =
2(4
3)
or
86,
m∠
B =
43,
an
d m
∠C
= 4
3 +
8 o
r 51.
So
lve
ea
ch
pro
ble
m.
1. In
tri
an
gle
DE
F,
m∠
E i
s th
ree t
imes
2. In
tri
an
gle
RS
T,
m∠
T i
s 5 m
ore
th
an
m
∠D
, an
d m
∠F
is
9 l
ess
th
an
m∠
E.
m
∠R
, an
d m
∠S
is
10 l
ess
th
an
m∠
T.
Wh
at
is t
he m
easu
re o
f each
an
gle
? W
hat
is t
he m
easu
re o
f each
an
gle
?
3
. In
tri
an
gle
JK
L,
m∠
K i
s fo
ur
tim
es
4. In
tri
an
gle
XY
Z,
m∠
Z i
s 2 m
ore
th
an
tw
ice
m∠
J,
an
d m
∠L
is
five t
imes
m∠
J.
m
∠X
, an
d m
∠Y
is
7 l
ess
th
an
tw
ice m
∠X
.
Wh
at
is t
he m
easu
re o
f each
an
gle
? W
hat
is t
he m
easu
re o
f each
an
gle
?
5. In
tri
an
gle
GH
I, m
∠H
is
20 m
ore
th
an
6. In
tri
an
gle
MN
O,
m∠
M i
s equ
al
to m
∠N
, m
∠G
, an
d m
∠G
is
8 m
ore
th
an
m∠
I.
an
d m
∠O
is
5 m
ore
th
an
th
ree t
imes
W
hat
is t
he m
easu
re o
f each
an
gle
?
m
∠N
. W
hat
is t
he m
easu
re o
f each
an
gle
?
7. In
tri
an
gle
ST
U,
m∠
U i
s h
alf
m∠
T,
8. In
tri
an
gle
PQ
R,
m∠
P i
s equ
al
to
an
d m
∠S
is
30 m
ore
th
an
m∠
T.
Wh
at
m
∠Q
, an
d m
∠R
is
24 l
ess
th
an
m∠
P.
is t
he m
easu
re o
f each
an
gle
? W
hat
is t
he m
easu
re o
f each
an
gle
?
9
. W
rite
you
r ow
n p
roble
ms
abou
t m
easu
res
of
tria
ngle
s.
4-2 Exam
ple
m∠
D =
27,
m∠
E =
81,
m∠
F =
72
m∠
R =
60,
m∠
S =
55,
m∠
T =
65
m∠
J =
18,
m∠
K =
72,
m∠
L =
90
m∠
X =
37,
m∠
Y =
67,
m∠
Z =
76
m∠
G =
56,
m∠
H =
76,
m∠
I =
48
m∠
M =
m∠
N =
35,
m∠
O =
110
m∠
S =
90,
m∠
T =
60,
m∠
U =
30
m∠
P =
m∠
Q =
68,
m∠
R =
44
See s
tud
en
ts’
wo
rk.
Lesson 4-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
17
G
len
co
e G
eo
me
try
Gra
ph
ing C
alc
ula
tor
Act
ivit
y
Cab
ri J
un
ior:
An
gle
s o
f Tri
an
gle
s
Cabri
Ju
nio
r ca
n b
e u
sed
to i
nvest
igate
th
e r
ela
tion
ship
s betw
een
th
e a
ngle
s in
a t
rian
gle
.
U
se
Ca
bri
Ju
nio
r t
o i
nv
esti
ga
te t
he
su
m o
f th
e m
ea
su
re
s o
f th
e
an
gle
s o
f a
tria
ng
le a
nd
to
in
ve
sti
ga
te t
he
me
asu
re
of
the
ex
terio
r a
ng
le i
n a
tria
ng
le i
n r
ela
tio
n t
o i
ts r
em
ote
in
terio
r a
ng
les.
Ste
p 1
U
se C
abri
Ju
nio
r to
dra
w a
tri
an
gle
.
• S
ele
ct F
2 T
ria
ng
le t
o d
raw
a t
rian
gle
.
• M
ove t
he c
urs
or
to w
here
you
wan
t th
e f
irst
vert
ex.
Th
en
pre
ss
EN
TE
R.
•
R
ep
eat
this
pro
ced
ure
to d
ete
rmin
e t
he n
ext
two v
ert
ices
of
the t
rian
gle
.
Ste
p 2
L
abel
each
vert
ex o
f th
e t
rian
gle
.
• S
ele
ct F
5 A
lph
-nu
m.
•
M
ove t
he c
urs
or
to a
vert
ex a
nd
pre
ss
EN
TE
R.
En
ter
A a
nd
pre
ss
EN
TE
R a
gain
.
• R
ep
eat
this
pro
ced
ure
to l
abel
vert
ex B
an
d v
ert
ex C
.
Ste
p 3
D
raw
an
exte
rior
an
gle
of
△A
BC
.
•
S
ele
ct F
2 L
ine
to d
raw
a l
ine t
hro
ugh
−−−
BC
.
•
S
ele
ct F
2 P
oin
t, P
oin
t o
n t
o d
raw
a p
oin
t on
$ %&
BC
so t
hat
C i
s betw
een
B a
nd
th
e n
ew
poin
t.
• S
ele
ct F
5 A
lph
-nu
m t
o l
abel
the p
oin
t D
.
Ste
p 4
F
ind
th
e m
easu
re o
f th
e t
hre
e i
nte
rior
an
gle
s of
△A
BC
an
d t
he
exte
rior
an
gle
, ∠
AC
D.
•
S
ele
ct F
5 M
ea
su
re
, A
ng
le.
•
T
o f
ind
th
e m
easu
re o
f ∠
AB
C,
sele
ct p
oin
ts A
, B
, an
d C
(wit
h t
he v
ert
ex B
as
the s
eco
nd
poin
t se
lect
ed
).
• U
se t
his
meth
od
to f
ind
th
e r
em
ain
ing a
ngle
measu
res.
Exerc
ises
An
aly
ze
yo
ur d
ra
win
g.
1
. U
se F
5 C
alc
ula
te t
o f
ind
th
e s
um
of
the m
easu
res
of
the t
hre
e i
nte
rior
an
gle
s.
2
. M
ak
e a
con
ject
ure
abou
t th
e s
um
of
the i
nte
rior
an
gle
s of
a t
rian
gle
.
3
. C
han
ge t
he d
imen
sion
s of
the t
rian
gle
by m
ovin
g p
oin
t A
. (P
ress
CLE
AR
so t
he p
oin
ter
beco
mes
a b
lack
arr
ow
. M
ove t
he p
oin
ter
close
to p
oin
t A
un
til
the a
rrow
beco
mes
tran
spare
nt
an
d p
oin
t A
is
bli
nk
ing.
Pre
ss
ALP
HA
to c
han
ge t
he a
rrow
to a
han
d.
Th
en
m
ove t
he p
oin
t.)
Is y
ou
r co
nje
ctu
re s
till
tru
e?
4
. W
hat
is t
he m
easu
re o
f ∠
AC
D?
Wh
at
is t
he s
um
of
the m
easu
res
of
the t
wo r
em
ote
in
teri
or
an
gle
s (∠
AB
C a
nd
∠B
AC
)?
5
. M
ak
e a
con
ject
ure
abou
t th
e r
ela
tion
ship
betw
een
th
e m
easu
re o
f an
exte
rior
an
gle
of
a
tria
ngle
an
d i
ts t
wo r
em
ote
in
teri
or
an
gle
s.
6
. C
han
ge t
he d
imen
sion
s of
the t
rian
gle
by m
ovin
g p
oin
t A
. Is
you
r co
nje
ctu
re s
till
tru
e?
4-2 Exam
ple
yes
Th
e s
um
of
the m
easu
res o
f th
e i
nte
rio
r an
gle
s o
f a t
rian
gle
is 1
80°
yes
See s
tud
en
ts’ w
ork
Th
e m
easu
re o
f an
exte
rio
r an
gle
is
eq
ual to
th
e s
um
of
the m
easu
res o
f it
s t
wo
rem
ote
in
teri
or
an
gle
s.
180°
Answers (Lesson 4-2)
Chapter 4 A7 Glencoe Geometry
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
18
G
len
co
e G
eo
me
try
Geo
mete
r’s
Sk
etc
hp
ad
Act
ivit
y
An
gle
s o
f Tri
an
gle
s
Th
e G
eom
ete
r’s
Sk
etc
hp
ad
can
be u
sed
to i
nvest
igate
th
e r
ela
tion
ship
s betw
een
th
e a
ngle
s in
a t
rian
gle
.
U
se
Th
e G
eo
me
ter’s
Sk
etc
hp
ad
to
in
ve
sti
ga
te t
he
su
m o
f th
e
me
asu
re
s o
f th
e a
ng
les o
f a
tria
ng
le a
nd
to
in
ve
sti
ga
te t
he
me
asu
re
of
the
ex
terio
r a
ng
le i
n a
tria
ng
le i
n r
ela
tio
n t
o i
ts r
em
ote
in
terio
r a
ng
les.
Ste
p 1
U
se T
he G
eom
ete
r’s
Sk
etc
hp
ad
to d
raw
a t
rian
gle
.
•
Con
stru
ct a
ray b
y s
ele
ctin
g t
he R
ay t
ool
from
th
e t
oolb
ar.
Fir
st,
clic
k w
here
you
wan
t th
e f
irst
poin
t. T
hen
cli
ck o
n a
seco
nd
poin
t to
dra
w t
he r
ay.
•
N
ext,
sele
ct t
he S
egm
en
t to
ol
from
th
e t
oolb
ar.
Use
th
e e
nd
poin
t of
the r
ay a
s th
e f
irst
poin
t fo
r th
e s
egm
en
t an
d c
lick
on
a s
eco
nd
poin
t to
con
stru
ct t
he
segm
en
t.
•
Con
stru
ct a
noth
er
segm
en
t jo
inin
g t
he s
eco
nd
poin
t of
the n
ew
segm
en
t to
a
poin
t on
th
e r
ay.
Ste
p 2
D
isp
lay t
he l
abels
for
each
poin
t.
Use
th
e p
oin
ter
tool
to s
ele
ct a
ll
fou
r p
oin
ts.
Dis
pla
y t
he l
abels
by s
ele
ctin
g S
ho
w L
ab
els
fro
m
the D
isp
lay
men
u.
Ste
p 3
F
ind
th
e m
easu
res
of
the t
hre
e
inte
rior
an
gle
s an
d t
he o
ne e
xte
rior
an
gle
. T
o f
ind
th
e m
easu
re o
f an
gle
A
BC
, u
se t
he S
ele
ctio
n A
rrow
tool
to s
ele
ct p
oin
ts A
, B
, an
d C
(w
ith
th
e v
ert
ex B
as
the s
eco
nd
poin
t se
lect
ed
). T
hen
, u
nd
er
the M
ea
su
re
m
en
u,
sele
ct A
ng
le.
Use
th
is m
eth
od
to
fin
d t
he r
em
ain
ing a
ngle
measu
res.
Exerc
ises
An
aly
ze
yo
ur d
ra
win
g.
1
. W
hat
is t
he s
um
of
the m
easu
res
of
the t
hre
e i
nte
rior
an
gle
s?
2
. M
ak
e a
con
ject
ure
abou
t th
e s
um
of
the i
nte
rior
an
gle
s of
a t
rian
gle
.
3
. C
han
ge t
he d
imen
sion
s of
the t
rian
gle
by s
ele
ctin
g p
oin
t A
wit
h t
he
poin
ter
tool
an
d m
ovin
g i
t. I
s you
r co
nje
ctu
re s
till
tru
e?
4
. W
hat
is t
he m
easu
re o
f ∠
AC
D?
Wh
at
is t
he s
um
of
the m
easu
res
of
the
two r
em
ote
in
teri
or
an
gle
s (∠
AB
C a
nd
∠B
AC
)?
5
. M
ak
e a
con
ject
ure
abou
t th
e r
ela
tion
ship
betw
een
th
e m
easu
re o
f an
exte
rior
an
gle
of
a t
rian
gle
an
d i
ts t
wo r
em
ote
in
teri
or
an
gle
s.
6
. C
han
ge t
he d
imen
sion
s of
the t
rian
gle
by s
ele
ctin
g p
oin
t A
wit
h t
he
poin
ter
tool
an
d m
ovin
g i
t. I
s you
r co
nje
ctu
re s
till
tru
e?
m/
AB
C =
60.00°
m/
BA
C =
58.19°
m/
AC
B =
61.81°
m/
AC
D =
118.19°
m/
AB
C +
m/
BA
C +
m/
AC
B
= 180.00°
m/
AB
C +
m/
BA
C =
118.19°
BC
D
A
4-2 Exam
ple
180
°
yes
See s
tud
en
ts’
wo
rk.
Th
e s
um
of
the m
easu
res o
f th
e i
nte
rio
r an
gle
s o
f a t
rian
gle
is 1
80
°.
Th
e m
easu
res o
f an
exte
rio
r an
gle
is e
qu
al to
th
e s
um
of
the m
easu
re o
f it
s t
wo
rem
ote
in
teri
or
an
gle
s.
yes
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 4-3
Ch
ap
ter
4
19
G
len
co
e G
eo
me
try
Co
ng
rue
nce
an
d C
orr
esp
on
din
g P
art
s T
rian
gle
s th
at
have t
he s
am
e s
ize a
nd
sam
e s
hap
e a
re
co
ng
ru
en
t tr
ian
gle
s.
Tw
o t
rian
gle
s are
con
gru
en
t if
an
d
on
ly i
f all
th
ree p
air
s of
corr
esp
on
din
g a
ngle
s are
con
gru
en
t an
d a
ll t
hre
e p
air
s of
corr
esp
on
din
g s
ides
are
con
gru
en
t. I
n
the f
igu
re,
△A
BC
" △
RS
T.
If
△XYZ
# △
RST
, n
am
e t
he
pa
irs o
f
co
ng
ru
en
t a
ng
les a
nd
co
ng
ru
en
t sid
es.
∠X
" ∠
R,
∠Y
" ∠
S,
∠Z
" ∠
T
−−
XY
" −
−
RS
, −
−
XZ
" −
−
RT
, −
−
YZ
" −
−
ST
Exerc
ises
Sh
ow
th
at
the
po
lyg
on
s a
re
co
ng
ru
en
t b
y i
de
nti
fyin
g a
ll c
on
gru
en
t co
rre
sp
on
din
g
pa
rts
. T
he
n w
rit
e a
co
ng
ru
en
ce
sta
tem
en
t.
1
.
CA
B
LJ
K
2.
C
D
A
B
3.
K J
L M
4
. F
GL
K JE
5
. B
D
CA
6
. R TU
S
△A
BC
# △
DE
F.
7
. F
ind
th
e v
alu
e o
f x.
8
. F
ind
th
e v
alu
e o
f y.
Y
XZ
T
SR
AC
B
R
TS
4-3
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Co
ng
ruen
t Tri
an
gle
s
Th
ird
An
gle
s
Th
eo
rem
If two a
ngles o
f one triangle a
re c
ongru
ent to
two a
ngles
of a s
econd triangle, th
en the third a
ngles o
f th
e triangles a
re c
ongru
ent.
Exam
ple
2x
+y
(2y
-5)°
75°
65°
40°
96.6
90.6
64.3
∠A
# ∠
J;
∠B
# ∠
K;
∠A
# ∠
D;
∠A
BC
# ∠
BC
D
∠
J #
∠L
; ∠
JK
M #
∠LM
K;
∠C
# ∠
L; −
−
AB
# −
−
JK
;
∠
AC
B #
∠C
BD
; −
−
AC
# −
−
BD
∠K
MJ #
∠M
KL; −
−
KJ #
−−
LM
−
−
BC
# −
−
KL ; −
−
AC
# −
−
JL
−
−
AB
# −
−
CD
−
−
KL #
−−
JM
△A
BC
# △
JK
L
△
AB
C #
△D
CB
△JK
M #
△LM
K
27.8
35
∠E
# ∠
J;
∠F #
∠K
;
∠
A #
∠D
;
∠
R #
∠T
;
∠G
# ∠
L; −
−
EF #
−−
JK
;
∠
AB
C #
∠D
CB
;
∠
RS
U #
∠T
SU
;
−
−
EG
# −
−
JL ; −
−
FG
# −
−
KL
∠
AC
B #
∠D
BC
;
∠
RU
S #
∠T
US
;
△FG
E #
△K
LJ
−
−
AB
# −
−
DC
; −
−
AC
# −
−
DB
;
−
−
RU
# −
−
TU
; −
−
RS #
−−
TS ;
−
−
BC
# −
−
CB
; △
AB
C #
△D
CB
−
−
SU
# −
−
SU
; △
RS
U #
△TS
U
Answers (Lesson 4-2 and Lesson 4-3)
Chapter 4 A8 Glencoe Geometry
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
20
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Co
ng
ruen
t Tri
an
gle
s
4-3 Exam
ple
Pro
ve
Tri
an
gle
s C
on
gru
en
t T
wo t
rian
gle
s are
con
gru
en
t if
an
d o
nly
if
their
co
rresp
on
din
g p
art
s are
con
gru
en
t. C
orr
esp
on
din
g p
art
s in
clu
de c
orr
esp
on
din
g a
ngle
s an
d
corr
esp
on
din
g s
ides.
Th
e p
hra
se “
if a
nd
on
ly i
f” m
ean
s th
at
both
th
e c
on
dit
ion
al
an
d i
ts
con
vers
e a
re t
rue.
For
tria
ngle
s, w
e s
ay,
“Corr
esp
on
din
g p
art
s of
con
gru
en
t tr
ian
gle
s are
co
ngru
en
t,”
or
CP
CT
C.
W
rit
e a
tw
o-c
olu
mn
pro
of.
G
ive
n:
−−
AB
" −
−−
CB
, −
−−
AD
" −
−−
CD
, ∠
BA
D "
∠B
CD
−
−−
BD
bis
ect
s ∠
AB
C
P
ro
ve
: △
AB
D "
△C
BD
P
ro
of:
Sta
tem
en
tR
ea
so
n
1.
−−
AB
" −
−−
CB
, −
−−
AD
" −
−−
CD
2.
−−
−
BD
" −
−−
BD
3.
∠B
AD
" ∠
BC
D
4.
∠A
BD
" ∠
CB
D
5.
∠B
DA
" ∠
BD
C
6.
△A
BD
" △
CB
D
1.
Giv
en
2.
Refl
exiv
e P
rop
ert
y o
f co
ngru
en
ce
3.
Giv
en
4.
Defi
nit
ion
of
an
gle
bis
ect
or
5.
Th
ird
An
gle
s T
heore
m
6.
CP
CT
C
Exerc
ises
Writ
e a
tw
o-c
olu
mn
pro
of.
1
. G
ive
n:
∠A
" ∠
C,
∠D
" ∠
B,
−−
−
AD
" −
−−
CB
, −
−
AE
" −
−−
CE
, −
−
AC
bis
ect
s −
−−
BD
Pro
ve
: △
AE
D "
△C
EB
Pro
of:
Sta
tem
en
tsR
easo
ns
1. ∠
A
∠C
, ∠
D
∠B
2. ∠
AE
D
∠C
EB
3.
−−
AD
−
−
CB
, −
−
AE
−
−
CE
4.
−−
DE
−
−
BE
5.
△A
ED
△
CE
B
1.
Giv
en
2.
Vert
ical
an
gle
s
3.
Giv
en
4.
Defi
nit
ion
of
seg
men
t b
isecto
r
5.
CP
CT
C
Writ
e a
pa
ra
gra
ph
pro
of.
2
. G
ive
n:
−−
−
BD
bis
ect
s ∠
AB
C a
nd ∠
AD
C,
−−
AB
" −
−−
CB
, −
−
AB
" −
−−
AD
, −
−−
CB
"
−−
−
DC
P
ro
ve
: △
AB
D "
△C
BD
We a
re g
iven
−−
BD
bis
ects
∠A
BC
an
d ∠
AD
C.
Th
ere
fore
∠
AB
D
∠C
BD
an
d ∠
AD
B
∠C
DB
by t
he d
efi
nit
ion
o
f an
gle
bis
ecto
rs.
By t
he T
hir
d A
ng
le T
heo
rem
, w
e
fin
d t
hat
∠A
∠
C.
We a
re g
iven
th
at
−−
AB
−
−
CB
, −
−
AB
−
−
AD
, an
d −
−
CB
−
−
DC
. U
sin
g t
he s
ub
sti
tuti
on
pro
pert
y,
we c
an
dete
rmin
e t
hat
−−
AD
−
−
CD
. F
inally,
−−
BD
−
−
BD
usin
g t
he R
efl
exiv
e P
rop
ert
y o
f co
ng
ruen
ce.
Th
ere
fore
△A
BD
△
CB
D b
y C
PC
TC
.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 4-3
Ch
ap
ter
4
21
G
len
co
e G
eo
me
try
4-3
Sh
ow
th
at
po
lyg
on
s a
re
co
ng
ru
en
t b
y i
de
nti
fyin
g a
ll c
on
gru
en
t co
rre
sp
on
din
g
pa
rts
. T
he
n w
rit
e a
co
ng
ru
en
ce
sta
tem
en
t.
1
.
L
P
J
S
V
T
2.
In t
he
fig
ure
, △
AB
C
△F
DE
.
3
. F
ind
th
e v
alu
e o
f x.
36
4
. F
ind
th
e v
alu
e o
f y.
48
5
. P
RO
OF W
rite
a t
wo-c
olu
mn
pro
of.
G
ive
n:
−−
−
AB
" −
−−
CB
, −
−−
AD
" −
−−
CD
, ∠
AB
D "
∠C
BD
,∠
AD
B "
∠C
DB
Pro
ve
: △
AB
D
△C
BD
Pro
of:
Sta
tem
en
ts
Reaso
ns
1.
−−
AB
−
−
CB
, −
−
AD
−
−
CD
1.
Giv
en
2.
∠A
BD
∠
CB
D,
∠A
DB
∠
CD
B
2.
Giv
en
3.
∠A
∠
C
3.
Th
ird
An
gle
Th
eo
rem
4.
△A
BD
△
CB
D
4.
CP
CT
C
Sk
ills
Pra
ctic
e
Co
ng
ruen
t Tri
an
gle
s
D
E G
F
108°
48°
y°
(2x
-y)°
AF
BC
ED
∠J
∠T;
∠P
∠
V;
∠E
DF
∠FD
G;
∠E
∠
G;
∠L
∠S
; −
−
JP
−
−
TV
∠
EFD
∠
GFD
; −
−
DE
−
−
DG
;
−−
PL
−−
VS
; −
−
JL
−−
TS
−
−
GF
−−
EF ;
△D
EF
△D
GF
△JP
L
△TV
S
Answers (Lesson 4-3)
Chapter 4 A9 Glencoe Geometry
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
22
G
len
co
e G
eo
me
try
Practi
ce
Co
ng
ruen
t Tri
an
gle
s
Sh
ow
th
at
the
po
lyg
on
s a
re
co
ng
ru
en
t b
y i
nd
en
tify
ing
all
co
ng
ru
en
t
co
rre
sp
on
din
g p
arts
. T
he
n w
rit
e a
co
ng
ru
en
ce
sta
tem
en
t.
1
.
D
R
S
CA
B
2.
M
N
L
P
Q
Po
lyg
on
ABCD
p
oly
go
n PQRS
.
3
. F
ind
th
e v
alu
e o
f x.
4
. F
ind
th
e v
alu
e o
f y.
5. P
RO
OF W
rite
a t
wo-c
olu
mn
pro
of.
Giv
en
: ∠
P !
∠R
, ∠
PS
Q !
∠R
SQ
, −
−−
PQ
! −
−−
RQ
, −
−
PS
! −
−
RS
Pro
ve
: △
PQ
S
△R
QS
Pro
of:
Sta
tem
en
ts
Reaso
ns
1.
∠P
∠
R,
∠P
SQ
∠
RS
Q
1.
Giv
en
2.
∠P
QS
∠
RQ
S
2.
Th
ird
An
gle
Th
eo
rem
3.
−−
PQ
−
−
RQ
, −
−
PS
−
−
RS
3.
Giv
en
4.
−−
QS
−
−
QS
4.
Refl
exiv
e P
rop
ert
y
5.
△P
QS
△
RQ
S
5.
CP
CT
C
6
. Q
UIL
TIN
G
a.
Ind
icate
th
e t
rian
gle
s th
at
ap
pear
to b
e c
on
gru
en
t.
b.
Nam
e t
he c
on
gru
en
t an
gle
s an
d c
on
gru
en
t si
des
of
a p
air
of
con
gru
en
t tr
ian
gle
s.
B
A I
E FH
G
CD
4-3
(2x
+4)° (3
y-
3)
80°
100°
10
12
∠A
∠
D;
∠B
∠
R
∠L
∠Q
; ∠
M
∠P
∠C
∠
S;
−−
AB
−
−
RD
∠
MN
L
∠P
NQ
;
−−
BC
−
−
RS
; −
−
AC
−
−
SD
−
−
LM
−
−
PQ
; −
−
MN
−
−
NP
△A
BC
△
DR
S
−−
LM
−
−
NQ
△LM
N
△Q
PN
△A
BI
△
EB
F,
△C
BD
△
HB
G
S
am
ple
an
sw
er:
∠A
∠
E,
∠A
BI
∠
EB
F,
∠I
∠
F;
−−
AB
−
−
EB
, −
−
BI
−
−
BF ,
−−
AI
−
−
EF
x =
48
y =
5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
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D
Lesson 4-3
Ch
ap
ter
4
23
G
len
co
e G
eo
me
try
Wo
rd
Pro
ble
m P
racti
ce
Co
ng
ruen
t Tri
an
gle
s
1
. P
ICT
UR
E H
AN
GIN
G C
an
dic
e h
un
g a
p
ictu
re t
hat
was
in a
tri
an
gu
lar
fram
e
on
her
bed
room
wall
.
BeforeA
B
C
A
B
C
After
On
e d
ay,
it f
ell
to t
he f
loor,
bu
t d
id n
ot
bre
ak
or
ben
d.
Th
e f
igu
re s
how
s th
e
obje
ct b
efo
re a
nd
aft
er
the f
all
. L
abel
the v
ert
ices
on
th
e f
ram
e a
fter
the f
all
acc
ord
ing t
o t
he “
befo
re”
fram
e.
2
. S
IER
PIN
SK
I’S
TR
IAN
GLE
T
he f
igu
re
belo
w i
s a p
ort
ion
of
Sie
rpin
ski’s
Tri
an
gle
. T
he t
rian
gle
has
the p
rop
ert
y
that
an
y t
rian
gle
mad
e f
rom
an
y
com
bin
ati
on
of
ed
ges
is e
qu
ilate
ral.
H
ow
man
y t
rian
gle
s in
th
is p
ort
ion
are
co
ngru
en
t to
th
e b
lack
tri
an
gle
at
the
bott
om
corn
er?
3
. Q
UIL
TIN
G S
tefa
n d
rew
th
is p
att
ern
for
a p
iece
of
his
qu
ilt.
It
is m
ad
e u
p o
f co
ngru
en
t is
osc
ele
s ri
gh
t tr
ian
gle
s. H
e
dre
w o
ne t
rian
gle
an
d t
hen
rep
eate
dly
d
rew
it
all
th
e w
ay a
rou
nd
.
45˚
Wh
at
are
th
e m
issi
ng m
easu
res
of
the
an
gle
s of
the t
rian
gle
?
4
. M
OD
ELS
D
an
a b
ou
gh
t a m
od
el
air
pla
ne
kit
. W
hen
he o
pen
ed
th
e b
ox,
these
tw
o
con
gru
en
t tr
ian
gu
lar
pie
ces
of
wood
fell
ou
t of
it.
A
B
CQ
R
P
Id
en
tify
th
e t
rian
gle
th
at
is c
on
gru
en
t to
△
AB
C.
5
. G
EO
GR
AP
HY
Ig
or
noti
ced
on
a m
ap
th
at
the t
rian
gle
wh
ose
vert
ices
are
th
e
sup
erm
ark
et,
th
e l
ibra
ry,
an
d t
he p
ost
off
ice (
△S
LP
) is
con
gru
en
t to
th
e
tria
ngle
wh
ose
vert
ices
are
Igor’
s h
om
e,
Jaco
b’s
hom
e,
an
d B
en
’s h
om
e (
△IJ
B).
T
hat
is,
△S
LP
! △
IJB
.
a.
Th
e d
ista
nce
betw
een
th
e
sup
erm
ark
et
an
d t
he p
ost
off
ice i
s 1 m
ile.
Wh
ich
path
alo
ng t
he t
rian
gle
△
IJB
is
con
gru
en
t to
th
is?
b.
Th
e m
easu
re o
f ∠
LP
S i
s 40.
Iden
tify
th
e a
ngle
th
at
is c
on
gru
en
t to
th
is
an
gle
in
△IJ
B.
4-3 1
1
45 a
nd
90
△Q
RP
the p
ath
betw
een
Ig
or’
s
ho
me a
nd
Ben
’s h
om
e
∠JB
I
Answers (Lesson 4-3)
Chapter 4 A10 Glencoe Geometry
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
24
G
len
co
e G
eo
me
try
En
rich
men
t4-3
Overl
ap
pin
g T
rian
gle
sS
om
eti
mes
wh
en
tri
an
gle
s overl
ap
each
oth
er,
th
ey s
hare
corr
esp
on
din
g p
art
s. W
hen
try
ing
to p
ictu
re t
he c
orr
esp
on
din
g p
art
s, t
ry r
ed
raw
ing t
he t
rian
gle
s se
para
tely
.
In t
he f
igu
re △
AB
D !
△D
CA
an
d m
∠B
DA
= 3
0°.
Fin
d m
∠C
AD
an
d m
∠C
DA
.
30°
Fir
st,
red
raw
△A
BD
an
d △
DC
A l
ike t
his
.
30°
Now
you
can
see t
hat
∠B
DA
! ∠
CA
D b
eca
use
th
ey a
re c
orr
esp
on
din
g p
art
s of
con
gru
en
t tr
ian
gle
s, s
o m
∠C
AD
= 3
0.
Usi
ng t
he T
rian
gle
Su
m T
heore
m,
180 -
(90 +
30)
= 6
0,
so
m∠
CD
A =
60
.
Exerc
ises
1
. In
th
e f
igu
re △
AB
C !
△C
DB
, m
∠A
BC
= 1
10,
an
d
m∠
D =
20.
Fin
d m
∠A
CB
an
d m
∠D
CB
.
2
. W
rite
a t
wo c
olu
mn
pro
of.
Giv
en
: ∠
A a
nd
∠B
are
rig
ht
an
gle
s, ∠
AC
D !
∠B
DC
,
−−
−
AD
! −
−−
BC
, −
−
AC
! −
−−
BD
Pro
ve
: △
AD
C !
△C
BD
Pro
of:
Sta
tem
en
ts
Reaso
ns
1.
∠A
an
d ∠
B a
re r
igh
t an
gle
s
1.
Giv
en
2.
∠A
" ∠
B
2.
All r
t #
are
"
3.
∠A
CD
" ∠
BD
C
3.
Giv
en
4.
∠A
DC
" ∠
BC
D
4.
Th
ird
An
gle
Th
eo
rem
5.
−−
AD
" −
−
BC
, −
−
AC
" −
−
BD
5.
Giv
en
6.
−−
DC
" −
−
CD
6.
Refl
exiv
e p
rop
ert
y
7.
△A
DC
" △
CB
D
7.
CP
CT
C
Exam
ple
m∠
AC
B =
m∠
DB
C =
50
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 4-4
Ch
ap
ter
4
25
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Pro
vin
g T
rian
gle
s C
on
gru
en
t—S
SS
, S
AS
SS
S P
ostu
late
Y
ou
kn
ow
th
at
two t
rian
gle
s are
con
gru
en
t if
corr
esp
on
din
g s
ides
are
co
ngru
en
t an
d c
orr
esp
on
din
g a
ngle
s are
con
gru
en
t. T
he S
ide-S
ide-S
ide (
SS
S)
Post
ula
te l
ets
you
sh
ow
th
at
two t
rian
gle
s are
con
gru
en
t if
you
kn
ow
on
ly t
hat
the s
ides
of
on
e t
rian
gle
are
con
gru
en
t to
th
e s
ides
of
the s
eco
nd
tri
an
gle
.
W
rit
e a
tw
o-c
olu
mn
pro
of.
Giv
en
: −
−
AB
! −
−−
DB
an
d C
is
the m
idp
oin
t of
−−
−
AD
.
Pro
ve
: △
AB
C !
△D
BC
B C
DA
Sta
tem
en
ts
Re
aso
ns
1.
−−
AB
! −
−−
DB
1
. G
iven
2.
C i
s th
e m
idp
oin
t of
−−
−
AD
. 2
. G
iven
3.
−−
AC
! −
−−
DC
3
. M
idp
oin
t T
heore
m
4.
−−
−
BC
! −
−−
BC
4
. R
efl
exiv
e P
rop
ert
y o
f !
5.
△A
BC
! △
DB
C
5.
SS
S P
ost
ula
te
Exerc
ises
Writ
e a
tw
o-c
olu
mn
pro
of.
1
. B
Y
CA
XZ
Giv
en
:A
B!
−−
XY
, −
−A
C !
−−
XZ
, −
−−
BC
!−
−Y
Z
Pro
ve
:△
AB
C!
△X
YZ
2
. T
U
RS
Giv
en
: −
−
RS
! −
−−
UT
, −
−
RT
! −
−−
US
Pro
ve
: △
RS
T !
△U
TS
4-4 SS
S P
os
tula
teIf
th
ree
sid
es o
f o
ne
tria
ng
le a
re c
on
gru
en
t to
th
ree
sid
es o
f a
se
co
nd
tria
ng
le,
the
n t
he
tria
ng
les a
re c
on
gru
en
t.
Exam
ple
Sta
tem
en
ts
Reaso
ns
Sta
tem
en
ts
Reaso
ns
1.
−−
AB
" −
−
XY
1.
Giv
en
−
−
AC
" −
−
XZ
−
−
BC
" −
−
YZ
2.
△A
BC
" △
XY
Z
2.
SS
S P
ost.
1.
−−
RS
" −
−
UT
1.
Giv
en
−
−
RT "
−−
US
2.
−−
ST "
−−
TS
2
. R
efl
. P
rop
.
3.
△R
ST "
△U
TS
3.
SS
S P
ost.
Answers (Lesson 4-3 and Lesson 4-4)
Chapter 4 A11 Glencoe Geometry
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
26
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Pro
vin
g T
rian
gle
s C
on
gru
en
t—S
SS
, S
AS
SA
S P
ostu
late
A
noth
er
way t
o s
how
th
at
two t
rian
gle
s are
con
gru
en
t is
to u
se t
he
Sid
e-A
ngle
-Sid
e (
SA
S)
Post
ula
te.
F
or e
ach
dia
gra
m,
de
term
ine
wh
ich
pa
irs o
f tr
ian
gle
s c
an
be
pro
ve
d c
on
gru
en
t b
y t
he
SA
S P
ostu
late
.
a
. A B
C
X YZ
b
. D
H
FE
G
J
c.
PQ
S
1
2R
In
△A
BC
, th
e a
ngle
is
not
T
he r
igh
t an
gle
s are
T
he i
ncl
ud
ed
an
gle
s,
“in
clu
ded
” by t
he s
ides
−−
AB
co
ngru
en
t an
d t
hey a
re t
he
∠
1 a
nd
∠2,
are
con
gru
en
t an
d −
−
AC
. S
o t
he t
rian
gle
s
in
clu
ded
an
gle
s fo
r th
e
beca
use
th
ey a
re
can
not
be p
roved
con
gru
en
t
co
ngru
en
t si
des.
alt
ern
ate
in
teri
or
an
gle
s by t
he S
AS
Post
ula
te.
△
DE
F $
△J
GH
by t
he
fo
r tw
o p
ara
llel
lin
es.
S
AS
Post
ula
te.
△
PS
R $
△R
QP
by t
he
S
AS
Post
ula
te.
Exerc
ises
4-4
SA
S P
os
tula
te
If t
wo
sid
es a
nd
th
e in
clu
de
d a
ng
le o
f o
ne
tria
ng
le a
re c
on
gru
en
t to
tw
o
sid
es a
nd
th
e in
clu
de
d a
ng
le o
f a
no
the
r tr
ian
gle
, th
en
th
e t
ria
ng
les a
re
co
ng
rue
nt.
Exam
ple
Writ
e t
he
sp
ecif
ied
ty
pe
of
pro
of.
1
. W
rite
a t
wo c
olu
mn
pro
of.
2
. W
rite
a t
wo-c
olu
mn
pro
of.
Giv
en
: N
P =
PM
, −−
− N
P ⊥
−−
PL
G
ive
n:
AB
= C
D,
−−
AB
‖ −
−−
CD
Pro
ve
: △
NP
L $
△M
PL
P
ro
ve
: △
AC
D $
△C
AB
Pro
of:
Pro
of:
3
. W
rite
a p
ara
gra
ph
pro
of.
Giv
en
: V
is
the m
idp
oin
t of
−−
YZ
.
V
is
the m
idp
oin
t of
−−
−
WX
.
Pro
ve
: △
XV
Z $
△W
VY
S
ince V
is t
he m
idp
oin
t o
f −
−
YZ a
nd
th
e m
idp
oin
t o
f −
−
WX ,
by t
he d
efi
nit
ion
of
mid
po
int,
Y
V =
VZ
an
d W
V =
VX
. S
ince ∠
YV
W a
nd
∠X
VZ
are
vert
ical
an
gle
s,
by t
he V
ert
ical
An
gle
Th
eo
rem
th
e a
ng
les a
re c
on
gru
en
t. T
here
fore
, b
y S
AS
, △
XV
Z $
△W
VY
Sta
tem
en
tsR
easo
ns
1.
NP
= P
M, −
−
NP
⊥ −
−
PL
2.
∠M
PL i
s a
rig
ht
an
gle
∠N
PL i
s a
rig
ht
an
gle
3.
∠M
PL $
∠N
PL
4.
PL =
PL
5.
△N
PL $
△M
PL
1.
Giv
en
2.
perp
en
dic
ula
r lin
es
form
rig
ht
an
gle
s
3.
all r
igh
t an
gle
s a
re
co
ng
ruen
t
4.
Refl
exiv
e P
rop
ert
y
of
co
ng
ruen
ce
5.
SA
S
Sta
tem
en
tsR
easo
ns
1.
AB
= C
D, −
−
AB
‖ −
−
CD
2.
∠B
AC
$ ∠
DC
A
3.
AC
= A
C
4.
△A
CD
$ △
CA
B
1.
Giv
en
2.
Alt
ern
ate
In
teri
or
An
gle
s T
heo
rem
3.
Re
fle
xiv
e P
rop
ert
y
of
co
ng
rue
nc
e
4.
SA
S
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 4-4
Ch
ap
ter
4
27
G
len
co
e G
eo
me
try
De
term
ine
wh
eth
er △
AB
C $
△K
LM
. E
xp
lain
.
1
. A
(-3,
3),
B(-
1,
3),
C(-
3,
1),
K(1
, 4),
L(3
, 4),
M(1
, 6)
2
. A
(-4,
-2),
B(-
4,
1),
C(-
1,
-1),
K(0
, -
2),
L(0
, 1),
M(4
, 1)
PR
OO
F W
rit
e t
he
sp
ecif
ied
ty
pe
of
pro
of.
3
. W
rite
a f
low
pro
of.
G
ive
n:
−−
PR
$ −
−−
DE
, −
−
PT
$ −−
− D
F
∠R
$ ∠
E,
∠T
$ ∠
F
P
ro
ve
: △
PR
T $
△D
EF
PR
$DE
Given
PT
$DF
Given
∠R
$∠E
Given
∠P
$∠D
Thir
d A
ng
le
Theo
rem
△PRT
$△DEF
SA
S
∠T
$∠F
Given
4
. W
rite
a t
wo-c
olu
mn
pro
of.
Giv
en
: −
−
AB
$ −−
− C
B ,
D i
s th
e m
idp
oin
t of
−−
AC
.
Pro
ve
: △
AB
D $
△C
BD
Pro
of:
Sta
tem
en
ts
Reaso
ns
1.
−−
AB
$ −
−
CB
, D
is t
he m
idp
oin
t o
f −
−
AC
1.
Giv
en
2.
−−
AD
$ −
−
DC
2.
Defi
nit
ion
of
Mid
po
int
3.
−−
BD
$ −
−
BD
3.
Refl
exiv
e P
rop
ert
y o
f C
on
gru
en
ce
4.
△A
BD
$ △
CB
D
4.
SS
S
T
R
P
F
E
D
4-4
Sk
ills
Pra
ctic
e
Pro
vin
g T
rian
gle
s C
on
gru
en
t—S
SS
, S
AS
AB
= 2
, K
L =
2,
BC
= 2
√ ( 2
, LM
= 2
√ ( 2
, A
C =
2,
KM
= 2
.
Th
e c
orr
esp
on
din
g s
ides h
ave t
he s
am
e m
easu
re a
nd
are
co
ng
ruen
t,
so
△A
BC
$ △
KLM
by S
SS
.
AB
= 3
, K
L =
3,
BC
= √
((
13 ,
LM
= 4
, A
C =
√ (
(
10 ,
KM
= 5
.
Th
e c
orr
esp
on
din
g s
ides a
re n
ot
co
ng
ruen
t, s
o △
AB
C i
s n
ot
co
ng
ruen
t to
△K
LM
.
Pro
of:
Answers (Lesson 4-4)
Chapter 4 A12 Glencoe Geometry
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
28
G
len
co
e G
eo
me
try
Practi
ce
Pro
vin
g T
rian
gle
s C
on
gru
en
t—S
SS
, S
AS
De
term
ine
wh
eth
er △
DE
F !
△P
QR
giv
en
th
e c
oo
rd
ina
tes o
f th
e v
erti
ce
s.
Ex
pla
in.
1
. D
(-6,
1),
E(1
, 2),
F(-
1,
-4),
P(0
, 5),
Q(7
, 6),
R(5
, 0)
2
. D
(-7,
-3),
E(-
4,
-1),
F(-
2,
-5),
P(2
, -
2),
Q(5
, -
4),
R(0
, -
5)
3
. W
rite
a f
low
pro
of.
Giv
en
: −
−
RS
" −
−
TS
V i
s th
e m
idp
oin
t of
−−
RT
.
Pro
ve
: △
RS
V "
△T
SV
De
term
ine
wh
ich
po
stu
late
ca
n b
e u
se
d t
o p
ro
ve
th
at
the
tria
ng
les a
re
co
ng
ru
en
t.
If i
t is
no
t p
ossib
le t
o p
ro
ve
co
ng
ru
en
ce
, w
rit
e n
ot
po
ssib
le.
4
.
5.
6.
7
. IN
DIR
EC
T M
EA
SU
RE
ME
NT
T
o m
easu
re t
he w
idth
of
a s
ink
hole
on
his
pro
pert
y,
Harm
on
mark
ed
off
con
gru
en
t tr
ian
gle
s as
show
n i
n t
he
dia
gra
m.
How
does
he k
now
th
at
the l
en
gth
s A'B' an
d A
B a
re e
qu
al?
A'
B'
AB
C
S
R V T
4-4 D
E =
5 √
$
2 ,
PQ
= 5
√ $
2 ,
EF =
2 √
$$
10 ,
QR
= 2
√ $
$
10 ,
DF =
5 √
$
2 ,
PR
= 5
√ $
2 .
△D
EF
! △
PQ
R b
y S
SS
sin
ce c
orr
esp
on
din
g s
ides h
ave t
he s
am
e
measu
re a
nd
are
co
ng
ruen
t.
DE
= √
$$
13 ,
PQ
= √
$$
13 ,
EF =
2 √
$
5 ,
QR
= √
$$
26 ,
DF =
√ $
$
29 ,
PR
= √
$$
13 .
Co
rresp
on
din
g s
ides a
re n
ot
co
ng
ruen
t, s
o △
DE
F i
s n
ot
co
ng
ruen
t to
△P
QR
.
Sin
ce ∠
AC
B a
nd
∠A
'CB
' are
vert
ical
an
gle
s,
they a
re c
on
gru
en
t. I
n t
he
fig
ure
, −
−
AC
! −
−−
A′C
an
d −
−
BC
! −
−−
B'C
. S
o △
AB
C !
△A
'B'C
by S
AS
. B
y C
PC
TC
, th
e l
en
gth
s A
'B' a
nd
AB
are
eq
ual.
Pro
of:
n
ot
po
ssib
le
SA
S o
r S
SS
S
SS
RS TS
Given
SV SV
Refl
exiv
e
Pro
per
ty
RV VT
Defi
nitio
n
of
mid
po
int
V is
the
mid
po
int
of RT.
Given
∆RSV
∆TSV
SS
S
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 4-4
Ch
ap
ter
4
29
G
len
co
e G
eo
me
try
Wo
rd
Pro
ble
m P
racti
ce
Pro
vin
g T
rian
gle
s C
on
gru
en
t—S
SS
, S
AS
1
. S
TIC
KS
T
yso
n h
ad
th
ree s
tick
s of
len
gth
s 24 i
nch
es,
28 i
nch
es,
an
d
30 i
nch
es.
Is
it p
oss
ible
to m
ak
e t
wo
non
con
gru
en
t tr
ian
gle
s u
sin
g t
he s
am
e
thre
e s
tick
s? E
xp
lain
.
2
. B
AK
ER
Y S
on
ia m
ad
e a
sh
eet
of
bak
lava.
Sh
e h
as
mark
ings
on
her
pan
so
th
at
she c
an
cu
t th
em
in
to l
arg
e
squ
are
s. A
fter
she c
uts
th
e p
ast
ry i
n
squ
are
s, s
he c
uts
th
em
dia
gon
all
y t
o
form
tw
o c
on
gru
en
t tr
ian
gle
s. W
hic
h
post
ula
te c
ou
ld y
ou
use
to p
rove t
he
two t
rian
gle
s co
ngru
en
t?
3
. C
AK
E C
arl
had
a p
iece
of
cak
e i
n t
he
shap
e o
f an
iso
scele
s tr
ian
gle
wit
h
an
gle
s 26,
77,
an
d 7
7.
He w
an
ted
to
div
ide i
t in
to t
wo e
qu
al
part
s, s
o h
e c
ut
it t
hro
ugh
th
e m
idd
le o
f th
e 2
6 a
ngle
to
the m
idp
oin
t of
the o
pp
osi
te s
ide.
He
says
that
beca
use
he i
s d
ivid
ing i
t at
the
mid
poin
t of
a s
ide,
the t
wo p
iece
s are
co
ngru
en
t. I
s th
is e
nou
gh
in
form
ati
on
? E
xp
lain
.
4
. T
ILE
S T
am
my i
nst
all
s bath
room
til
es.
H
er
curr
en
t jo
b r
equ
ires
tile
s th
at
are
equ
ilate
ral
tria
ngle
s an
d a
ll t
he t
iles
have t
o b
e c
on
gru
en
t to
each
oth
er.
Sh
e
has
a b
ig s
ack
of
tile
s all
in
th
e s
hap
e
of
equ
ilate
ral
tria
ngle
s. A
lth
ou
gh
sh
e
kn
ow
s th
at
all
th
e t
iles
are
equ
ilate
ral,
sh
e i
s n
ot
sure
th
ey a
re a
ll t
he s
am
e
size.
Wh
at
mu
st s
he m
easu
re o
n e
ach
ti
le t
o b
e s
ure
th
ey a
re c
on
gru
en
t?
Exp
lain
.
5. IN
VE
ST
IGA
TIO
NA
n i
nvest
igato
r at
a
crim
e s
cen
e f
ou
nd
a t
rian
gu
lar
pie
ce o
f to
rn f
abri
c. T
he i
nvest
igato
r re
mem
bere
d t
hat
on
e o
f th
e s
usp
ect
s h
ad
a t
rian
gu
lar
hole
in
th
eir
coat.
P
erh
ap
s it
was
a m
atc
h.
Un
fort
un
ate
ly,
to a
void
tam
peri
ng w
ith
evid
en
ce,
the
invest
igato
r d
id n
ot
wan
t to
tou
ch t
he
fabri
c an
d c
ou
ld n
ot
fit
it t
o t
he c
oat
dir
ect
ly.
a.
If t
he i
nvest
igato
r m
easu
res
all
th
ree
sid
e l
en
gth
s of
the f
abri
c an
d t
he
hole
, ca
n t
he i
nvest
igato
r m
ak
e a
co
ncl
usi
on
abou
t w
heth
er
or
not
the
hole
cou
ld h
ave b
een
fil
led
by t
he
fabri
c?
b.
If t
he i
nvest
igato
r m
easu
res
two s
ides
of
the f
abri
c an
d t
he i
ncl
ud
ed
an
gle
an
d t
hen
measu
res
two s
ides
of
the
hole
an
d t
he i
ncl
ud
ed
an
gle
can
he
dete
rmin
e i
f it
is
a m
atc
h?
Exp
lain
.
4-4 N
o.
Th
e s
ticks d
o n
ot
ch
an
ge
siz
e,
so
an
y a
rran
gem
en
t w
ill
yie
ld a
tri
an
gle
co
ng
ruen
t to
th
e
on
e b
efo
re.
eit
her
SS
S o
r S
AS
Sam
ple
an
sw
er:
No
. T
hat
is
on
ly e
xp
lain
ing
on
e s
ide t
hat
is
co
ng
ruen
t o
n t
he t
rian
gle
s.
He
need
s t
o s
ho
w t
wo
mo
re s
ides
of
the t
rian
gle
are
co
ng
ruen
t to
pro
ve t
he p
ieces a
re c
on
gru
en
t.
yes
Sh
e j
ust
need
s t
o m
easu
re o
ne
sid
e o
f each
tile b
ecau
se t
hey
are
all e
qu
ilate
ral
tria
ng
les.
On
ly i
f th
e t
wo
sid
es a
nd
in
clu
ded
an
gle
of
on
e t
rian
gle
are
th
e c
orr
esp
on
din
g s
ides
an
d i
nclu
ded
an
gle
of
the
oth
er
tria
ng
le.
Answers (Lesson 4-4)
Chapter 4 A13 Glencoe Geometry
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
30
G
len
co
e G
eo
me
try
En
rich
men
t4-4
Co
ng
ruen
t Tri
an
gle
s i
n t
he C
oo
rdin
ate
Pla
ne
If y
ou
kn
ow
th
e c
oord
inate
s of
the v
ert
ices
of
two t
rian
gle
s in
th
e c
oord
inate
p
lan
e,
you
can
oft
en
deci
de w
heth
er
the t
wo t
rian
gle
s are
con
gru
en
t. T
here
m
ay b
e m
ore
th
an
on
e w
ay t
o d
o t
his
.
1
. C
on
sid
er
△A
BD
an
d △
CD
B w
hose
vert
ices
have c
oord
inate
s A
(0,
0),
B
(2,
5),
C(9
, 5),
an
d D
(7,
0).
Bri
efl
y d
esc
ribe h
ow
you
can
use
wh
at
you
k
now
abou
t co
ngru
en
t tr
ian
gle
s an
d t
he c
oord
inate
pla
ne t
o s
how
th
at
△A
BD
! △
CD
B.
You
may w
ish
to m
ak
e a
sk
etc
h t
o h
elp
get
you
sta
rted
.
2
. C
on
sid
er
△P
QR
an
d △
KL
M w
hose
vert
ices
are
th
e f
oll
ow
ing p
oin
ts.
P
(1,
2)
Q
(3,
6)
R
(6,
5)
K
(-2,
1)
L
(-6,
3)
M
(-5,
6)
B
riefl
y d
esc
ribe h
ow
you
can
sh
ow
th
at
△P
QR
! △
KL
M.
3
. If
you
kn
ow
th
e c
oord
inate
s of
all
th
e v
ert
ices
of
two t
rian
gle
s, i
s it
a
lwa
ys
poss
ible
to t
ell
wh
eth
er
the t
rian
gle
s are
con
gru
en
t? E
xp
lain
.
Sam
ple
an
sw
er:
Sh
ow
th
at
the s
lop
es o
f −
−
AD
an
d −
−
BC
are
eq
ual. C
on
clu
de
that
−−
BC
||
−−
AD
. U
se t
he a
ng
le r
ela
tio
nsh
ips f
or
para
llel
lin
es a
nd
a
tran
svers
al
to c
on
clu
de t
hat
∠C
BD
is c
on
gru
en
t to
∠B
DA
. C
ou
nt
the
len
gth
s o
f th
e h
ori
zo
nta
l seg
men
ts t
o c
on
clu
de t
hat
BC
= A
D.
Usin
g
these f
acts
an
d t
he f
act
that
−−
BD
is a
co
mm
on
sid
e f
or
the t
rian
gle
s t
o
co
nclu
de t
hat
△A
BD
$ △
CD
B b
y S
AS
.
Use t
he D
ista
nce F
orm
ula
to
fin
d t
he l
en
gth
s o
f th
e s
ides o
f b
oth
tr
ian
gle
s.
Co
nclu
de t
hat
△P
QR
$ △
KLM
by S
SS
.
Yes;
yo
u c
an
use t
he D
ista
nce F
orm
ula
an
d S
SS
.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 4-5
Ch
ap
ter
4
31
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Pro
vin
g T
rian
gle
s C
on
gru
en
t—A
SA
, A
AS
AS
A P
ostu
late
T
he A
ngle
-Sid
e-A
ngle
(A
SA
) P
ost
ula
te l
ets
you
sh
ow
th
at
two t
rian
gle
s are
con
gru
en
t.
W
rit
e a
tw
o c
olu
mn
pro
of.
G
ive
n:
−−
AB
‖ −
−−
CD
∠C
BD
! ∠
AD
B
P
ro
ve
: △
AB
D !
△C
DB
Sta
tem
en
tsR
ea
so
ns
1.
−−
AB
‖ −
−−
CD
2.
∠C
BD
! ∠
AD
B
3.
∠A
BD
! ∠
BD
C
4.
BD
= B
D
5.
△A
BD
! △
CD
B
1.
Giv
en
2.
Giv
en
3.
Alt
ern
ate
In
teri
or
An
gle
s T
heore
m
4.
Refl
exiv
e P
rop
ert
y o
f co
ngru
en
ce
5.
AS
A
4-5 AS
A P
os
tula
teIf
tw
o a
ng
les a
nd
th
e in
clu
de
d s
ide
of
on
e t
ria
ng
le a
re c
on
gru
en
t to
tw
o a
ng
les a
nd
the
in
clu
de
d s
ide
of
an
oth
er
tria
ng
le,
the
n t
he
tria
ng
les a
re c
on
gru
en
t.
Exam
ple
Exerc
ises
PR
OO
F W
rit
e t
he
sp
ecif
ied
ty
pe
of
pro
of.
1
. W
rite
a t
wo c
olu
mn
pro
of.
2
. W
rite
a p
ara
gra
ph
pro
of.
S
V
U
RT
AC
B
E
D
Giv
en
: ∠
S !
∠V
,
Giv
en
: −
−−
CD
bis
ect
s −
−
AE
, −
−
AB
‖ −
−−
CD
T
is
the m
idp
oin
t of
−−
SV
∠
E !
∠B
CA
Pro
ve
: △
RT
S !
△U
TV
P
ro
ve
: △
AB
C !
△C
DE
Pro
of:
We k
no
w t
hat
∠E
$ ∠
BC
A
an
d −−
− C
D b
isects
−−
AE
. S
ince −−
− C
D
bis
ects
−−
AE
, b
y t
he d
efi
nit
ion
of
bis
ecto
r, A
C =
CE
. W
e a
re a
lso
g
iven
th
at
−−−
AB
‖ −−
− C
D ,
fro
m t
his
we
can
dete
rmin
e t
hat
∠A
is
co
ng
ruen
t to
∠D
CE
by t
he
Alt
ern
ate
In
teri
or
An
gle
T
heo
rem
. F
rom
th
is w
e k
no
w
that
∠A
BC
$ ∠
CD
E b
y t
he
AS
A T
heo
rem
.
Sta
tem
en
tsR
easo
ns
1.
∠S
$ ∠
V
T i
s t
he
mid
po
int
of
−−
SV
2.
ST
= T
V
3.
∠R
TS
$ ∠
VTU
4.
△R
TS
$ △
UTV
1.
Giv
en
2.
defi
nit
ion
of
Mid
po
int
3.
Vert
ical
An
gle
th
eo
rem
.
4.
AS
A
Answers (Lesson 4-4 and Lesson 4-5)
Chapter 4 A14 Glencoe Geometry
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
32
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Pro
vin
g T
rian
gle
s C
on
gru
en
t—A
SA
, A
AS
AA
S T
he
ore
m
An
oth
er
way t
o s
how
th
at
two t
rian
gle
s are
con
gru
en
t is
th
e A
ngle
-A
ngle
-Sid
e (
AA
S)
Th
eore
m.
You
now
have f
ive w
ays
to s
how
th
at
two t
rian
gle
s are
con
gru
en
t.
• d
efi
nit
ion
of
tria
ngle
con
gru
en
ce
• A
SA
Post
ula
te
• S
SS
Post
ula
te
• A
AS
Th
eore
m
• S
AS
Post
ula
te
In
th
e d
iag
ra
m,
∠BCA
! ∠
DCA
. W
hic
h s
ide
s
are
co
ng
ru
en
t? W
hic
h a
dd
itio
na
l p
air
of
co
rre
sp
on
din
g p
arts
ne
ed
s t
o b
e c
on
gru
en
t fo
r t
he
tria
ng
les t
o b
e c
on
gru
en
t b
y
the
AA
S T
he
ore
m?
−−
AC
" −
−
AC
by t
he R
efl
exiv
e P
rop
ert
y o
f co
ngru
en
ce.
Th
e c
on
gru
en
t an
gle
s ca
nn
ot
be ∠
1 a
nd
∠2,
beca
use
−−
AC
wou
ld b
e t
he i
ncl
ud
ed
sid
e.
If ∠
B "
∠D
, th
en
△A
BC
" △
AD
C b
y t
he A
AS
Th
eore
m.
Exerc
ises
PR
OO
F W
rit
e t
he
sp
ecif
ied
ty
pe
of
pro
of.
1
. W
rite
a t
wo c
olu
mn
pro
of.
G
ive
n:
−−−
BC
‖ −
−
EF
A
B =
ED
∠
C "
∠F
P
ro
ve
: △
AB
D "
△D
EF
Sta
tem
en
tsR
easo
ns
1.
−−
BC
‖ −
−
EF
A
B =
ED
∠
C !
∠F
2.
∠A
BD
! ∠
DE
F
3.
△A
BD
! △
DE
F
1.
Giv
en
2.
Co
rresp
on
din
g A
ng
les T
heo
rem
3.
AA
S
2
. W
rite
a f
low
pro
of.
G
ive
n:
∠S
" ∠
U;
−−
TR
bis
ect
s ∠
ST
U.
P
ro
ve
: ∠
SR
T "
∠U
RT
P
roo
f:
S
RT
U
4-5 AA
S T
he
ore
mIf
tw
o a
ng
les a
nd
a n
on
inclu
de
d s
ide
of
on
e t
ria
ng
le a
re c
on
gru
en
t to
th
e c
orr
esp
on
din
g
two
an
gle
s a
nd
sid
e o
f a
se
co
nd
tria
ng
le,
the
n t
he
tw
o t
ria
ng
les a
re c
on
gru
en
t.
Exam
ple
D
C1 2
A
B
Given
Given
RT
RT
Refl
. P
rop. of
Def
.of
∠ b
isec
tor
TR
bis
ects ∠
ST
U.
SR
T
U
RT
∠ST
R
∠U
TR
AA
S
∠SR
T
∠U
RT
CP
CTC
∠S
∠U
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 4-5
Ch
ap
ter
4
33
G
len
co
e G
eo
me
try
PR
OO
F W
rit
e a
flo
w p
ro
of.
1
. G
ive
n:
∠N
" ∠
L
−
−
JK
" −
−−
MK
P
ro
ve
: △
JK
N "
△M
KL
2
. G
ive
n:
−−
AB
" −−
− C
B
∠
A "
∠C
−
−−
DB
bis
ect
s ∠
AB
C.
P
ro
ve
: −−
− A
D "
−−
−
CD
3
. W
rite
a p
ara
gra
ph
pro
of.
G
ive
n:
−−
−
DE
‖ −−
− F
G
∠E
" ∠
G
P
ro
ve
: △
DF
G "
△F
DE
FG
DE
AC
B D
N
J
M
KL
4-5
Sk
ills
Pra
ctic
e
Pro
vin
g T
rian
gle
s C
on
gru
en
t—A
SA
, A
AS
Pro
of:
P
roo
f: S
ince i
t is
giv
en
th
at
−−
DE
‖ −
−
FG
, it
fo
llo
ws t
hat
∠E
DF !
∠G
FD
,
b
ecau
se a
lt.
int.
& o
f p
ara
llel
lin
es a
re !
. It
is g
iven
th
at
∠E
! ∠
G.
By t
he
Refl
exiv
e
P
rop
ert
y,
−−
DF !
−−
FD
. S
o △
DFG
! △
FD
E b
y A
AS
.
∠N
"∠
L
Given
JK
"M
K
Given
∠JK
N"
∠M
KL
Ver
tica
l & a
re '
.
△JK
N"
△M
KL
AA
S
Pro
of:
∠A
"∠
C
Given
AB
"C
B
Given
CP
CT
C
AD
'C
D
DB
bis
ects ∠
AB
C.
Given
△A
BD
"△
CB
D
AS
A
∠A
BD
"∠
CB
D
Def
. o
f ∠
bis
ecto
r
Answers (Lesson 4-5)
Chapter 4 A15 Glencoe Geometry
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
34
G
len
co
e G
eo
me
try
PR
OO
F W
rit
e t
he
sp
ecif
ied
ty
pe
of
pro
of.
1
. W
rite
a f
low
pro
of.
G
ive
n:
S i
s th
e m
idp
oin
t of
−−−
QT
.
−−
−
QR
‖ −
−−
TU
P
ro
ve
: △
QS
R $
△T
SU
2
. W
rite
a p
ara
gra
ph
pro
of.
G
ive
n:
∠D
$ ∠
F
−−
−
GE
bis
ect
s ∠
DE
F.
P
ro
ve
: −
−−
DG
$ −−
− F
G
AR
CH
ITE
CT
UR
E
Fo
r E
xe
rcis
es 3
an
d 4
, u
se
th
e f
oll
ow
ing
in
form
ati
on
.
An
arc
hit
ect
use
d t
he w
ind
ow
desi
gn
in
th
e d
iagra
m w
hen
rem
od
eli
ng
an
art
stu
dio
. −
−
AB
an
d −−
− C
B e
ach
measu
re 3
feet.
3
. S
up
pose
D i
s th
e m
idp
oin
t of
−−
AC
. D
ete
rmin
e w
heth
er
△A
BD
$ △
CB
D.
Ju
stif
y y
ou
r an
swer.
4
. S
up
pose
∠A
$ ∠
C.
Dete
rmin
e w
heth
er
△A
BD
$ △
CB
D.
Ju
stif
y y
ou
r an
swer.
DB
AC
D
G
F
E
U
QS
R
T
4-5
Practi
ce
Pro
vin
g T
rian
gle
s C
on
gru
en
t—A
SA
, A
AS
Pro
of:
Pro
of:
Sin
ce i
t is
giv
en
th
at
−−
GE
bis
ects
∠D
EF,
∠D
EG
# ∠
FE
G b
y t
he
defi
nit
ion
of
an
an
gle
bis
ecto
r. I
t is
giv
en
th
at
∠D
# ∠
F.
By t
he
Refl
exiv
e P
rop
ert
y,
−−
GE
# −
−
GE
. S
o △
DE
G #
△FE
G b
y A
AS
. T
here
fore
−−
DG
# −
−
FG
by C
PC
TC
.
Sin
ce D
is t
he m
idp
oin
t o
f −
−
AC
, −
−
AD
# −
−
CD
by t
he M
idp
oin
t T
heo
rem
.
−−
AB
# −
−
CB
by t
he d
efi
nit
ion
of
co
ng
ruen
t seg
men
ts.
By t
he R
efl
exiv
e
Pro
pert
y −
−
BD
# −
−
BD
. S
o △
AB
D #
△C
BD
by S
SS
.
We a
re g
iven
−−
AB
# −
−
CB
an
d ∠
A #
∠C
. −
−
BD
# −
−
BD
by t
he R
efl
exiv
e
Pro
pert
y.
Sin
ce S
SA
can
no
t b
e u
sed
to
pro
ve t
hat
tria
ng
les a
re
co
ng
ruen
t, w
e c
an
no
t say w
heth
er
△A
BD
# △
CB
D.
∠Q
∠T
Given
QR
||TU
Given
Mid
poin
t Theo
rem
Alt. In
t. &
are
.
QS TS
S is
the
mid
po
int
of QT
.
QSR
TSU
AS
A
∠QSR
∠TSU
Ver
tica
l & a
re
.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Lesson 4-5
Ch
ap
ter
4
35
G
len
co
e G
eo
me
try
1
. D
OO
R S
TO
PS
T
wo d
oor
stop
s h
ave
cross
-sect
ion
s th
at
are
rig
ht
tria
ngle
s.
Th
ey b
oth
have a
20 a
ngle
an
d t
he
len
gth
of
the s
ide b
etw
een
th
e 9
0 a
nd
20 a
ngle
s are
equ
al.
Are
th
e c
ross
-
sect
ion
s co
ngru
en
t? E
xp
lain
.
2
. M
AP
PIN
G T
wo p
eop
le d
eci
de t
o t
ak
e a
walk
. O
ne p
ers
on
is
in B
om
bay a
nd
th
e
oth
er
is i
n M
ilw
au
kee.
Th
ey s
tart
by
walk
ing s
traig
ht
for
1 k
ilom
ete
r. T
hen
they b
oth
tu
rn r
igh
t at
an
an
gle
of
110,
an
d c
on
tin
ue t
o w
alk
str
aig
ht
again
.
Aft
er
a w
hil
e,
they b
oth
tu
rn r
igh
t
again
, bu
t th
is t
ime a
t an
an
gle
of
120.
Th
ey e
ach
walk
str
aig
ht
for
a w
hil
e i
n
this
new
dir
ect
ion
un
til
they e
nd
up
wh
ere
th
ey s
tart
ed
. E
ach
pers
on
walk
ed
in a
tri
an
gu
lar
path
at
their
loca
tion
.
Are
th
ese
tw
o t
rian
gle
s co
ngru
en
t?
Exp
lain
.
3
. C
ON
ST
RU
CT
ION
T
he r
ooft
op
of
An
gelo
’s h
ou
se c
reate
s an
equ
ilate
ral
tria
ngle
wit
h t
he a
ttic
flo
or.
An
gelo
wan
ts t
o d
ivid
e h
is a
ttic
in
to 2
equ
al
part
s. H
e t
hin
ks
he s
hou
ld d
ivid
e i
t by
pla
cin
g a
wall
fro
m t
he c
en
ter
of
the r
oof
to t
he f
loor
at
a 9
0 a
ngle
. If
An
gelo
does
this
, th
en
each
sect
ion
wil
l sh
are
a s
ide
an
d h
ave c
orr
esp
on
din
g 9
0 a
ngle
s. W
hat
els
e m
ust
be e
xp
lain
ed
to p
rove t
hat
the
two t
rian
gu
lar
sect
ion
s are
con
gru
en
t?
90˚
90˚
4
. LO
GIC
W
hen
Caro
lyn
fin
ish
ed
her
mu
sica
l tr
ian
gle
cla
ss,
her
teach
er
gave
each
stu
den
t in
th
e c
lass
a c
ert
ific
ate
in
the s
hap
e o
f a g
old
en
tri
an
gle
. E
ach
stu
den
t re
ceiv
ed
a d
iffe
ren
t sh
ap
ed
tria
ngle
. C
aro
lyn
lost
her
tria
ngle
on
her
way h
om
e.
Late
r sh
e s
aw
part
of
a
gold
en
tri
an
gle
un
der
a g
rate
.
Is e
nou
gh
of
the t
rian
gle
vis
ible
to a
llow
Caro
lyn
to d
ete
rmin
e t
hat
the t
rian
gle
is
ind
eed
hers
? E
xp
lain
.
5
. P
AR
K M
AIN
TE
NA
NC
E P
ark
off
icia
ls
need
a t
rian
gu
lar
tarp
to c
over
a f
ield
shap
ed
lik
e a
n e
qu
ilate
ral
tria
ngle
200
feet
on
a s
ide.
a.
Su
pp
ose
you
kn
ow
th
at
a t
rian
gu
lar
tarp
has
two 6
0 a
ngle
s an
d o
ne s
ide
of
len
gth
200 f
eet.
Wil
l th
is t
arp
cover
the f
ield
? E
xp
lain
.
b.
Su
pp
ose
you
kn
ow
th
at
a t
rian
gu
lar
tarp
has
thre
e 6
0 a
ngle
s. W
ill
this
tarp
nece
ssari
ly c
over
the f
ield
?
Exp
lain
.
4-5
Wo
rld
Pro
ble
m P
rati
ce
Pro
vin
g T
rian
gle
s C
on
gru
en
t—A
SA
, A
AS
yes,
by A
SA
Sam
ple
an
sw
er:
Becau
se t
he
ori
gin
al
tria
ng
le w
as e
qu
ilate
ral,
on
e m
ore
pair
of
an
gle
s i
s
co
ng
ruen
t. T
he t
rian
gle
s a
re
co
ng
ruen
t b
y A
AS
.
No
; th
ere
are
eq
uia
ng
ula
r
tria
ng
les o
f all s
izes.
yes,
by A
AS
Yes,
if s
he k
no
ws t
he m
easu
res
of
her
lost
tria
ng
le.
Sin
ce t
wo
an
gle
s a
nd
th
e i
nclu
ded
sid
e
betw
een
th
em
can
be s
een
, sh
e
can
dete
rmin
e i
f it
is c
on
gru
en
t
by t
he A
SA
Th
eo
rem
.
Yes,
becau
se t
wo
kn
ow
n
an
gle
s a
re 6
0,
the t
hir
d m
ust
be a
lso
. A
SA
or
AA
S c
an
be
used
to
sh
ow
th
at
it w
ill
co
ver.
Answers (Lesson 4-5)
Chapter 4 A16 Glencoe Geometry
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NA
ME
DA
TE
PE
RIO
D
Ch
ap
ter
4
36
G
len
co
e G
eo
me
try
En
rich
men
t4-5
Wh
at
Ab
ou
t A
AA
to
Pro
ve T
rian
gle
Co
ng
ruen
ce?
You
have l
earn
ed
abou
t th
e S
SS
an
d A
SA
post
ula
tes
to p
rove t
hat
two t
rian
gle
s are
con
gru
en
t. D
o y
ou
th
ink
th
at
tria
ngle
s ca
n b
e p
roven
con
gru
en
t by A
AA
?
Invest
igate
th
e A
AS
th
eore
m.
AA
S r
ep
rese
nts
th
e c
on
ject
ure
that
two t
rian
gle
s are
con
gru
en
t if
th
ey h
ave t
wo p
air
s of
corr
esp
on
din
g a
ngle
s co
ngru
en
t an
d c
orr
esp
on
din
g
non
-in
clu
ded
sid
es
con
gru
en
t. F
or
exam
ple
, in
th
e
tria
ngle
s at
the r
igh
t, ∠
A !
∠Y
, ∠
B !
∠X
, an
d −
−
AC
! −
−
YZ
.
1
. R
efe
r to
th
e d
raw
ing a
bove.
Wh
at
is t
he m
easu
re o
f th
e m
issi
ng a
ngle
in
th
e f
irst
tria
ngle
? W
hat
is t
he m
easu
re o
f th
e m
issi
ng a
ngle
in
th
e s
eco
nd
tri
an
gle
?
2
. W
ith
th
e i
nfo
rmati
on
fro
m E
xerc
ise 1
, w
hat
post
ula
te c
an
you
now
use
to p
rove t
hat
the
two t
rian
gle
s are
con
gru
en
t?
Now
in
vest
igate
th
e A
AA
con
ject
ure
. A
AA
rep
rese
nts
th
e c
on
ject
ure
th
at
two t
rian
gle
s are
con
gru
en
t if
th
ey h
ave t
hre
e p
air
s of
corr
esp
on
din
g a
ngle
s co
ngru
en
t. T
he t
rian
gle
s belo
w
dem
on
stra
te A
AA
.
25˚
25˚
70˚
70˚
85˚
85˚
3
. A
re t
he t
rian
gle
s above c
on
gru
en
t? E
xp
lain
.
4
. C
an
you
dra
w t
wo t
rian
gle
s th
at
have a
ll t
hre
e a
ngle
s co
ngru
en
t th
at
are
not
con
gru
en
t?
Can
AA
A b
e u
sed
to p
rove t
hat
two t
rian
gle
s are
con
gru
en
t?
5
. W
hat
chara
cteri
stic
s d
o t
hese
tw
o t
rian
gle
s d
raw
n b
y A
AA
seem
to p
oss
ess
?
89˚
89˚
66˚
66˚
A
BC
5 in
5 in
X
Y
Z
25;
25
AS
A
No
, th
e s
ides h
ave d
iffe
ren
t le
ng
ths.
yes;
no
Th
ey h
ave t
he s
am
e s
hap
e,
bu
t d
iffe
ren
t siz
es.
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-6
Ch
ap
ter
4
37
G
len
co
e G
eo
me
try
Exerc
ises
ALG
EB
RA
F
ind
th
e v
alu
e o
f e
ach
va
ria
ble
.
1
.
R
P
Q2
x°
40
°
2.
S
V
T3
x-
6
2x
+ 6
3.
W
YZ
3x°
4
.
2x°
(6x
+6)°
5.
5x°
6.
RS
T 3x°
x°
7
. P
RO
OF W
rite
a t
wo-c
olu
mn
pro
of.
G
ive
n:
∠1 !
∠2
P
ro
ve
: −
−
AB
! −
−−
CB
Pro
of:
Sta
tem
en
ts
Reaso
ns
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Iso
scele
s a
nd
Eq
uilate
ral
Tri
an
gle
s
Pro
pe
rtie
s o
f Is
osc
ele
s T
ria
ng
les
An
iso
sce
les t
ria
ng
le h
as
two c
on
gru
en
t si
des
call
ed
th
e l
eg
s.
Th
e a
ngle
form
ed
by t
he l
egs
is c
all
ed
th
e v
erte
x a
ng
le.
Th
e o
ther
two
an
gle
s are
call
ed
ba
se
an
gle
s.
You
can
pro
ve a
th
eore
m a
nd
its
con
vers
e a
bou
t is
osc
ele
s tr
ian
gle
s.
•
If
tw
o s
ides
of
a t
rian
gle
are
con
gru
en
t, t
hen
th
e a
ngle
s op
posi
te
those
sid
es
are
con
gru
en
t. (
Iso
sce
les T
ria
ng
le T
he
ore
m)
•
If
tw
o a
ngle
s of
a t
rian
gle
are
con
gru
en
t, t
hen
th
e s
ides
op
posi
te
those
an
gle
s are
con
gru
en
t. (
Co
nv
erse
of
Iso
sce
les T
ria
ng
le
Th
eo
re
m)
If −
−
AB
! −
−−
CB
, th
en ∠
A !
∠C
.
If ∠
A !
∠C
, th
en −
−
AB
! −
−−
CB
.
A
B
C
F
ind
x,
giv
en
−−
BC
# −
−
BA
.
B
AC( 4
x+
5) °
( 5x
- 1
0) °
BC
= B
A,
so
m∠
A =
m∠
C
Isos.
Triangle
Theore
m
5x -
10 =
4x +
5
Substitu
tion
x -
10 =
5
subtr
act
4x f
rom
each s
ide.
x =
15
Add 1
0 t
o e
ach s
ide.
Exam
ple
1Exam
ple
2
Fin
d x
.
RT
S 3x
- 1
3
2x
m∠
S =
m∠
T,
so
SR
= T
R
Convers
e o
f Is
os. △
Thm
.
3x -
13 =
2x
Substitu
tion
3x =
2x +
13
A
dd 1
3 t
o e
ach s
ide.
x =
13
Subtr
act
2x f
rom
each s
ide.
B
AC
D
E
13
2
4-6 1
. ∠
1 #
∠2
1. G
iven
2.
∠2 #
∠3
2. V
ert
ical
an
gle
s a
re c
on
gru
en
t.3.
∠1 #
∠3
3. T
ran
sit
ive P
rop
ert
y o
f #
4.
−−
AB
# −
−
CB
4. C
on
vers
e o
f Is
os.
Tri
an
gle
Th
m.
12
936
35
12
15
Answers (Lesson 4-5 and Lesson 4-6)
Chapter 4 A17 Glencoe Geometry
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Ch
ap
ter
4
38
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Iso
scele
s a
nd
Eq
uilate
ral
Tri
an
gle
s
Pro
pe
rtie
s o
f E
qu
ila
tera
l T
ria
ng
les
An
eq
uil
ate
ra
l tr
ian
gle
has
thre
e c
on
gru
en
t si
des.
Th
e I
sosc
ele
s T
rian
gle
Th
eore
m l
ead
s to
tw
o c
oro
llari
es
abou
t equ
ilate
ral
tria
ngle
s.
1.
A t
ria
ng
le is e
qu
ilate
ral if a
nd
on
ly if
it is e
qu
ian
gu
lar.
2.
Ea
ch
an
gle
of
an
eq
uila
tera
l tr
ian
gle
me
asu
res 6
0°.
P
ro
ve
th
at
if a
lin
e i
s p
ara
lle
l to
on
e s
ide
of
an
eq
uil
ate
ra
l tr
ian
gle
, th
en
it
form
s a
no
the
r e
qu
ila
tera
l tr
ian
gle
.
Pro
of:
Sta
tem
en
ts
Re
aso
ns
1.
△A
BC
is
equ
ilate
ral;
−−−
PQ
‖ −−
− B
C .
1.
Giv
en
2.
m∠
A =
m∠
B =
m∠
C =
60
2.
Each
∠ o
f an
equ
ilate
ral
△ m
easu
res
60
°.
3.
∠1 &
∠B
, ∠
2 &
∠C
3.
If ‖
lin
es,
th
en
corr
es.
' a
re &
.
4.
m∠
1 =
60,
m∠
2 =
60
4.
Su
bst
itu
tion
5.
△A
PQ
is
equ
ilate
ral.
5
. If
a △
is
equ
ian
gu
lar,
th
en
it
is e
qu
ilate
ral.
Exerc
ises
ALG
EB
RA
F
ind
th
e v
alu
e o
f e
ach
va
ria
ble
.
1
. D
FE
6x°
2.
G
JH
6x
- 5
5x
3.
3x°
4
.
4x
40
60
°
5.
X
ZY
4x
- 4
3x
+ 8
60
°
6.
4x°
60°
7
. P
RO
OF W
rite
a t
wo-c
olu
mn
pro
of.
G
ive
n:
△A
BC
is
equ
ilate
ral;
∠1 &
∠2.
P
ro
ve
: ∠
AD
B &
∠C
DB
P
roo
f:
S
tate
men
ts
Reaso
ns
A D C
B1 2
A
B
PQ
C
12
4-6 Exam
ple
1
. △
AB
C i
s e
qu
ilate
ral.
1.
Giv
en
2
. −
−
AB
$ −
−
CB
; ∠
A $
∠C
2.
An
eq
uilate
ral
△ h
as $
sid
es a
nd
$ a
ng
les.
3
. ∠
1 $
∠2
3.
Giv
en
4
. △
AB
D $
△C
BD
4.
AS
A P
ostu
late
5
. ∠
AD
B $
∠C
DB
5.
CP
CT
C
10
12
15
10
520
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-6
Ch
ap
ter
4
39
G
len
co
e G
eo
me
try
Sk
ills
Pra
ctic
e
Iso
scele
s a
nd
Eq
uilate
ral
Tri
an
gle
s
D
C
B
AE
Re
fer t
o t
he
fig
ure
at
the
rig
ht.
1
. If
−−
AC
& −−
− A
D ,
nam
e t
wo c
on
gru
en
t an
gle
s.
2
. If
−−−
BE
& −−
− B
C ,
nam
e t
wo c
on
gru
en
t an
gle
s.
3
. If
∠E
BA
& ∠
EA
B,
nam
e t
wo c
on
gru
en
t se
gm
en
ts.
4
. If
∠C
ED
& ∠
CD
E,
nam
e t
wo c
on
gru
en
t se
gm
en
ts.
Fin
d e
ach
me
asu
re
.
5
. m
∠A
BC
6. m
∠E
DF
60
°
40°
ALG
EB
RA
F
ind
th
e v
alu
e o
f e
ach
va
ria
ble
.
7
.
2x
+4
3x
-10
8.
(2x
+3
)°
9
. P
RO
OF W
rite
a t
wo-c
olu
mn
pro
of.
G
ive
n:
−−
−
CD
& −
−−
CG
−−
−
DE
& −−
− G
F
P
ro
ve
: −−
− C
E &
−−
CF
D E F G
C
4-6 ∠
AC
D $
∠C
DA
∠B
EC
$ ∠
BC
E
−−
EB
$ −
−
EA
−−
CE
$
−−
CD
P
roo
f:
S
tate
men
ts
Reaso
ns
1
. −
−
CD
$ −
−
CG
1.
Giv
en
3
. −
−
DE
$ −
−
GF
3.
Giv
en
4
. △
CD
E $
△C
GF
4.
SA
S
5
. −
−
CE
$ −
−
CF
5.
CP
CT
C
2
. ∠
D $
∠G
2
. If
2 s
ides o
f a △
are
$,
then
th
e &
op
po
sit
e
tho
se s
ides a
re $
.
60
70
14
21
Answers (Lesson 4-6)
Chapter 4 A18 Glencoe Geometry
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Ch
ap
ter
4
40
G
len
co
e G
eo
me
try
Practi
ce
Iso
scele
s a
nd
Eq
uilate
ral
Tri
an
gle
s
Lin
co
ln H
aw
ks
E DBC
A12
3 4
UR
TV
S
Re
fer t
o t
he
fig
ure
at
the
rig
ht.
1
. If
−−−
RV
" −
−
RT
, n
am
e t
wo c
on
gru
en
t an
gle
s.
∠R
TV
! ∠
RV
T
2
. If
−−
RS
" −
−
SV
, n
am
e t
wo c
on
gru
en
t an
gle
s.
∠S
VR
! ∠
SR
V
3
. If
∠S
RT
" ∠
ST
R,
nam
e t
wo c
on
gru
en
t se
gm
en
ts. −
−
ST !
−−
SR
4
. If
∠S
TV
" ∠
SV
T,
nam
e t
wo c
on
gru
en
t se
gm
en
ts.
−−
ST
! −
−
SV
Fin
d e
ach
me
asu
re
.
5
. m
∠K
ML
6. m
∠H
MG
7. m
∠G
HM
8. If
m∠
HJ
M =
145,
fin
d m
∠M
HJ
.
9. If
m∠
G =
67,
fin
d m
∠G
HM
.
10
. P
RO
OF W
rite
a t
wo-c
olu
mn
pro
of.
G
ive
n:
−−
−
DE
‖ −−
− B
C
∠1 "
∠2
P
ro
ve
: −
−
AB
" −
−
AC
11
. S
PO
RT
S A
pen
nan
t fo
r th
e s
port
s te
am
s at
Lin
coln
Hig
h
Sch
ool
is i
n t
he s
hap
e o
f an
iso
scele
s tr
ian
gle
. If
th
e m
easu
re
of
the v
ert
ex a
ngle
is
18,
fin
d t
he m
easu
re o
f each
base
an
gle
.
4-6
50°
81,
81
P
roo
f:
S
tate
men
ts
R
easo
ns
1. G
iven
2. C
orr
. $
are
!.
3. G
iven
4. C
on
gru
en
ce o
f $
is t
ran
sit
ive.
5. I
f 2 $
of
a △
are
!,
then
th
e s
ide
o
pp
osit
e t
ho
se s
ides a
re !
1. −
−
DE
‖
−−
BC
2. ∠
1 !
∠4
∠
2 !
∠3
3. ∠
1 !
∠2
4. ∠
3 !
∠4
5. −
−
AB
! −
−
AC
60
70
40
17.5
46
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-6
Ch
ap
ter
4
41
G
len
co
e G
eo
me
try
Wo
rd
Pro
ble
m P
racti
ce
Iso
scele
s a
nd
Eq
uilate
ral
Tri
an
gle
s
1
1
. T
RIA
NG
LE
S A
t an
art
su
pp
ly s
tore
, tw
o
dif
fere
nt
tria
ngu
lar
rule
rs a
re a
vail
able
.
On
e h
as
an
gle
s 45°,
45°,
an
d 9
0°.
Th
e
oth
er
has
an
gle
s 30°,
60°,
an
d 9
0°.
Wh
ich
tri
an
gle
is
isosc
ele
s?
2. R
ULE
RS
A
fold
able
ru
ler
has
two h
inges
that
div
ide t
he r
ule
r in
to t
hir
ds.
If
the
en
ds
are
fold
ed
up
un
til
they t
ou
ch,
wh
at
kin
d o
f tr
ian
gle
resu
lts?
3
. H
EX
AG
ON
S Ju
an
ita p
lace
d o
ne e
nd
of
each
of
6 b
lack
sti
cks
at
a c
om
mon
poin
t
an
d t
hen
sp
ace
d t
he o
ther
en
ds
even
ly
aro
un
d t
hat
poin
t. S
he c
on
nect
ed
th
e
free e
nd
s of
the s
tick
s w
ith
lin
es.
Th
e
resu
lt w
as
a r
egu
lar
hexagon
. T
his
con
stru
ctio
n s
how
s th
at
a r
egu
lar
hexagon
can
be m
ad
e f
rom
six
con
gru
en
t
tria
ngle
s. C
lass
ify t
hese
tri
an
gle
s.
Exp
lain
.
4
. P
AT
HS
A
marb
le p
ath
is
con
stru
cted
ou
t of
severa
l co
ngru
en
t is
osc
ele
s
tria
ngle
s. T
he v
ert
ex a
ngle
s are
all
20.
Wh
at
is t
he m
easu
re o
f an
gle
1 i
n t
he
figu
re?
5. B
RID
GE
S E
very
day,
cars
dri
ve t
hro
ugh
isosc
ele
s tr
ian
gle
s w
hen
th
ey g
o o
ver
the
Leon
ard
Zak
im B
rid
ge i
n B
ost
on
. T
he
ten
-lan
e r
oad
way f
orm
s th
e b
ase
s of
the
tria
ngle
s.
a.
Th
e a
ngle
labele
d A
in
th
e p
ictu
re
has
a m
easu
re o
f 67.
Wh
at
is t
he
measu
re o
f ∠
B?
b.
Wh
at
is t
he m
easu
re o
f ∠
C?
c.
Nam
e t
he t
wo c
on
gru
en
t si
des.
the 4
5°-
45
°- 9
0°
tria
ng
le
A
B
C
−−
AB
! −
−
BC
4-6 e
qu
ilate
ral
tria
ng
le
Sam
ple
an
sw
er:
Sin
ce t
wo
sid
es
are
co
ng
ruen
t, t
hey a
re i
so
scele
s.
Becau
se t
he s
ticks a
re s
paced
even
ly,
the v
ert
ex a
ng
les (
at
the
cen
ter)
are
all 3
60
−
6
or
60.
Th
us t
he
base a
ng
les m
ust
be
(18
0 -
60
) −
6
or
60.
Sin
ce a
ll a
ng
les o
f an
y o
ne
tria
ng
le i
n t
he f
igu
re a
re 6
0,
all
sid
es a
re e
qu
al
an
d t
he t
rian
gle
s
are
eq
uilate
ral.
80 4
6
67
Answers (Lesson 4-6)
Chapter 4 A19 Glencoe Geometry
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Ch
ap
ter
4
42
G
len
co
e G
eo
me
try
En
rich
men
t
Tri
an
gle
Ch
allen
ges
Som
e p
roble
ms
incl
ud
e d
iagra
ms.
If
you
are
not
sure
how
to s
olv
e t
he p
roble
m,
begin
by
usi
ng t
he g
iven
in
form
ati
on
. F
ind
th
e m
easu
res
of
as
man
y a
ngle
s as
you
can
, w
riti
ng e
ach
m
easu
re o
n t
he d
iagra
m.
Th
is m
ay g
ive y
ou
m
ore
clu
es
to t
he s
olu
tion
.
1
. G
ive
n:
BE
= B
F,
∠B
FG
! ∠
BE
F !
∠
BE
D,
m∠
BF
E =
82 a
nd
A
BF
G a
nd
BC
DE
each
have
op
posi
te s
ides
para
llel
an
d
con
gru
en
t.
Fin
d m
∠A
BC
. A
GD
FE
CB
3
. G
ive
n:
m∠
UZ
Y =
90,
m∠
ZW
X =
45,
△Y
ZU
! △
VW
X,
UV
XY
is
a
squ
are
(all
sid
es
con
gru
en
t, a
ll
an
gle
s ri
gh
t an
gle
s).
Fin
d m
∠W
ZY
.
2
. G
ive
n:
AC
= A
D,
an
d −
−
AB
⊥
−−
−
BD
, m
∠D
AC
= 4
4 a
nd
−−
−
CE
bis
ect
s ∠
AC
D.
Fin
d m
∠D
EC
. A
DC
B
E
4
. G
ive
n:
m∠
N =
120,
−−
−
JN
! −
−−
MN
, △
JN
M !
△K
LM
.
Fin
d m
∠J
KM
.
J
K
L
MN
UV
W
XY
Z
4-6
15
78
148
45
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-7
Ch
ap
ter
4
43
G
len
co
e G
eo
me
try
Ide
nti
fy C
on
gru
en
ce T
ran
sfo
rma
tio
ns
A c
on
gru
en
ce
tra
nsfo
rm
ati
on
is
a
tran
sform
ati
on
wh
ere
th
e o
rigin
al
figu
re,
or
pre
image,
an
d t
he t
ran
sform
ed
fig
ure
, or
image,
figu
re a
re s
till
con
gru
en
t. T
he t
hre
e t
yp
es
of
con
gru
en
ce t
ran
sform
ati
on
s are
re
fle
cti
on
(or
flip
), t
ra
nsla
tio
n (
or
slid
e),
an
d r
ota
tio
n (
or
turn
).
Id
en
tify
th
e t
yp
e o
f co
ng
ru
en
ce
tra
nsfo
rm
ati
on
sh
ow
n a
s a
reflection
, translation
, o
r rotation
.
x
y
Ea
ch
ve
rte
x a
nd
its
im
ag
e a
re t
he
sa
me
dis
tan
ce
fro
m t
he
y-a
xis
.
Th
is is a
re
fle
ctio
n.
x
y
Ea
ch
ve
rte
x a
nd
its
im
ag
e a
re in
the
sa
me
po
sitio
n,
just
two
un
its
do
wn
. T
his
is a
tra
nsla
tio
n.
x
y
Ea
ch
ve
rte
x a
nd
its
im
ag
e a
re t
he
sa
me
dis
tan
ce
fro
m t
he
orig
in.
Th
e a
ng
les f
orm
ed
by e
ach
pa
ir o
f
co
rre
sp
on
din
g p
oin
ts a
nd
th
e
orig
in a
re c
on
gru
en
t. T
his
is a
rota
tio
n.
Exerc
ises
Ide
nti
fy t
he
ty
pe
of
co
ng
ru
en
ce
tra
nsfo
rm
ati
on
sh
ow
n a
s a
reflection
, translation
,
or rotation
.
1
.
x
y
2.
x
y
3.
x
y
4
.
x
y
5.
x
y
6.
x
y
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Co
ng
ruen
ce T
ran
sfo
rmati
on
s
Exam
ple
4-7
refl
ecti
on
tran
sla
tio
nro
tati
on
ro
tati
on
refl
ecti
on
or t
ran
sla
tio
nrefl
ecti
on
Answers (Lesson 4-6 and Lesson 4-7)
Chapter 4 A20 Glencoe Geometry
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Ch
ap
ter
4
44
G
len
co
e G
eo
me
try
Ve
rify
Co
ng
rue
nce
Y
ou
can
veri
fy t
hat
refl
ect
ion
s, t
ran
slati
on
s, a
nd
rota
tion
s of
tria
ngle
s p
rod
uce
con
gru
en
t tr
ian
gle
s u
sin
g S
SS
.
V
erif
y c
on
gru
en
ce
aft
er a
tra
nsfo
rm
ati
on
.
△W
XY
wit
h v
ert
ices
W(3
, -
7),
X(6
, -
7),
Y(6
, -
2)
is a
tra
nsf
orm
ati
on
of
△R
ST
wit
h v
ert
ices
R(2
, 0),
S(5
, 0),
an
d T
(5,
5).
Gra
ph
th
e
ori
gin
al
figu
re a
nd
its
im
age.
Iden
tify
th
e t
ran
sform
ati
on
an
d
veri
fy t
hat
it i
s a c
on
gru
en
ce t
ran
sform
ati
on
.
Gra
ph
each
fig
ure
. U
se t
he D
ista
nce
Form
ula
to s
how
th
e s
ides
are
con
gru
en
t an
d t
he t
rian
gle
s are
con
gru
en
t by S
SS
.
RS
= 3
, S
T =
5,
TR
= √
##
##
##
#
( 5
- 2
) 2 +
(5 -
0) 2
= √
##
34
WX
= 3
, X
Y =
5,
YW
= √
##
##
##
##
#
( 6
- 3
) 2 +
(-
2-
(-
7) )
2 =
√ #
#
34
−−
RS
% −−
− W
X ,
−−
ST
, %
−−
XY
, −
−
TR
% −−
− Y
W
By S
SS
, △
RS
T %
△W
XY
.
Exerc
ises
CO
OR
DIN
AT
E G
EO
ME
TR
Y G
ra
ph
ea
ch
pa
ir o
f tr
ian
gle
s w
ith
th
e g
ive
n v
erti
ce
s.
Th
en
id
en
tify
th
e t
ra
nsfo
rm
ati
on
, a
nd
ve
rif
y t
ha
t it
is a
co
ng
ru
en
ce
tra
nsfo
rm
ati
on
.
1
. A
(−3,
1),
B(−
1,
1),
C(−
1,
4)
2. Q
(−3,
0),
R(−
2,
2),
S(−
1,
0)
D
(3,
1),
E(1
, 1),
F(1
, 4)
T
(2,
−4),
U(3
, −
2),
V(4
, −
4)
x
y
x
y
4-7
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Co
ng
ruen
ce T
ran
sfo
rmati
on
s
Exam
ple
x
y
refl
ecti
on
; A
B =
2,
BC
= 3
,
CA
= √
""
13 ,
DE
= 2
, E
F =
3,
FD
= √
""
13 t
hu
s −
−
AB
$ −
−
DE
,
−−
BC
$ −
−
EF ,
−−
AC
$ −
−
DF B
y S
SS
,
△A
BC
$ △
DE
F
tran
sla
tio
n;
QR
= √
"
5 ,
RS
= √
"
5 ,
SQ
= 2
, TU
= √
"
5 ,
UV
= √
"
5 ,
VT
= 2
th
us,
−−
QR
$ −
−
TU
, −
−
RS
$ −
−
UV
,
−−
SQ
$ −
−
VT .
By S
SS
, △
QR
S $
△TU
V.
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-7
Ch
ap
ter
4
45
G
len
co
e G
eo
me
try
4-7
Ide
nti
fy t
he
ty
pe
of
co
ng
ru
en
ce
tra
nsfo
rm
ati
on
sh
ow
n a
s a
refl
ecti
on
, tr
an
sla
tio
n,
or r
ota
tio
n.
1
.
x
y
2.
x
y
r
ota
tio
n
re
flecti
on
CO
OR
DIN
AT
E G
EO
ME
TR
Y Id
en
tify
ea
ch
tra
nsfo
rm
ati
on
an
d v
erif
y t
ha
t it
is a
co
ng
ru
en
ce
tra
nsfo
rm
ati
on
.
3
.
x
y
4.
x
y
T
ran
sla
tio
n;
AB
= √
"
5 ,
BC
= √
""
13 ,
CA
= 4
, D
E =
√ "
5 ,
EF =
√ "
"
13 ,
FD
= 4
. △
AB
C $
△D
EF b
y S
SS
.
R
efl
ecti
on
; Q
R =
5,
RS
= 4
,
SQ
= 3
, TU
= 5
, U
V =
4,
VT =
3.
△Q
RS
$ △
TU
V b
y S
SS
.
CO
OR
DIN
AT
E G
EO
ME
TR
Y G
ra
ph
ea
ch
pa
ir o
f tr
ian
gle
s w
ith
th
e g
ive
n v
erti
ce
s.
Th
en
, id
en
tify
th
e t
ra
nsfo
rm
ati
on
, a
nd
ve
rif
y t
ha
t it
is a
co
ng
ru
en
ce
tra
nsfo
rm
ati
on
.
5
. A
(1,
3),
B(1
, 1),
C(4
, 1);
6. J
(−3,
0),
K(−
2,
4),
L(−
1,
0);
D
(3,
-1),
E(1
, -
1),
F(1
, -
4)
Q
(2,
−4),
R(3
, 0),
S(4
, −
4)
x
y
x
y
4-7
Sk
ills
Pra
ctic
e
Co
ng
ruen
ce T
ran
sfo
rmati
on
s
Tra
nsla
tio
n;
JK
= √
""
17 ,
KL =
√ "
"
17 ,
LJ =
2,
QR
= √
""
17 ,
RS
= √
""
17 ,
SQ
= 2
. △
JK
L $
△Q
RS
by S
SS
.
rota
tio
n;
AB
= 2
, B
C =
3,
CA
= √
##
13 ,
DE
= 2
, E
F =
3,
FD
= √
##
13
△A
BC
$ △
DE
F b
y S
SS
.
Answers (Lesson 4-7)
Chapter 4 A21 Glencoe Geometry
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Ch
ap
ter
4
46
G
len
co
e G
eo
me
try
Ide
nti
fy t
he
ty
pe
of
co
ng
ru
en
ce
tra
nsfo
rm
ati
on
sh
ow
n a
s a
reflection
, translation
,
or rotation
.
1
.
x
y
2.
x
y
3
. Id
en
tify
th
e t
yp
e o
f co
ngru
en
ce t
ran
sform
ati
on
sh
ow
n a
s a reflection
, translation
, or rotation
, an
d v
eri
fy t
hat
it
is a
con
gru
en
ce t
ran
sform
ati
on
.
R
ota
tio
n;
AB
= √
""
13 ,
BC
= 3
√ "
2 ,
CA
= 5
, AD
= √
""
13
D
E =
3 √
"
2 ,
EA
= 5
, △
AB
C $
△A
DE
by S
SS
.
4
. ∆ABC
has
vert
ices A
(−4,
2),
B(−
2,
0),
C(−
4,
-2).
∆DEF
has
vert
ices D
(4,
2),
E(2
, 0),
F(4
, −
2).
Gra
ph
th
e o
rigin
al
figu
re
an
d i
ts i
mage.
Th
en
id
en
tify
th
e t
ran
sform
ati
on
an
d v
eri
fy
that
it i
s a c
on
gru
en
ce t
ran
sform
ati
on
.
R
efl
ecti
on
; A
B =
2 √
"
2 ,
BC
= 2
√ "
2 ,
CA
= 4
, D
E =
2 √
"
2 ,
EF =
2 √
"
2 ,
FD
= 4
. △
AB
C $
△D
EF b
y S
SS
.
5
. S
TE
NC
ILS
C
arl
y i
s p
lan
nin
g o
n s
ten
cili
ng a
patt
ern
of
flow
ers
alo
ng t
he c
eil
ing i
n h
er
bed
room
. S
he w
an
ts a
ll o
f th
e f
low
ers
to l
ook
exact
ly t
he s
am
e.
Wh
at
typ
e o
f co
ngru
en
ce
tran
sform
ati
on
sh
ou
ld s
he u
se?
Wh
y?
T
ran
sla
tio
n;
A t
ran
sla
tio
n c
on
tain
s c
on
gru
en
t sh
ap
es w
ith
th
e s
am
e
ori
en
tati
on
.
4-7
Practi
ce
Co
ng
ruen
ce T
ran
sfo
rmati
on
s
x
y
tran
sla
tio
nro
tati
on
x
y
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-7
Ch
ap
ter
4
47
G
len
co
e G
eo
me
try
4-7
Wo
rd
Pro
ble
m P
racti
ce
Co
ng
ruen
ce T
ran
sfo
rmati
on
s
1
. Q
UIL
TIN
G Y
ou
an
d y
ou
r tw
o f
rien
ds
vis
it a
cra
ft f
air
an
d n
oti
ce a
qu
ilt,
part
of
wh
ose
patt
ern
is
show
n b
elo
w.
You
r fi
rst
frie
nd
says
the p
att
ern
is
rep
eate
d
by t
ran
slati
on
. Y
ou
r se
con
d f
rien
d s
ays
the p
att
ern
is
rep
eate
d b
y r
ota
tion
. W
ho
is c
orr
ect
?
B
oth
fri
en
ds a
re c
orr
ect.
Th
e
tria
ng
les a
re b
ein
g r
ota
ted
an
d
the h
exag
on
s a
re b
ein
g
tran
sla
ted
.
2
. R
EC
YC
LIN
G T
he i
nte
rnati
on
al
sym
bol
for
recy
clin
g i
s sh
ow
n b
elo
w.
Wh
at
typ
e
of
con
gru
en
ce t
ran
sform
ati
on
does
this
il
lust
rate
?
R
ota
tio
n
3
. A
NA
TO
MY
C
arl
noti
ces
that
wh
en
he
hold
s h
is h
an
ds
palm
up
in
fro
nt
of
him
, th
ey l
ook
th
e s
am
e.
Wh
at
typ
e o
f co
ngru
en
ce t
ran
sform
ati
on
does
this
il
lust
rate
?
R
efl
ecti
on
4
. S
TA
MP
S A
sh
eet
of
post
age s
tam
ps
con
tain
s 20 s
tam
ps
in 4
row
s of
5
iden
tica
l st
am
ps
each
. W
hat
typ
e o
f co
ngru
en
ce t
ran
sform
ati
on
does
this
il
lust
rate
?
T
ran
sla
tio
n
5
. M
OS
AIC
S José
cu
t re
ctan
gle
s ou
t of
tiss
ue p
ap
er
to c
reate
a p
att
ern
. H
e c
ut
ou
t fo
ur
con
gru
en
t blu
e r
ect
an
gle
s an
d
then
cu
t ou
t fo
ur
con
gru
en
t re
d
rect
an
gle
s th
at
were
sli
gh
tly s
mall
er
to
create
th
e p
att
ern
sh
ow
n b
elo
w.
a.
Wh
ich
con
gru
en
ce t
ran
sform
ati
on
d
oes
this
patt
ern
ill
ust
rate
?
Ro
tati
on
b.
If t
he w
hole
squ
are
were
rota
ted
90,
wou
ld t
he t
ran
sform
ati
on
sti
ll b
e t
he
sam
e?
Yes
Answers (Lesson 4-7)
Chapter 4 A22 Glencoe Geometry
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Ch
ap
ter
4
48
G
len
co
e G
eo
me
try
4-7
En
rich
men
t
Tra
nsfo
rmati
on
s i
n T
he C
oo
rdin
ate
Pla
ne
Th
e f
oll
ow
ing s
tate
men
t te
lls
on
e w
ay t
o r
ela
te p
oin
ts t
o
corr
esp
on
din
g t
ran
slate
d p
oin
ts i
n t
he c
oord
inate
pla
ne.
(x,
y)
→ (
x +
6,
y -
3)
Th
is c
an
be r
ead
, “T
he p
oin
t w
ith
coord
inate
s (x
, y)
is
map
ped
to t
he p
oin
t w
ith
coord
inate
s (x
+ 6
, y -
3).
” W
ith
th
is t
ran
sform
ati
on
, fo
r exam
ple
, (3
, 5)
is m
ap
ped
or
moved
to (
3 +
6,
5 -
3)
or
(9,
2).
Th
e f
igu
re s
how
s h
ow
th
e
tria
ngle
AB
C i
s m
ap
ped
to t
rian
gle
XY
Z.
1
. D
oes
the t
ran
sform
ati
on
above a
pp
ear
to b
e a
con
gru
en
ce t
ran
sform
ati
on
? E
xp
lain
you
r an
swer.
Dra
w t
he
tra
nsfo
rm
ati
on
im
ag
e f
or e
ach
fig
ure
. T
he
n t
ell
wh
eth
er t
he
tra
nsfo
rm
ati
on
is o
r i
s n
ot
a c
on
gru
en
ce
tra
nsfo
rm
ati
on
.
2
. (x
, y)
→ (
x -
4,
y)
3. (x
, y)
→ (
x +
5,
y +
4)
4
. (x
, y)
→ (
-x ,
-y)
5. (x
, y)
→ (
- 1
−
2 x
, y)
x
y
Ox
y
O
x
y
Ox
y
O
x
y
B
A
C
X
ZY
O
(x,
y)#
(x
+ 6
, y
- 3
)
yes
yes
yes
no
Yes;
the t
ran
sfo
rmati
on
slid
es t
he f
igu
re t
o t
he l
ow
er
rig
ht
wit
ho
ut
ch
an
gin
g i
ts s
ize o
r sh
ap
e.
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-8
Ch
ap
ter
4
49
G
len
co
e G
eo
me
try
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
Tri
an
gle
s a
nd
Co
ord
inate
Pro
of
Po
siti
on
an
d L
ab
el
Tri
an
gle
s A
coord
inate
pro
of
use
s p
oin
ts,
dis
tan
ces,
an
d s
lop
es
to
pro
ve g
eom
etr
ic p
rop
ert
ies.
Th
e f
irst
ste
p i
n w
riti
ng a
coord
inate
pro
of
is t
o p
lace
a f
igu
re
on
th
e c
oord
inate
pla
ne a
nd
label
the v
ert
ices.
Use
th
e f
oll
ow
ing g
uid
eli
nes.
1.
Use
th
e o
rig
in a
s a
ve
rte
x o
r ce
nte
r o
f th
e f
igu
re.
2.
Pla
ce
at
lea
st
on
e s
ide
of
the
po
lyg
on
on
an
axis
.
3.
Ke
ep
th
e f
igu
re in
th
e f
irst
qu
ad
ran
t if p
ossib
le.
4.
Use
co
ord
ina
tes t
ha
t m
ake
co
mp
uta
tio
ns a
s s
imp
le a
s p
ossib
le.
Po
sit
ion
an
iso
sce
les t
ria
ng
le o
n t
he
co
ord
ina
te p
lan
e s
o t
ha
t it
s s
ide
s a
re
a u
nit
s l
on
g a
nd
on
e s
ide
is o
n t
he
po
sit
ive
x-a
xis
.
Sta
rt w
ith
R(0
, 0).
If
RT
is
a,
then
an
oth
er
vert
ex i
s T
(a,
0).
For
vert
ex S
, th
e x
-coord
inate
is
a −
2 .
Use
b f
or
the y
-coord
inate
,
so t
he v
ert
ex i
s S
( a
−
2 ,
b) .
Exerc
ises
Na
me
th
e m
issin
g c
oo
rd
ina
tes o
f e
ach
tria
ng
le.
1
.
x
y
B( 2
p, 0
)
C( ?
,q)
A( 0
, 0)
2
.
x
y
S( 2
a, 0
)
T( ?
, ?)
R( 0
, 0)
3
.
x
y
G( 2
g, 0
)
F( ?
,b)
E( ?
, ?)
Po
sit
ion
an
d l
ab
el
ea
ch
tria
ng
le o
n t
he
co
ord
ina
te p
lan
e.
4
. is
osc
ele
s tr
ian
gle
5
. is
osc
ele
s ri
gh
t △
DE
F
6. equ
ilate
ral
tria
ngle
△
RS
T w
ith
base
−−
RS
w
ith
legs
e u
nit
s lo
ng
△
EQ
I w
ith
vert
ex
4a
un
its
lon
g
Q
(0,
√ &
3
b)
an
d s
ides
2b u
nit
s lo
ng
x
y
I(b,
0)
E( –
b, 0
)
Q( 0
,√
3b)
x
y
E( e
, 0)
F( e
,e)
D( 0
, 0)
x
yT
( 2a,
b)
R( 0
, 0)
S( 4
a, 0
)
x
y
T( a
, 0)
R( 0
, 0
)S(a – 2
,b )
Sam
ple
an
sw
ers
are
giv
en
.
C
(p,
q)
T(2
a,
2a)
E(-
2g
, 0);
F(0
, b
)
Exam
ple
4-8
Answers (Lesson 4-7 and Lesson 4-8)
Chapter 4 A23 Glencoe Geometry
Co
pyrig
ht ©
Gle
nc
oe
/Mc
Gra
w-H
ill, a d
ivis
ion
of T
he
Mc
Gra
w-H
ill Co
mp
an
ies, In
c.
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Ch
ap
ter
4
50
G
len
co
e G
eo
me
try
Writ
e C
oo
rd
ina
te
Pro
ofs
Coord
inate
pro
ofs
can
be u
sed
to p
rove t
heore
ms
an
d t
o
veri
fy p
rop
ert
ies.
Man
y c
oord
inate
pro
ofs
use
th
e D
ista
nce
Form
ula
, S
lop
e F
orm
ula
, or
Mid
poin
t T
heore
m.
P
ro
ve
th
at
a s
eg
me
nt
fro
m t
he
ve
rte
x
an
gle
of
an
iso
sce
les t
ria
ng
le t
o t
he
mid
po
int
of
the
ba
se
is p
erp
en
dic
ula
r t
o t
he
ba
se
.
Fir
st,
posi
tion
an
d l
abel
an
iso
scele
s tr
ian
gle
on
th
e c
oord
inate
p
lan
e.
On
e w
ay i
s to
use
T(a
, 0),
R(-
a,
0),
an
d S
(0,
c).
Th
en
U(0
, 0)
is t
he m
idp
oin
t of
−−
RT
.
Giv
en
: Is
osc
ele
s △
RS
T;
U i
s th
e m
idp
oin
t of
base
−−
RT
.
Pro
ve
: −−
− S
U ⊥
−−
RT
Pro
of:
U i
s th
e m
idp
oin
t of
−−
RT
so t
he c
oord
inate
s of
U a
re ( -
a +
a
−
2
, 0
+ 0
−
2
) = (
0,
0).
Th
us
−−−
SU
lie
s on
the y
-axis
, an
d △
RS
T w
as
pla
ced
so −
−
RT
lie
s on
th
e x
-axis
. T
he a
xes
are
perp
en
dic
ula
r, s
o
−−−
SU
⊥ −
−
RT
.
Exerc
ise
PR
OO
F
Writ
e a
co
ord
ina
te p
ro
of
for t
he
sta
tem
en
t.
Pro
ve t
hat
the s
egm
en
ts j
oin
ing t
he m
idp
oin
ts o
f th
e s
ides
of
a r
igh
t tr
ian
gle
form
a r
igh
t tr
ian
gle
.
x
y
C( 2
a,
0)
B( 0
, 2
b)
P
Q
R
A( 0
, 0)
x
y
T( a
, 0)
U( 0
, 0)
R( –
a,
0)
S( 0
,c)
Sam
ple
an
sw
er:
Po
sit
ion
an
d l
ab
el
rig
ht
△A
BC
wit
h t
he c
oo
rdin
ate
s
A(0
, 0),
B(0
, 2b
), a
nd
C(2
a,
0).
Th
e m
idp
oin
t P
of
BC
is
( 0 +
2a
−
2
, 2
b +
0
−
2
) = (
a,
b).
Th
e m
idp
oin
t Q
of
AC
is ( 0
+ 2
a
−
2
, 2
+ 0
−
2
) = (
a,
0).
Th
e m
idp
oin
t R
of
AB
is
( 0 +
0
−
2
, 0
+ 2
b
−
2
) = (
0,
b).
Th
e s
lop
e o
f −
−
RP
is b
- b
−
a -
0 =
0
−
a =
0,
so
th
e s
eg
men
t is
ho
rizo
nta
l.
Th
e s
lop
e o
f −
−
PQ
is
b -
0
−
a -
a =
b
−
0 ,
wh
ich
is u
nd
efi
ned
, so
th
e s
eg
men
t is
vert
ical.
∠R
PQ
is a
rig
ht
an
gle
becau
se a
ny h
ori
zo
nta
l lin
e i
s p
erp
en
dic
ula
r to
an
y
vert
ical
lin
e.
△P
RQ
has a
rig
ht
an
gle
, so
△P
RQ
is a
rig
ht
tria
ng
le.
4-8
Stu
dy
Gu
ide a
nd
In
terv
en
tio
n
(con
tin
ued
)
Tri
an
gle
s a
nd
Co
ord
inate
Pro
of
Exam
ple
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-8
Ch
ap
ter
4
51
G
len
co
e G
eo
me
try
Po
sit
ion
an
d l
ab
el
ea
ch
tria
ng
le o
n t
he
co
ord
ina
te p
lan
e.
1
. ri
gh
t △
FG
H w
ith
legs
2. is
osc
ele
s △
KL
P w
ith
3
. is
osc
ele
s △
AN
D w
ith
a u
nit
s an
d b
un
its
lon
g
base
−−
KP
6b
un
its
lon
g
base
−−−
AD
5a
un
its
lon
g
x
y F( 0
,a)
G( 0
, 0)
H( b
, 0)
x
yL
( 3b,
c)
K( 0
, 0)
P( 6
b,
0)
x
y
N(5 – 2
a,
b)
A( 0
, 0)
D( 5
a,
0)
Na
me
th
e m
issin
g c
oo
rd
ina
tes o
f e
ach
tria
ng
le.
4
.
x
y
B( 2
a,
0)
A( 0
, ?)
C( 0
, 0)
5
.
x
y
Y( 2
b,
0)
Z( ?
, ?)
X( 0
, 0)
6
.
x
y
N( 3
b,
0)
M( ?
, ?)
O( 0
, 0)
7
.
x
y
Q( ?
, ?)
R( 2
a,
b)
P( 0
, 0)
8
.
x
y
P( 7
b,
0)
R( ?
, ?)
N( 0
, 0)
9
.
x
y
U( a
, 0)
T( ?
, ?)
S( –
a,
0)
10
. P
RO
OF W
rite
a c
oord
inate
pro
of
to p
rove t
hat
in a
n i
sosc
ele
s ri
gh
t tr
ian
gle
, th
e s
egm
en
t fr
om
th
e v
ert
ex o
f th
e r
igh
t an
gle
to t
he m
idp
oin
t of
the h
yp
ote
nu
se i
s p
erp
en
dic
ula
r to
th
e h
yp
ote
nu
se.
G
ive
n:
isosc
ele
s ri
gh
t △
AB
C w
ith
∠A
BC
th
e r
igh
t an
gle
an
d M
th
e m
idp
oin
t of
−−
AC
P
ro
ve
: −−
− B
M ⊥
−−
AC
x
y
C( 2
a,
0)
A( 0
, 2
a)
M
B( 0
, 0
)
Sam
ple
an
sw
ers
are
giv
en
.
2
a
Z(b
, b
√ %
3 )
M(0
, c)
Q
(4a,
0)
R ( 7
−
2 b
, c
) T
(∅,
b )
P
roo
f:
T
he M
idp
oin
t F
orm
ula
sh
ow
s t
hat
the c
oo
rdin
ate
s o
f
M
are
( 0 +
2a
−
2
, 2
a +
0
−
2
) or
(a,
a).
Th
e s
lop
e o
f −
−
AC
is
2a
- 0
−
0 -
2a
= -
1.
Th
e s
lop
e o
f −
−
BM
is
a -
0
−
a -
0 =
1.
Th
e p
rod
uct
o
f th
e s
lop
es i
s -
1,
so
−−
BM
⊥ −
−
AC
.
4-8
Sk
ills
Pra
ctic
e
Tri
an
gle
s a
nd
Co
ord
inate
Pro
of
Answers (Lesson 4-8)
Chapter 4 A24 Glencoe Geometry
Answers
Co
pyri
gh
t ©
Gle
nco
e/M
cG
raw
-Hill, a
div
isio
n o
f T
he
Mc
Gra
w-H
ill C
om
pa
nie
s,
Inc
.
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Ch
ap
ter
4
52
G
len
co
e G
eo
me
try
Po
sit
ion
an
d l
ab
el
ea
ch
tria
ng
le o
n t
he
co
ord
ina
te p
lan
e.
1
. equ
ilate
ral
△S
WY
wit
h
2. is
osc
ele
s △
BL
P w
ith
3. is
osc
ele
s ri
gh
t △
DG
J
si
des
1 −
4 a
un
its
lon
g
base
−−
BL
3b u
nit
s lo
ng
w
ith
hyp
ote
nu
se −
−
DJ
an
d
legs
2a
un
its
lon
g
x
y D( 0
, 2
a)
G( 0
, 0)
J( 2
a,
0)
x
yP
(3 – 2b,
c)
B( 0
, 0)
L( 3
b, 0
)x
yY
(1 – 8a,
b)
W(1 – 4
a, 0)
S( 0
, 0
)
Na
me
th
e m
issin
g c
oo
rd
ina
tes o
f e
ach
tria
ng
le.
4
.
x
yS
( ?, ?)
J( 0
, 0
)R
( b
, 0)
1 – 3
5
.
x
y
C( ?
, 0
)
E( 0
, ?
)
B( –
3a, 0
)
6
.
x
y
P( 2
b,
0)
M( 0
, ?)
N( ?
, 0)
NE
IGH
BO
RH
OO
DS
F
or E
xe
rcis
es 7
an
d 8
, u
se
th
e f
oll
ow
ing
in
form
ati
on
.
Kari
na l
ives
6 m
iles
east
an
d 4
mil
es
nort
h o
f h
er
hig
h s
chool.
Aft
er
sch
ool
she w
ork
s p
art
ti
me a
t th
e m
all
in
a m
usi
c st
ore
. T
he m
all
is
2 m
iles
west
an
d 3
mil
es
nort
h o
f th
e s
chool.
7
. P
ro
of W
rite
a c
oord
inate
pro
of
to p
rove t
hat
Kari
na’s
hig
h s
chool,
her
hom
e,
an
d t
he
mall
are
at
the v
ert
ices
of
a r
igh
t tr
ian
gle
.
G
ive
n:
△S
KM
P
ro
ve
: △
SK
M i
s a r
igh
t tr
ian
gle
.
8
. F
ind
th
e d
ista
nce
betw
een
th
e m
all
an
d K
ari
na’s
hom
e.
x
y S( 0
, 0)
K( 6
, 4)
M( –
2, 3
)
S
( 1
−
6 b
, c
) C
(3a,
0),
E(0
, c)
M(0
, c
), N
(-2b
, 0)
P
roo
f:
S
lop
e o
f S
K =
4 -
0
−
6 -
0 o
r 2
−
3
S
lop
e o
f S
M =
3
- 0
−
-2
- 0
or
- 3
−
2
S
ince t
he s
lop
e o
f −
−
SM
is t
he n
eg
ati
ve r
ecip
rocal
of
the
s
lop
e o
f −
−
SK
, −
−
SM
⊥ −
−
SK
. T
here
fore
, △
SK
M i
s r
igh
t tr
ian
gle
.
K
M =
√ %
%%
%%
%%
%%
(-
2 -
6) 2
+ (
3 -
4) 2
= √
%%
%
64 +
1 =
√ %
%
65 o
r ≈
8.1
miles
4-8
Practi
ce
Tri
an
gle
s a
nd
Co
ord
inate
Pro
of
Sam
ple
an
sw
ers
are
g
iven
.
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-8
Ch
ap
ter
4
53
G
len
co
e G
eo
me
try
1
. S
HE
LV
ES
M
art
ha h
as
a s
helf
bra
cket
shap
ed
lik
e a
rig
ht
isosc
ele
s tr
ian
gle
. S
he w
an
ts t
o k
now
th
e l
en
gth
of
the
hyp
ote
nu
se r
ela
tive t
o t
he s
ides.
Sh
e
does
not
have a
ru
ler,
bu
t re
mem
bers
th
e D
ista
nce
Form
ula
. S
he p
lace
s th
e
bra
cket
on
a c
oord
inate
gri
d w
ith
th
e
righ
t an
gle
at
the o
rigin
. T
he l
en
gth
of
each
leg i
s a
. W
hat
are
th
e c
oord
inate
s of
the v
ert
ices
mak
ing t
he t
wo a
cute
an
gle
s?(a
, 0)
an
d (
0,
a)
2
. FLA
GS
A
fla
g i
s sh
ap
ed
lik
e a
n i
sosc
ele
s tr
ian
gle
. A
desi
gn
er
wou
ld
lik
e t
o m
ak
e a
dra
win
g o
f th
e
flag o
n a
coord
inate
pla
ne.
Sh
e p
osi
tion
s it
so t
hat
the
base
of
the t
rian
gle
is
on
th
e
y-a
xis
wit
h o
ne e
nd
poin
t lo
cate
d a
t (0
, 0).
Sh
e l
oca
tes
the t
ip o
f th
e f
lag a
t (a
, b − 2
).
Wh
at
are
th
e c
oord
inate
s of
the t
hir
d v
ert
ex?
(0,
b)
3
. B
ILLIA
RD
S T
he f
igu
re s
how
s a s
itu
ati
on
on
a b
illi
ard
table
.
(–b,
?)
y
x
O
( 0,
0)
Wh
at
are
th
e c
oord
inate
s of
the c
ue b
all
befo
re i
t is
hit
an
d t
he p
oin
t w
here
th
e
cue b
all
hit
s th
e e
dge o
f th
e t
able
?
Sam
ple
an
sw
er:
(-
2b
, 0),
(-
b,
a)
4
. T
EN
TS
T
he e
ntr
an
ce t
o M
att
’s t
en
t m
ak
es
an
iso
scele
s tr
ian
gle
. If
pla
ced
on
a c
oord
inate
gri
d w
ith
th
e b
ase
on
th
e
x–axis
an
d t
he l
eft
corn
er
at
the o
rigin
, th
e r
igh
t co
rner
wou
ld b
e a
t (6
, 0)
an
d
the v
ert
ex a
ngle
wou
ld b
e a
t (3
, 4).
P
rove t
hat
it i
s an
iso
scele
s tr
ian
gle
.
Sam
ple
an
sw
er:
Bo
th s
eg
men
ts
of
the t
rian
gle
fro
m t
he b
ase t
o
the v
ert
ex a
re 5
un
its l
on
g.
Sin
ce
the t
wo
sid
es a
re c
on
gru
en
t, i
t is
an
iso
scele
s t
rian
gle
.
5
. D
RA
FT
ING
An
en
gin
eer
is d
esi
gn
ing
a r
oad
way.
Th
ree r
oad
s in
ters
ect
to
form
a t
rian
gle
. T
he e
ngin
eer
mark
s tw
o p
oin
ts o
f th
e t
rian
gle
at
(-5,
0)
an
d
(5,
0)
on
a c
oord
inate
pla
ne.
a.
Desc
ribe t
he s
et
of
poin
ts i
n t
he
coord
inate
pla
ne t
hat
cou
ld n
ot
be
use
d a
s th
e t
hir
d v
ert
ex o
f th
e
tria
ngle
.
the x
-axis
b.
Desc
ribe t
he s
et
of
poin
ts i
n t
he
coord
inate
pla
ne t
hat
wou
ld m
ak
e
the v
ert
ex o
f an
iso
scele
s tr
ian
gle
to
geth
er
wit
h t
he t
wo c
on
gru
en
t si
des.
the y
-axis
excep
t fo
r th
e o
rig
in
c.
Desc
ribe t
he s
et
of
poin
ts i
n t
he
coord
inate
pla
ne t
hat
wou
ld m
ak
e a
ri
gh
t tr
ian
gle
wit
h t
he o
ther
two
poin
ts i
f th
e r
igh
t an
gle
is
loca
ted
at
(-5,
0).
the p
oin
ts w
ith
x-c
oo
rdin
ate
-
5,
excep
t fo
r (-
5,
0)
Wo
rd
Pro
ble
m P
racti
ce
Tri
an
gle
s a
nd
Co
ord
inate
Pro
of
4-8
Answers (Lesson 4-8)
Chapter 4 A25 Glencoe Geometry
Co
pyrig
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oe
/Mc
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ill, a d
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w-H
ill Co
mp
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ies, In
c.
NA
ME
DA
TE
PE
RIO
D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Ch
ap
ter
4
54
G
len
co
e G
eo
me
try
Recta
ng
le P
ara
do
x
Use
gri
d p
ap
er
to d
raw
th
e t
wo r
ect
an
gle
s belo
w.
Cu
t th
em
in
to p
iece
s alo
ng t
he
dash
ed
lin
es.
Th
en
rearr
an
ge t
he p
iece
s to
form
a n
ew
rect
an
gle
.
1
. M
ak
e a
sk
etc
h o
f th
e n
ew
rect
an
gle
. L
abel
the w
hole
nu
mber
len
gth
s
2
. W
hat
is t
he c
om
bin
ed
are
a o
f th
e t
wo o
rigin
al
rect
an
gle
s?
3
9 u
nit
s2
3
. W
hat
is t
he a
rea o
f th
e n
ew
rect
an
gle
?
4
0 u
nit
s2
4
. M
ak
e a
con
ject
ure
as
to w
hy t
here
is
a d
iscr
ep
an
cy b
etw
een
th
e a
nsw
ers
to
Exerc
ises
2 a
nd
3.
S
ee s
tud
en
ts’
an
sw
ers
.
5
. U
se a
str
aig
ht
ed
ge t
o p
reci
sely
dra
w t
he f
ou
r sh
ap
es
that
mak
e u
p t
he l
arg
er
rect
an
gle
. W
hat
does
you
dra
win
g s
how
?
I
t sh
ow
s a
gap
. S
o,
the r
ecta
ng
les d
on
’t q
uit
e
co
me t
og
eth
er
like t
hey l
oo
k l
ike t
hey w
ill.
6
. H
ow
cou
ld y
ou
use
slo
pe t
o s
how
th
at
you
dra
win
g
is a
ccu
rate
?
T
he s
lop
e o
f th
e b
ott
om
ed
ge o
f th
e t
op
left
pie
ce a
nd
th
e s
lop
e o
f th
e
bo
tto
m e
dg
e o
f th
e a
dja
cen
t p
iece a
re n
ot
the s
am
e s
o t
he t
wo
pie
ces d
o
no
t fi
t to
geth
er
to m
ake a
str
aig
ht
lin
e a
s t
he m
od
el
su
gg
ests
th
ey w
ou
ld.
53
5
5
3
3
3
3
5
3
35
5
4-8
En
ric
hm
en
t
Answers (Lesson 4-8)
Chapter 4 A26 Glencoe Geometry
Copyright
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-Hill
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cG
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-Hill
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panie
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Answers
Chapter 4 A27 Glencoe Geometry
Chapter 4 Assessment Answer KeyQuiz 1 (Lessons 4-1 and 4-2) Quiz 3 (Lessons 4-5 and 4-6) Mid-Chapter Test
Page 57 Page 58 Page 59
Quiz 2 (Lessons 4-3 and 4-4) Quiz 4 (Lessons 4-7 and 4-8)
Page 57 Page 58
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
obtuse scalene
B
x = 2, AB = 15,
BC = 15, AC = 10
AB = √ "" 53 , BC = 2 √ "" 10 ,
AC = √ "" 41 ; scalene
37
42
132
73
73
30
1.
2.
3.
4.
5.
△KMN $ △KML
∠O
x = 8; y = 14
SSS
∠Q $ ∠S
1.
2.
3.
4.
5.
Def. of angle bisector
AAS
60
45
120
1.
2.
3.
4.
5.
translation
x
y
J(a–2, 0)
D(0, a)
L(0, 0)
(a √ " 3 , 0)
AC = √ "" 34 , AB = 6,
and CB = √ "" 34
−−
AC $ −−
CB
1.
2.
3.
4.
D
H
A
G
5.
6.
7.
8.
AB = √ "" 41 ,
BC = √ "" 29 ,
AC = 5 √ " 2 ; scalene
SAS
3
congruent
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Chapter 4 A28 Glencoe Geometry
Chapter 4 Assessment Answer KeyVocabulary Test Form 1
Page 60 Page 61 Page 62
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
false, equiangular triangle
true
congruence transformation
exterior
included angle
flow proof
right triangle
included side
coordinate proof
vertex angle
a statement that can easily be
proved using a theorem
two triangles in which all
corresponding parts are
congruent
1.
2.
3.
4.
5.
6.
7.
8.
C
J
A
H
B
J
C
G
9.
10.
11.
12.
13.
C
F
C
F
B
isoscelesB:
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Chapter 4 A29 Glencoe Geometry
Chapter 4 Assessment Answer KeyForm 2A Form 2B
Page 63 Page 64 Page 65 Page 66
9.
10.
11.
12.
13.
B
H
B
F
A
B: A(0, 0), C(-a, 0)
1.
2.
3.
4.
5.
6.
7.
8.
D
G
D
F
C
G
D
J
9.
10.
11.
12.
13.
C
H
D
F
D
B: -2
1.
2.
3.
4.
5.
6.
7.
8.
C
H
A
J
B
J
D
F
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Chapter 4 A30 Glencoe Geometry
Chapter 4 Assessment Answer KeyForm 2C
Page 67 Page 68
1.
2.
3.
4.
5.
6.
7.
8.
acute scalene
x = 3, AB =
BC = AC = 24
EF = FG = 4 √ " 2 ,
EG = 8, isosceles
70
140
50
△DFG $ △BAC,∠D $ ∠B, ∠F $
∠A, ∠G $ ∠C
reflection;
AB = √ " 29 , BC = √ " 10 ,
CA = √ "" 17 , QT = √ " 29 ,
ST = √ " 10 , QS = √ " 17 ,
9.
10.
11.
12.
13.
14.
B:
45, 45, 90
Isosceles △
Theorem, AAS
50
x = 4
P ( b − 2 , b −
2 ) , AB = 2CP
△XYZ $ △MNO
x
y
C(b–2, c)
B(b, 0)A(0, 0)
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Answers
Chapter 4 A31 Glencoe Geometry
Chapter 4 Assessment Answer KeyForm 2D
Page 69 Page 70
1.
2.
3.
4.
5.
6.
7.
8.
9.
obtuse isosceles
x = 5, AB =
BC = 30, AC = 40
EF = FG = 5,
EG = 5 √ " 2 ,isosceles
110
70
30
△ABC $ △FDE,
∠A $ ∠F, ∠B $
∠D, ∠C $ ∠E
45, 45, 90
rotation; JK = √ "" 10 ,
KL = √ "" 37 , LJ = √ "" 29 ,
WX = √ "" 10 , XY =
√ "" 37 ,
YW = √ " 29
10.
11.
12.
13.
14.
B:
Isosceles △ Theorem, AAS
50
x = 6
M ( a −
2 , 0) , N (0,
b −
2 )
△ABC $ △DEF
x
y
M(3a, 0)
L(1.5a, b)
K(0, 0)
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Chapter 4 A32 Glencoe Geometry
Chapter 4 Assessment Answer KeyForm 3
Page 71 Page 72
1.
2.
3.
4.
5.
6.
7.
8.
x = 4,
AB = BC = 78, AC = 100
AB = 5, BC = 10, AC = 3 √ " 5 ; scalene
obtuse
20
90
40
reflection
GH = JK = 2 √ " 5 , IG = LJ
= 2 √ " 5 , IH = LK = 2 √ " 2 ;
△GHI $ △JKL by SSS.
AAS
9.
10.
11.
12.
13.
B:
Isosceles
Triangle Theorem
SAS
x = 3
Hypotenuse-leg
congruence for
right △s
m∠1 = m∠3 = m∠4 = m∠6 =
m∠9 = 20, m∠2 = m∠5 = 40,
m∠8 = 60, m∠7 = 140
x
y
B(a + b, 0)
C(a + b, c)2
A(0, 0)
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Chapter 4 A33 Glencoe Geometry
Chapter 4 Assessment Answer KeyExtended-Response Test, Page 73
Scoring Rubric
Score General Description Specific Criteria
4 Superior
A correct solution that
is supported by well-
developed, accurate
explanations
• Shows thorough understanding of the concepts of using
the Distance Formula to classify triangles and verify
congruence, finding missing angles, solving algebraic
equations in isosceles and equilateral triangles, proving
triangles congruent, congruence and writing coordinate
proofs.
• Uses appropriate strategies to solve problems
• Written explanations are exemplary.
• Figures are accurate and appropriate.
• Goes beyond requirements of some or all problems.
3 Satisfactory
A generally correct solution,
but may contain minor flaws
in reasoning or computation
• Shows understanding of the concepts of using the Distance
Formula to classify triangles and verify congruence, finding
missing angles, solving algebraic equations in isosceles and
equilateral triangles, proving triangles congruent, congruence
and writing proofs.
• Uses appropriate strategies to solve problems
• Computations are mostly correct.
• Written explanations are effective.
• Figures are mostly accurate and appropriate.
• Satisfies all requirements of all problems.
2 Nearly Satisfactory
A partially correct
interpretation and/or
solution to the problem
• Shows understanding of most of the concepts of using the
Distance Formula to classify triangles and verify congruence,
finding missing angles, solving algebraic equations in
isosceles and equilateral triangles, proving triangles
congruent, congrueence and writing proofs.
• May not use appropriate strategies to solve problems
• Computations are mostly correct.
• Written explanations are satisfactory.
• Figures are mostly accurate.
• Satisfies the requirements of most of the problems.
1 Nearly Unsatisfactory
A correct solution with no
supporting evidence or
explanation
• Final computation is correct.
• No written explanations or work is shown to substantiate the
final computation.
• Figures may be accurate but lack detail or explanation.
• Satisfies minimal requirements of some of the problems.
0 Unsatisfactory
An incorrect solution
indicating no mathematical
understanding of the concept
or task, or no solution is given
• Shows little or no understanding of most of the concepts
of using the Distance Formula to classify triangles and
verify congruence, finding missing angles, solving algebraic
equations in isosceles and equilateral triangles, proving
triangles congruent, congruence and writing proofs.
• Does not use appropriate strategies to solve problems
• Computations are incorrect.
• Written explanations are unsatisfactory.
• Figures are inaccurate or inappropriate.
• Does not satisfy the requirements of the problems.
• No answer may be given.
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Chapter 4 A34 Glencoe Geometry
Chapter 4 Assessment Answer KeyExtended-Response Test, Page 73
Sample Answers
1. a. The figure is an acute isosceles triangle.
b. 9x + 4 + 2(20x - 10) = 180
9x + 4 + 40x - 20 = 180
49x - 16 = 180
49x = 196
x = 4
2. a. To determine whether a triangle is equilateral, the coordinates should be graphed first to see if they form a triangle. Then list the segments which form the triangle and use the Distance Formula to find their lengths. A triangle is equilateral only if all three sides have equal measures.
b. This triangle is not equilateral since AB = AC = 5 and BC = 5 √ " 2 , which makes this an isosceles triangle.
3. Since ∠DBA, ∠ABC, and ∠EBC form a straight line, the sum of the angle measures is 180. Therefore m∠ABC = 180 - 40 - 62 or 78. Then since the sum of the measures of the angles of a triangle is 180, m∠1 = 180 - 78 - 58 or 44. Lastly, since ∠2 is an exterior angle, its measure is equivalent to the sum of the measures of the two remote interior angles, 58 + 78, which equals 136.
4. a. SSS postulate
b. ∠J $ ∠G, ∠D $ ∠E, ∠L $ ∠S
−−−
DJ $ −−−
EG , −−−
DL $ −−−
ES , −−
JL $ −−−
GS
5. Statements Reasons
1. −−−
AB || −−−
DE 1. Given
2. ∠ABC $ ∠DEC 2. Alt. int. % are $.
3. −−−
AD bisects −−−
BE . 3. Given
4. −−−
BC $ −−−
EC 4. Definition of segment bisector
5. ∠ACB $ ∠DCE 5. Vert. % are $.
6. △ABC $ △DEC 6. ASA
In addition to the scoring rubric found on page A33, the following sample answers
may be used as guidance in evaluating open-ended assessment items.
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Chapter 4 A35 Glencoe Geometry
Chapter 4 Assessment Answer KeyStandardized Test Practice
Page 74 Page 75
1.
2. F G H J
3. A B C D
4. F G H J
5.
6. F G H J
7. A B C D
A B C D
A B C D
8.
9.
10.
11.
12.
13.
14. F G H J
A B C D
F G H J
A B C D
F G H J
A B C D
F G H J
15.
16. 9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
6
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
1 8
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c.
Chapter 4 A36 Glencoe Geometry
Chapter 4 Assessment Answer KeyStandardized Test Practice (continued)
Page 76
17.
18.
19.
20.
21.
22a.
22b.
22c.
29 ft
35
62
40
∠P ! ∠H, ∠Q ! ∠G,
∠R ! ∠B, −−
PQ ! −−
HG ,
−−
QR ! −−
GB , −−
PR ! −−
HB
−−
FD
right triangle
15