Chapter 4Resource Masters
Geometry
Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.
Study Guide and Intervention Workbook 0-07-860191-6Skills Practice Workbook 0-07-860192-4Practice Workbook 0-07-860193-2Reading to Learn Mathematics Workbook 0-07-861061-3
ANSWERS FOR WORKBOOKS The answers for Chapter 4 of these workbookscan be found in the back of this Chapter Resource Masters booklet.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe’s Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 0-07-860181-9 GeometryChapter 4 Resource Masters
1 2 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03
© Glencoe/McGraw-Hill iii Glencoe Geometry
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Proof Builder . . . . . . . . . . . . . . . . . . . . . . ix
Lesson 4-1Study Guide and Intervention . . . . . . . . 183–184Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 185Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 186Reading to Learn Mathematics . . . . . . . . . . 187Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 188
Lesson 4-2Study Guide and Intervention . . . . . . . . 189–190Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 191Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 192Reading to Learn Mathematics . . . . . . . . . . 193Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 194
Lesson 4-3Study Guide and Intervention . . . . . . . . 195–196Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 197Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Reading to Learn Mathematics . . . . . . . . . . 199Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 200
Lesson 4-4Study Guide and Intervention . . . . . . . . 201–202Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 203Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 204Reading to Learn Mathematics . . . . . . . . . . 205Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 206
Lesson 4-5Study Guide and Intervention . . . . . . . . 207–208Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 209Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 210Reading to Learn Mathematics . . . . . . . . . . 211Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 212
Lesson 4-6Study Guide and Intervention . . . . . . . . 213–214Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 215Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 216Reading to Learn Mathematics . . . . . . . . . . 217Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 218
Lesson 4-7Study Guide and Intervention . . . . . . . . 219–220Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 221Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 222Reading to Learn Mathematics . . . . . . . . . . 223Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 224
Chapter 4 AssessmentChapter 4 Test, Form 1 . . . . . . . . . . . . 225–226Chapter 4 Test, Form 2A . . . . . . . . . . . 227–228Chapter 4 Test, Form 2B . . . . . . . . . . . 229–230Chapter 4 Test, Form 2C . . . . . . . . . . . 231–232Chapter 4 Test, Form 2D . . . . . . . . . . . 233–234Chapter 4 Test, Form 3 . . . . . . . . . . . . 235–236Chapter 4 Open-Ended Assessment . . . . . . 237Chapter 4 Vocabulary Test/Review . . . . . . . 238Chapter 4 Quizzes 1 & 2 . . . . . . . . . . . . . . . 239Chapter 4 Quizzes 3 & 4 . . . . . . . . . . . . . . . 240Chapter 4 Mid-Chapter Test . . . . . . . . . . . . 241Chapter 4 Cumulative Review . . . . . . . . . . . 242Chapter 4 Standardized Test Practice . 243–244
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A32
© Glencoe/McGraw-Hill iv Glencoe Geometry
Teacher’s Guide to Using theChapter 4 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 4 Resource Masters includes the core materials neededfor 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 in theGeometry TeacherWorks CD-ROM.
Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.
WHEN TO USE Give these pages tostudents before beginning Lesson 4-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.
Vocabulary Builder Pages ix–xinclude another student study tool thatpresents up to fourteen of the key theoremsand postulates from the chapter. Studentsare to write each theorem or postulate intheir own words, including illustrations ifthey choose to do so. You may suggest thatstudents highlight or star the theorems orpostulates with which they are not familiar.
WHEN TO USE Give these pages tostudents before beginning Lesson 4-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toupdate it as they complete each lesson.
Study Guide and InterventionEach lesson in Geometry addresses twoobjectives. There is one Study Guide andIntervention master for each objective.
WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.
Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.
WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.
WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.
© Glencoe/McGraw-Hill v Glencoe Geometry
Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.
Assessment OptionsThe assessment masters in the Chapter 4Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.
• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriateintervals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Geometry. It can also beused as a test. This master includes free-response questions.
• The Standardized Test Practice offerscontinuing review of geometry conceptsin various formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and short-responsequestions. Bubble-in and grid-in answersections are provided on the master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questionsthat appear in the Student Edition onpages 232–233. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided forthe assessment masters in this booklet.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
44
© Glencoe/McGraw-Hill vii Glencoe Geometry
Voca
bula
ry B
uild
erThis 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. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
acute triangle
base angles
congruence transformation
kuhn·GROO·uhns
congruent triangles
coordinate proof
corollary
equiangular triangle
equilateral triangle
exterior angle
(continued on the next page)
© Glencoe/McGraw-Hill viii Glencoe Geometry
Vocabulary Term Found on Page Definition/Description/Example
flow proof
included angle
included side
isosceles triangle
obtuse triangle
remote interior angles
right triangle
scalene triangle
SKAY·leen
vertex angle
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
44
Learning to Read MathematicsProof Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
44
© Glencoe/McGraw-Hill ix Glencoe Geometry
Proo
f Bu
ilderThis is a list of key theorems and postulates you will learn in Chapter 4. As you
study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.
Theorem or Postulate Found on Page Description/Illustration/Abbreviation
Theorem 4.1Angle Sum Theorem
Theorem 4.2Third Angle Theorem
Theorem 4.3Exterior Angle Theorem
Theorem 4.4
Theorem 4.5Angle-Angle-Side Congruence (AAS)
Theorem 4.6Leg-Leg Congruence (LL)
Theorem 4.7Hypotenuse-Angle Congruence (HA)
(continued on the next page)
© Glencoe/McGraw-Hill x Glencoe Geometry
Theorem or Postulate Found on Page Description/Illustration/Abbreviation
Theorem 4.8Leg-Angle Congruence (LA)
Theorem 4.9Isosceles Triangle Theorem
Theorem 4.10
Postulate 4.1Side-Side-Side Congruence (SSS)
Postulate 4.2Side-Angle-Side Congruence (SAS)
Postulate 4.3Angle-Side-Angle Congruence (ASA)
Postulate 3.4Hypotenuse-Leg Congruence(HL)
Learning to Read MathematicsProof Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
44
Study Guide and InterventionClassifying Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
© Glencoe/McGraw-Hill 183 Glencoe Geometry
Less
on
4-1
Classify Triangles by Angles One way to classify a triangle is by the measures of its angles.
• 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.
• 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.
Classify each triangle.
a.
All three angles are congruent, so all three angles have measure 60°.The triangle is an equiangular triangle.
b.
The triangle has one angle that is obtuse. It is an obtuse triangle.
c.
The triangle has one right angle. It is a right triangle.
Classify each triangle as acute, equiangular, obtuse, or right.
1. 2. 3.
4. 5. 6.60�
28� 92�F D
B
45�
45�90�X Y
W
65� 65�
50�
U V
T
60� 60�
60�
Q
R S
120�
30� 30�N O
P
67�
90� 23�
K
L M
90�
60� 30�
G
H J
25�35�
120�
D F
E
60�
A
B C
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 184 Glencoe Geometry
Classify Triangles by Sides You can classify a triangle by the measures of its 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.
• If no two sides of a triangle are congruent, then the triangle is a scalene triangle.
Classify each triangle.
a. b. c.
Two sides are congruent. All three sides are The triangle has no pairThe triangle is an congruent. The triangle of congruent sides. It is isosceles triangle. is an equilateral triangle. a scalene triangle.
Classify each triangle as equilateral, isosceles, or scalene.
1. 2. 3.
4. 5. 6.
7. Find the measure of each side of equilateral �RST with RS � 2x � 2, ST � 3x,and TR � 5x � 4.
8. Find the measure of each side of isosceles �ABC with AB � BC if AB � 4y,BC � 3y � 2, and AC � 3y.
9. Find the measure of each side of �ABC with vertices A(�1, 5), B(6, 1), and C(2, �6).Classify the triangle.
D E
F
x
x x8x
32x
32x
B
CA
UW
S
12 17
19Q O
MG
K I18
18 182
1
3��G C
A
2312
15X V
TN
R PL J
H
Study Guide and Intervention (continued)
Classifying Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
ExampleExample
ExercisesExercises
Skills PracticeClassifying Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
© Glencoe/McGraw-Hill 185 Glencoe Geometry
Less
on
4-1
Use a protractor to classify each triangle as acute, equiangular, obtuse, or right.
1. 2. 3.
4. 5. 6.
Identify the indicated type of triangles.
7. right 8. isosceles
9. scalene 10. obtuse
ALGEBRA Find x and the measure of each side of the triangle.
11. �ABC is equilateral with AB� 3x � 2, BC � 2x � 4, and CA � x � 10.
12. �DEF is isosceles, �D is the vertex angle, DE � x � 7, DF � 3x � 1, and EF � 2x � 5.
Find the measures of the sides of �RST and classify each triangle by its sides.
13. R(0, 2), S(2, 5), T(4, 2)
14. R(1, 3), S(4, 7), T(5, 4)
E CD
A B
© Glencoe/McGraw-Hill 186 Glencoe Geometry
Use a protractor to classify each triangle as acute, equiangular, obtuse, or right.
1. 2. 3.
Identify the indicated type of triangles if A�B� � A�D� � B�D� � D�C�, B�E� � E�D�, A�B� ⊥ B�C�, and E�D� ⊥ D�C�.
4. right 5. obtuse
6. scalene 7. isosceles
ALGEBRA Find x and the measure of each side of the triangle.
8. �FGH is equilateral with FG � x � 5, GH � 3x � 9, and FH � 2x � 2.
9. �LMN is isosceles, �L is the vertex angle, 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
Practice Classifying Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
Reading to Learn MathematicsClassifying Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
© Glencoe/McGraw-Hill 187 Glencoe Geometry
Less
on
4-1
Pre-Activity Why are triangles important in construction?
Read the introduction to Lesson 4-1 at the top of page 178 in your textbook.
• Why are triangles used for braces in construction rather than other shapes?
• Why do you think that isosceles triangles are used more often thanscalene triangles in construction?
Reading the Lesson1. Supply the correct numbers to complete each sentence.
a. In an obtuse triangle, there are acute angle(s), right angle(s), and
obtuse angle(s).
b. In an acute triangle, there are acute angle(s), right angle(s), and
obtuse angle(s).
c. In a right triangle, there are acute angle(s), right angle(s), and
obtuse angle(s).
2. Determine whether each statement is always, sometimes, or never true.
a. A right triangle is scalene.
b. An obtuse triangle is isosceles.
c. An equilateral triangle is a right triangle.
d. An equilateral triangle is isosceles.
e. An acute triangle is isosceles.
f. A scalene triangle is obtuse.
3. Describe each triangle by as many of the following words as apply: acute, obtuse, right,scalene, isosceles, or equilateral.
a. b. c.
Helping You Remember4. A good way to remember a new mathematical term is to relate it to a nonmathematical
definition of the same word. How is the use of the word acute, when used to describeacute pain, related to the use of the word acute when used to describe an acute angle oran acute triangle?
5
34
135�80�
70�
30�
© Glencoe/McGraw-Hill 188 Glencoe Geometry
Reading MathematicsWhen you read geometry, you may need to draw a diagram to make the texteasier 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 isgreater than the x-coordinate of A. Both coordinates of C are greaterthan the corresponding coordinates of B. Is triangle 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 ABCmust be obtuse.
Answer each question. Draw a simple triangle on the grid above 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. Thex-coordinates of R and S are the y-coordinates of J and K are thesame. The y-coordinate of T is same. The x-coordinates of K and Lbetween the y-coordinates of R are the same. Is triangle JKL acute,and S. The x-coordinate of T is less right, or obtuse?than the x-coordinate of R. Is angleR of triangle RST acute, right, or obtuse?
3. Consider three noncollinear points, 4. Consider three points, G, H, and ID, 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, andopposites. 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-coordinateequilateral? of G. Is triangle GHI scalene,
isosceles, or equilateral?
BA
Q
x
y
O
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-14-1
ExampleExample
Study Guide and InterventionAngles of Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
© Glencoe/McGraw-Hill 189 Glencoe Geometry
Less
on
4-2
Angle Sum Theorem If the measures of two angles of a triangle are known,the measure of the third angle can always be found.
Angle Sum The sum of the measures of the angles of a triangle is 180.Theorem 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 Angle Sum
Theorem
25 � 35 � m�T � 180 Substitution
60 � m�T � 180 Add.
m�T � 120 Subtract 60
from each side.
35�
25�R T
S
Find the missing angle measures.
m�1 � m�A � m�B � 180 Angle Sum Theorem
m�1 � 58 � 90 � 180 Substitution
m�1 � 148 � 180 Add.
m�1 � 32 Subtract 148 from
each side.
m�2 � 32 Vertical angles are
congruent.
m�3 � m�2 � m�E � 180 Angle Sum Theorem
m�3 � 32 � 108 � 180 Substitution
m�3 � 140 � 180 Add.
m�3 � 40 Subtract 140 from
each side.
58�
90�
108�
12 3
E
DAC
B
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find the measure of each numbered angle.
1. 2.
3. 4.
5. 6. 20�
152�
DG
A
130�60�
1 2
S
R
TW
Q
O
NM
P58�
66�
50�
321
V
W T
U
30�
60�
2
1
S
Q R30�
1
90�
62�
1 N
M
P
© Glencoe/McGraw-Hill 190 Glencoe Geometry
Exterior Angle Theorem At each vertex of a triangle, the angle formed by one sideand an extension of the other side is called an exterior angle of the triangle. For eachexterior angle of a triangle, the remote interior angles are the interior angles that are notadjacent to that exterior angle. In the diagram below, �B and �A are the remote interiorangles for exterior �DCB.
Exterior AngleThe measure of an exterior angle of a triangle is equal to
Theoremthe sum of the measures of the two remote interior angles.m�1 � m�A � m�B
AC
B
D1
Study Guide and Intervention (continued)
Angles of Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
Find m�1.
m�1 � m�R � m�S Exterior Angle Theorem
� 60 � 80 Substitution
� 140 Add.
R T
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�
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find the measure of each numbered angle.
1. 2.
3. 4.
Find x.
5. 6. 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
B C D
A
25�
35�
12Y Z W
X
65�
50�
1
Skills PracticeAngles of Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
© Glencoe/McGraw-Hill 191 Glencoe Geometry
Less
on
4-2
Find the missing angle measures.
1. 2.
Find the measure of each angle.
3. m�1
4. m�2
5. m�3
Find the measure of each angle.
6. m�1
7. m�2
8. m�3
Find the measure of each angle.
9. m�1
10. m�2
11. m�3
12. m�4
13. m�5
Find the measure of each angle.
14. m�1
15. m�2 63�
1
2D
A C
B
80�
60�
40�
105�
1 4 52
3
150�55�
70�
1 2
3
85� 55�
40�
1 2
3
146�
TIGERS80�
73�
© Glencoe/McGraw-Hill 192 Glencoe Geometry
Find the missing angle measures.
1. 2.
Find the measure of each angle.
3. m�1
4. m�2
5. m�3
Find the measure of each angle.
6. m�1
7. m�4
8. m�3
9. m�2
10. m�5
11. m�6
Find the measure of each angle if �BAD and �BDC are right angles and m�ABC � 84.
12. m�1
13. m�2
14. CONSTRUCTION The diagram shows an example of the Pratt Truss used in bridgeconstruction. Use the diagram to find m�1.
145�1
64�1
2A
BC
D
118�36�
68�
70�
65�
82�
1
2
3 4
5
6
58�
39�
35�
12
3
40� 55�
72�
?
Practice Angles of Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
Reading to Learn MathematicsAngles of Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
© Glencoe/McGraw-Hill 193 Glencoe Geometry
Less
on
4-2
Pre-Activity How are the angles of triangles used to make kites?
Read the introduction to Lesson 4-2 at the top of page 185 in your textbook.
The frame of the simplest kind of kite divides the kite into four triangles.Describe these four triangles and how they are related to each other.
Reading the Lesson
1. Refer to the figure.
a. Name the three interior angles of the triangle. (Use threeletters to name each angle.)
b. Name three exterior angles of the triangle. (Use three lettersto name each angle.)
c. Name the remote interior angles of �EAB.
d. Find the measure of each angle without using a protractor.
i. �DBC ii. �ABC iii. �ACF iv. �EAB
2. Indicate whether each statement is true or false. If the statement is false, replace theunderlined word or number with a word or number that will make the statement true.
a. The acute angles of a right triangle are .
b. The sum of the measures of the angles of any triangle is .
c. A triangle can have at most one right angle or angle.
d. If two angles of one triangle are congruent to two angles of another triangle, then thethird angles of the triangles are .
e. The measure of an exterior angle of a triangle is equal to the of themeasures of the two remote interior angles.
f. If the measures of two angles of a triangle are 62 and 93, then the measure of thethird angle is .
g. An angle of a triangle forms a linear pair with an interior angle of thetriangle.
Helping You Remember
3. Many students remember mathematical ideas and facts more easily if they see themdemonstrated visually rather than having them stated in words. Describe a visual wayto demonstrate the Angle Sum Theorem.
exterior
35
difference
congruent
acute
100
supplementary
39�
23�
EA B D
CF
© Glencoe/McGraw-Hill 194 Glencoe Geometry
Finding Angle Measures in TrianglesYou can use algebra to solve problems involving triangles.
In triangle ABC, m�A, is twice m�B, and m�Cis 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 � 1802x � x � (x � 8) � 180
4x � 8 � 1804x � 172x � 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 twicem�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.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-24-2
ExampleExample
Study Guide and InterventionCongruent Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
© Glencoe/McGraw-Hill 195 Glencoe Geometry
Less
on
4-3
Corresponding Parts of Congruent TrianglesTriangles 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 � �TX�Y� � R�S�, X�Z� � R�T�, Y�Z� � S�T�
Identify the congruent triangles in each figure.
1. 2. 3.
Name the corresponding congruent angles and sides for the congruent triangles.
4. 5. 6. R
T
U S
B D
CA
F G L K
JE
K
J
L
MC
D
A
B
CA
B
LJ
K
Y
XZ
T
SR
AC
B
R
T
S
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 196 Glencoe Geometry
Identify Congruence Transformations If two triangles are congruent, you canslide, flip, or turn one of the triangles and they will still be congruent. These are calledcongruence transformations because they do not change the size or shape of the figure.It is common to use prime symbols to distinguish between an original �ABC and atransformed �A�B�C�.
Name the congruence transformation that produces �A�B�C� from �ABC.The congruence transformation is a slide.�A � �A�; �B � �B�; �C ��C�;A�B� � A���B���; A�C� � A���C���; B�C� � B���C���
Describe the congruence transformation between the two triangles as a slide, aflip, or a turn. Then name the congruent triangles.
1. 2.
3. 4.
5. 6.
x
y
OM
N
P
N�
P�
x
y
OA� B�
C�
A B
C
x
y
OB�B
A
Cx
y
O
Q� P�
P
Q
x
y
ON�
M�P�
N
MP
x
y
OR
S�T�
S
T
x
y
O
A�
B�B
C�A C
Study Guide and Intervention (continued)
Congruent Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
ExampleExample
ExercisesExercises
Skills PracticeCongruent Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
© Glencoe/McGraw-Hill 197 Glencoe Geometry
Less
on
4-3
Identify the congruent triangles in each figure.
1. 2.
3. 4.
Name the congruent angles and sides for each pair of congruent triangles.
5. �ABC � �FGH
6. �PQR � �STU
Verify that each of the following transformations preserves congruence, and namethe congruence transformation.
7. �ABC � �A�B�C� 8. �DEF � �D�E�F �
x
y
OD�
E�
F�D
E
F
x
y
OA�
B�
C�
A
B
C
D
E
G
FRP
Q
S
WY
XC
AB
L
P
J
S
V
T
© Glencoe/McGraw-Hill 198 Glencoe Geometry
Identify the congruent triangles in each figure.
1. 2.
Name the congruent angles and sides for each pair of congruent triangles.
3. �GKP � �LMN
4. �ANC � �RBV
Verify that each of the following transformations preserves congruence, and namethe congruence transformation.
5. �PST � �P�S�T� 6. �LMN � �L�M�N�
QUILTING For Exercises 7 and 8, refer to the quilt design.
7. Indicate the triangles that appear to be congruent.
8. Name the congruent angles and congruent sides of a pair of congruent triangles.
B
A
I
E
FH G
C D
x
y
O
M�
N�L�
M
NL
x
y
O
S�
T�P�
S
TP
MN
L
P
QD
R
SCA
B
Practice Congruent Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
Reading to Learn MathematicsCongruent Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
© Glencoe/McGraw-Hill 199 Glencoe Geometry
Less
on
4-3
Pre-Activity Why are triangles used in bridges?
Read the introduction to Lesson 4-3 at the top of page 192 in your textbook.
In the bridge shown in the photograph in your textbook, diagonal braceswere used to divide squares into two isosceles right triangles. Why do youthink these braces are used on the bridge?
Reading the Lesson1. If �RST � �UWV, complete each pair of congruent parts.
�R � � �W �T �
R�T� � � U�W� � W�V�
2. Identify the congruent triangles in each diagram.
a. b.
c. d.
3. Determine whether each statement says that congruence of triangles is reflexive,symmetric, or transitive.
a. If the first of two triangles is congruent to the second triangle, then the secondtriangle is congruent to the first.
b. If there are three triangles for which the first is congruent to the second and the secondis congruent to the third, then the first triangle is congruent to the third.
c. Every triangle is congruent to itself.
Helping You Remember4. A good way to remember something is to explain it to someone else. Your classmate Ben is
having trouble writing congruence statements for triangles because he thinks he has tomatch up three pairs of sides and three pairs of angles. How can you help him understandhow to write correct congruence statements more easily?
R T
US
V
N O P
QM
S
P R
Q
CA
B
D
© Glencoe/McGraw-Hill 200 Glencoe Geometry
Transformations in The Coordinate Plane
The following statement tells one way to map preimage points to image 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 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 youranswer.
Draw the transformation image for each figure. Then tell whether thetransformation is or is not a congruence transformation.
2. (x, y) → (x � 4, y) 3. (x, y) → (x � 8, y � 7)
4. (x, y) → (�x , �y) 5. (x, y) → ���12�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)
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-34-3
Study Guide and InterventionProving Congruence—SSS, SAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
© Glencoe/McGraw-Hill 201 Glencoe Geometry
Less
on
4-4
SSS Postulate You know that two triangles are congruent if corresponding sides arecongruent and corresponding angles are congruent. The Side-Side-Side (SSS) Postulate letsyou show that two triangles are congruent if you know only that the sides of one triangleare congruent to the sides of the second triangle.
SSS PostulateIf the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.
Write a two-column proof.Given: A�B� � D�B� and C is the midpoint of A�D�.Prove: �ABC � �DBC
Statements Reasons
1. A�B� � D�B� 1. Given
2. C is the midpoint of A�D�. 2. Given
3. A�C� � D�C� 3. Definition of midpoint
4. B�C� � B�C� 4. Reflexive Property of �5. �ABC � �DBC 5. SSS Postulate
Write a two-column proof.
B
CDA
ExampleExample
ExercisesExercises
1.
Given: A�B� � X�Y�, A�C� � X�Z�, B�C� � Y�Z�Prove: �ABC � �XYZ
Statements Reasons
1.A�B� � X�Y� 1.Given
2.�ABC � �XYZ 2. SSS Post.
2.
Given: R�S� � U�T�, R�T� � U�S�Prove: �RST � �UTS
Statements Reasons
1.R�S� � U�T� 1. Given
2.S�T� � T�S� 2. Refl. Prop.3.�RST � �UTS 3. SSS Post.
T U
R S
B Y
CA XZ
© Glencoe/McGraw-Hill 202 Glencoe Geometry
SAS Postulate Another way to show that two triangles are congruent is to use the Side-Angle-Side (SAS) Postulate.
SAS PostulateIf 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.
For each diagram, determine which pairs of triangles can beproved congruent by the SAS Postulate.
a. b. c.
In �ABC, the angle is not The right angles are The included angles, �1 “included” by the sides A�B� congruent and they are the and �2, are congruent and A�C�. 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.
For each figure, determine which pairs of triangles can be proved congruent bythe SAS Postulate.
1. 2. 3.
4. 5. 6.
J H
GF
K
C
BA
D
V
T
W
M
P L
N
M
X
W Z
YQT
P
UN M
R
P Q
S
1
2 R
D H
F E
G
J
A
B C
X
Y Z
Study Guide and Intervention (continued)
Proving Congruence—SSS, SAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
ExampleExample
ExercisesExercises
Skills PracticeProving Congruence—SSS, SAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
© Glencoe/McGraw-Hill 203 Glencoe Geometry
Less
on
4-4
Determine whether �ABC � �KLM given the coordinates of the vertices. 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)
3. Write a flow proof.Given: P�R� � D�E�, P�T� � D�F�
�R � �E, �T � �FProve: �PRT � �DEF
Determine which postulate can be used to prove that the triangles are congruent.If it is not possible to prove that they are congruent, write not possible.
4. 5. 6.
PR � DEGiven
PT � DF Given
�R � �E Given
�P � �D Third AngleTheorem
�PRT � �DEF SAS
�T � �F Given
T
R
P
F
E
D
© Glencoe/McGraw-Hill 204 Glencoe Geometry
Determine 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: R�S� � T�S�
V is the midpoint of R�T�.Prove: �RSV � �TSV
Determine which postulate can be used to prove that the triangles are congruent.If it is not possible to prove that they are congruent, 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 thediagram. How does he know that the lengths A�B� and AB are equal?
A�B�
A B
C
RS � TSGiven
SV � SVReflexiveProperty
RV � VTDefinitionof midpoint
V is themidpoint of RT. Given
�RSV � �TSV SSS
S
R
V
T
Practice Proving Congruence—SSS, SAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
Reading to Learn MathematicsProving Congruence—SSS, SAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
© Glencoe/McGraw-Hill 205 Glencoe Geometry
Less
on
4-4
Pre-Activity How do land surveyors use congruent triangles?
Read the introduction to Lesson 4-4 at the top of page 200 in your textbook.
Why do you think that land surveyors would use congruent right trianglesrather than other congruent triangles to establish property boundaries?
Reading the Lesson
1. Refer to the figure.
a. Name the sides of �LMN for which �L is the included angle.
b. Name the sides of �LMN for which �N is the included angle.
c. Name the sides of �LMN for which �M is the included angle.
2. Determine whether you have enough information to prove that the two triangles in eachfigure are congruent. If so, write a congruence statement and name the congruencepostulate that you would use. If not, write not possible.
a. b.
c. E�H� and D�G� bisect each other. d.
Helping You Remember
3. Find three words that explain what it means to say that two triangles are congruent andthat can help you recall the meaning of the SSS Postulate.
GE
F
HD
R
T
SU
G
FD
E
C
A
DB
L
N
M
© Glencoe/McGraw-Hill 206 Glencoe Geometry
Congruent Parts of Regular Polygonal RegionsCongruent figures are figures that have exactly the same size and shape. There are manyways to divide regular polygonal regions into congruent parts. Three ways to divide anequilateral triangular region are shown. You can verify that the parts are congruent bytracing one part, then rotating, sliding, or reflecting that part on top of the other parts.
1. Divide each square into four congruent parts. Use three different ways.
2. Divide each pentagon into five congruent parts. Use three different ways.
3. Divide each hexagon into six congruent parts. Use three different ways.
4. What hints might you give another student who is trying to divide figures like those into congruent parts?
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-44-4
Study Guide and InterventionProving Congruence—ASA, AAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
© Glencoe/McGraw-Hill 207 Glencoe Geometry
Less
on
4-5
ASA Postulate The Angle-Side-Angle (ASA) Postulate lets you show that two trianglesare congruent.
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.
Find the missing congruent parts so that the triangles can beproved congruent by the ASA Postulate. Then write the triangle congruence.
a.
Two pairs of corresponding angles are congruent, �A � �D and �C � �F. If theincluded sides A�C� and D�F� are congruent, then �ABC � �DEF by the ASA Postulate.
b.
�R � �Y and S�R� � X�Y�. If �S � �X, then �RST� �YXW by the ASA Postulate.
What corresponding parts must be congruent in order to prove that the trianglesare congruent by the ASA Postulate? Write the triangle congruence statement.
1. 2. 3.
4. 5. 6.
A C
B
E
D
S
V
U
R T
D
A B
C
DC
EA
B
YW
X
ZE A
BD
C
R T W Y
S X
A C
B
D F
E
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 208 Glencoe Geometry
AAS Theorem Another way to show that two triangles are congruent is the Angle-Angle-Side (AAS) Theorem.
AAS TheoremIf two angles and a nonincluded side of one triangle are congruent to the corresponding twoangles and side of a second triangle, then the two triangles are congruent.
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 Postulate?A�C� � A�C� by the Reflexive Property of congruence. The congruent angles cannot be �1 and �2, because A�C� would be the included side.If �B � �D, then �ABC � �ADC by the AAS Theorem.
In Exercises 1 and 2, draw and label �ABC and �DEF. Indicate which additionalpair of corresponding parts needs to be congruent for the triangles to becongruent by the AAS Theorem.
1. �A � �D; �B � �E 2. BC � EF; �A � �D
3. Write a flow proof.Given: �S � �U; T�R� bisects �STU.Prove: �SRT � �URT
Given
Given
RT � RT Refl. Prop. of �
Def.of � bisector
TR bisects �STU.
�SRT � �URT
�STR � �UTR
AAS�SRT � �URTCPCTC
�S � �U
S
R T
U
B
A
C
E
D
F
CA
B
FD
E
D
C12A
B
Study Guide and Intervention (continued)
Proving Congruence—ASA, AAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
ExampleExample
ExercisesExercises
Skills PracticeProving Congruence—ASA, AAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
© Glencoe/McGraw-Hill 209 Glencoe Geometry
Less
on
4-5
Write a flow proof.
1. Given: �N � �LJ�K� � M�K�
Prove: �JKN � �MKL
2. Given: A�B� � C�B��A � �CD�B� bisects �ABC.
Prove: A�D� � C�D�
3. Write a paragraph proof.
Given: D�E� || F�G��E � �G
Prove: �DFG � �FDE
FG
D E
�A � �C
GivenAB � CB
Given CPCTCAD � CD
DB bisects �ABC. Given
�ABD � �CBDASA
�ABD � �CBDDef. of � bisector
A C
B
D
�N � �LGiven
JK � MK Given
�JKN � �MKL Vertical � are �.
�JKN � �MKLAAS
N
J
M
K L
© Glencoe/McGraw-Hill 210 Glencoe Geometry
1. Write a flow proof.Given: S is the midpoint of Q�T�.
Q�R� || T�U�Prove: �QSR � �TSU
2. Write a paragraph proof.
Given: �D � �FG�E� bisects �DEF.
Prove: D�G� � F�G�
ARCHITECTURE For Exercises 3 and 4, use the following information.An architect used the window design in the diagram when remodeling an art studio. A�B� and C�B� each measure 3 feet.
3. Suppose D is the midpoint of A�C�. 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
�Q � �T
Given
QR || TU Given
Def.of midpoint
Alt. Int. � are �.
QS � TS S is the midpoint of QT.
�QSR � �TSUASA
�QSR � �TSUVertical � are �.
UQ S
RT
Practice Proving Congruence—ASA, AAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
Reading to Learn MathematicsProving Congruence—ASA, AAS
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
© Glencoe/McGraw-Hill 211 Glencoe Geometry
Less
on
4-5
Pre-Activity How are congruent triangles used in construction?Read the introduction to Lesson 4-5 at the top of page 207 in your textbook.Which of the triangles in the photograph in your textbook appear to becongruent?
Reading the Lesson1. Explain in your own words the difference between how the ASA Postulate and the AAS
Theorem are used to prove that two triangles are congruent.
2. Which of the following conditions are sufficient to prove that two triangles are congruent?A. Two sides of one triangle are congruent to two sides of the other triangle.B. The three sides of one triangles are congruent to the three sides of the other triangle.C. The three angles of one triangle are congruent to the three angles of the other triangle.D. All six corresponding parts of two triangles are congruent.E. Two angles and the included side of one triangle are congruent to two sides and the
included angle of the other triangle.F. Two sides and a nonincluded angle of one triangle are congruent to two sides and a
nonincluded angle of the other triangle.G. Two angles and a nonincluded side of one triangle are congruent to two angles and
the corresponding nonincluded side of the other triangle.H. Two sides and the included angle of one triangle are congruent to two sides and the
included angle of the other triangle.I. Two angles and a nonincluded side of one triangle are congruent to two angles and a
nonincluded side of the other triangle.
3. Determine whether you have enough information to prove that the two triangles in eachfigure are congruent. If so, write a congruence statement and name the congruencepostulate or theorem that you would use. If not, write not possible.
a. b. T is the midpoint of R�U�.
Helping You Remember4. A good way to remember mathematical ideas is to summarize them in a general statement.
If you want to prove triangles congruent by using three pairs of corresponding parts,what is a good way to remember which combinations of parts will work?
R
S
T
U
V
A DCB
E
© Glencoe/McGraw-Hill 212 Glencoe Geometry
Congruent Triangles in the Coordinate PlaneIf you know the coordinates of the vertices of two triangles in the coordinateplane, you can often decide whether the two triangles are congruent. Theremay 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 youknow 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.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-54-5
Study Guide and InterventionIsosceles Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
© Glencoe/McGraw-Hill 213 Glencoe Geometry
Less
on
4-6
Properties of Isosceles Triangles An isosceles triangle has two congruent sides.The angle formed by these sides is called the vertex angle. The other two angles are calledbase 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. If A�B� � C�B�, then �A � �C.
If �A � �C, then A�B� � C�B�.
A
B
C
Find x.
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.
B
A
C (4x � 5)�
(5x � 10)�
Find x.
m�S � m�T, soSR � TR. Converse of Isos. � Thm.
3x � 13 � 2x Substitution
3x � 2x � 13 Add 13 to each side.
x � 13 Subtract 2x from each side.
R T
S
3x � 13
2x
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find x.
1. 2. 3.
4. 5. 6.
7. Write a two-column proof.Given: �1 � �2Prove: A�B� � C�B�
Statements Reasons
B
A C D
E
1 32
R S
T3x�
x�D
BG
L3x�
30�D
T Q
PK
(6x � 6)� 2x�
W
Y Z3x�
S
V
T 3x � 6
2x � 6R
P
Q2x �
40�
© Glencoe/McGraw-Hill 214 Glencoe Geometry
Properties of Equilateral Triangles An equilateral triangle has three congruentsides. The Isosceles Triangle Theorem can be used to prove two properties of equilateraltriangles.
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; P�Q� || B�C�. 1. Given2. m�A � m�B � m�C � 60 2. Each � of an equilateral � measures 60°.3. �1 � �B, �2 � �C 3. If || lines, then corres. �s are �.4. m�1 � 60, m�2 � 60 4. Substitution5. �APQ is equilateral. 5. If a � is equiangular, then it is equilateral.
Find x.
1. 2. 3.
4. 5. 6.
7. Write a two-column proof.Given: �ABC is equilateral; �1 � �2.Prove: �ADB � �CDB
Proof:
Statements Reasons
A
D
C
B12
R
O
HM 60�4x�
X
Z Y4x � 4
3x � 8 60�
P Q
LV R60�
4x 40
L
N
M
K
�KLM is equilateral.
3x�G
J H
6x � 5 5x
D
F E6x�
A
B
P Q
C
1 2
Study Guide and Intervention (continued)
Isosceles Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
ExampleExample
ExercisesExercises
Skills PracticeIsosceles Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
© Glencoe/McGraw-Hill 215 Glencoe Geometry
Less
on
4-6
Refer to the figure.
1. If A�C� � A�D�, name two congruent angles.
2. If B�E� � B�C�, name two congruent angles.
3. If �EBA � �EAB, name two congruent segments.
4. If �CED � �CDE, name two congruent segments.
�ABF is isosceles, �CDF is equilateral, and m�AFD � 150.Find each measure.
5. m�CFD 6. m�AFB
7. m�ABF 8. m�A
In the figure, P�L� � R�L� and L�R� � B�R�.
9. If m�RLP � 100, find m�BRL.
10. If m�LPR � 34, find m�B.
11. Write a two-column proof.
Given: C�D� � C�G�D�E� � G�F�
Prove: C�E� � C�F�
DE
FG
C
R P
BL
D
C
F
B
35�
A E
D
C
B
A E
© Glencoe/McGraw-Hill 216 Glencoe Geometry
Refer to the figure.
1. If R�V� � R�T�, name two congruent angles.
2. If R�S� � S�V�, name two congruent angles.
3. If �SRT � �STR, name two congruent segments.
4. If �STV � �SVT, name two congruent segments.
Triangles GHM and HJM are isosceles, with G�H� � M�H�and H�J� � M�J�. Triangle KLM is equilateral, and m�HMK � 50.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. Write a two-column proof.
Given: D�E� || B�C��1 � �2
Prove: A�B� � A�C�
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. Lincoln Hawks
E
D B
C
A1
23
4
G
M
LK
J
H
U
R
TV
S
Practice Isosceles Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
Reading to Learn MathematicsIsosceles Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
© Glencoe/McGraw-Hill 217 Glencoe Geometry
Less
on
4-6
Pre-Activity How are triangles used in art?
Read the introduction to Lesson 4-6 at the top of page 216 in your textbook.
• Why do you think that isosceles and equilateral triangles are used moreoften than scalene triangles in art?
• Why might isosceles right triangles be used in art?
Reading the Lesson1. Refer to the figure.
a. What kind of triangle is �QRS?
b. Name the legs of �QRS.
c. Name the base of �QRS.
d. Name the vertex angle of �QRS.
e. Name the base angles of �QRS.
2. Determine whether each statement is always, sometimes, or never true.
a. If a triangle has three congruent sides, then it has three congruent angles.
b. If a triangle is isosceles, then it is equilateral.
c. If a right triangle is isosceles, then it is equilateral.
d. The largest angle of an isosceles triangle is obtuse.
e. If a right triangle has a 45° angle, then it is isosceles.
f. If an isosceles triangle has three acute angles, then it is equilateral.
g. The vertex angle of an isosceles triangle is the largest angle of the triangle.
3. Give the measures of the three angles of each triangle.
a. an equilateral triangle
b. an isosceles right triangle
c. an isosceles triangle in which the measure of the vertex angle is 70
d. an isosceles triangle in which the measure of a base angle is 70
e. an isosceles triangle in which the measure of the vertex angle is twice the measure ofone of the base angles
Helping You Remember4. If a theorem and its converse are both true, you can often remember them most easily by
combining them into an “if-and-only-if” statement. Write such a statement for the IsoscelesTriangle Theorem and its converse.
R
Q
S
© Glencoe/McGraw-Hill 218 Glencoe Geometry
Triangle ChallengesSome problems include diagrams. If you are not sure how to solve theproblem, begin by using the given information. Find the measures of as manyangles as you can, writing each measure on the diagram. This may give youmore clues to the solution.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-64-6
1. Given: BE � BF, �BFG � �BEF ��BED, m�BFE � 82 andABFG and BCDE each haveopposite sides parallel andcongruent.
Find m�ABC.
3. Given: m�UZY � 90, m�ZWX � 45,�YZU � �VWX, UVXY is asquare (all sides congruent, allangles right angles).
Find m�WZY.
2. Given: AC � AD, and A�B��B�D�,m�DAC � 44 andC�E� bisects �ACD.
Find m�DEC.
4. Given: m�N � 120, J�N� � M�N�,�JNM � �KLM.
Find m�JKM.J
K
L
MN
A
D C
BE
U VW
XYZ
A
G DF E
CB
Study Guide and InterventionTriangles and Coordinate Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
© Glencoe/McGraw-Hill 219 Glencoe Geometry
Less
on
4-7
Position and Label Triangles A coordinate proof uses points, distances, and slopes toprove geometric properties. The first step in writing a coordinate proof is to place a figure onthe 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 the computations as simple as possible.
Position an equilateral 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 �a2�. Use b for the y-coordinate,
so the vertex is S��a2�, b�.
Find the missing coordinates of each triangle.
1. 2. 3.
Position and label each triangle on the coordinate plane.
4. isosceles triangle 5. isosceles right �DEF 6. equilateral triangle �EQI�RST with base R�S� with legs e units long with vertex Q(0, a) and4a units long sides 2b units long
x
y
I(b, 0)E(–b, 0)
Q(0, a)
x
y
E(e, 0)
F(e, e)
D(0, 0)x
y T(2a, b)
R(0, 0) S(4a, 0)
x
y
G(2g, 0)
F(?, b)
E(?, ?)x
y
S(2a, 0)
T(?, ?)
R(0, 0)x
y
B(2p, 0)
C(?, q)
A(0, 0)
x
y
T(a, 0)R(0, 0)
S�a–2, b�
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 220 Glencoe Geometry
Write Coordinate Proofs Coordinate proofs can be used to prove theorems and toverify properties. Many coordinate proofs use the Distance Formula, Slope Formula, orMidpoint 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 R�T�.
Given: Isosceles �RST; U is the midpoint of base R�T�.Prove: S�U� ⊥ R�T�
Proof:U is the midpoint of R�T� so the coordinates of U are ���a
2� a�, �
0 �2
0�� � (0, 0). Thus S�U� lies on
the y-axis, and �RST was placed so R�T� lies on the x-axis. The axes are perpendicular, so S�U� ⊥ R�T�.
Prove that the segments joining the midpoints of the sides of a right triangle forma right triangle.
C(2a, 0)
B(0, 2b)
P
Q
R
A(0, 0)
x
y
T(a, 0)U(0, 0)R(–a, 0)
S(0, c)
Study Guide and Intervention (continued)
Triangles and Coordinate Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
ExampleExample
ExercisesExercises
Skills PracticeTriangles and Coordinate Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
© Glencoe/McGraw-Hill 221 Glencoe Geometry
Less
on
4-7
Position and label each triangle on the coordinate plane.
1. right �FGH with legs 2. isosceles �KLP with 3. isosceles �AND witha units and b units base K�P� 6b units long base A�D� 5a long
Find the missing coordinates of each triangle.
4. 5. 6.
7. 8. 9.
10. Write a coordinate proof to prove that in an isosceles right triangle, the segment fromthe vertex of the right angle to the midpoint of the hypotenuse is perpendicular to thehypotenuse.
Given: isosceles right �ABC with �ABC the right angle and M the midpoint of A�C�Prove: B�M� ⊥ A�C�
C(2a, 0)
A(0, 2a)
M
B(0, 0)
x
y
U(a, 0)
T(?, ?)
S(–a, 0)x
y
P(7b, 0)
R(?, ?)
N(0, 0)x
y
Q(?, ?)
R(2a, b)
P(0, 0)
x
y
N(3b, 0)
M(?, ?)
O(0, 0)x
y
Y(2b, 0)
Z(?, ?)
X(0, 0)x
y
B(2a, 0)
A(0, ?)
C(0, 0)
x
yN �5–
2a, b�
A(0, 0) D(5a, 0)x
y L(3b, c)
K(0, 0) P(6b, 0)x
y
F(0, a)
G(0, 0) H(b, 0)
© Glencoe/McGraw-Hill 222 Glencoe Geometry
Position and label each triangle on the coordinate plane.
1. equilateral �SWY with 2. isosceles �BLP with 3. isosceles right �DGJsides �
14�a long base B�L� 3b units long with hypotenuse D�J� and
legs 2a units long
Find the missing coordinates of each triangle.
4. 5. 6.
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 parttime at the mall in a music store. The mall is 2 miles west and 3 miles north of the school.
7. Write a coordinate proof to prove that Karina’s high school, her home, and the mall areat the vertices of a right triangle.
Given: �SKMProve: �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)
x
y
P(2b, 0)
M(0, ?)
N(?, 0)x
y
C(?, 0)
E(0, ?)
B(–3a, 0)x
y S(?, ?)
J(0, 0) R�1–3b, 0�
x
yD(0, 2a)
G(0, 0) J(2a, 0)x
yP �3–
2b, c�
B(0, 0) L(3b, 0)x
y Y �1–8a, b�
W �1–4a, 0�S(0, 0)
Practice Triangles and Coordinate Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
Reading to Learn MathematicsTriangles and Coordinate Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
© Glencoe/McGraw-Hill 223 Glencoe Geometry
Less
on
4-7
Pre-Activity How can the coordinate plane be useful in proofs?
Read the introduction to Lesson 4-7 at the top of page 222 in your textbook.
From the coordinates of A, B, and C in the drawing in your textbook, whatdo you know about �ABC?
Reading the Lesson
1. Find the missing coordinates of each triangle.
a. b.
2. Refer to the figure.
a. Find the slope of S�R� and the slope of S�T�.
b. Find the product of the slopes of S�R� and S�T�. What does this tell you about S�R� and S�T�?
c. What does your answer from part b tell you about �RST ?
d. Find SR and ST. What does this tell you about S�R� and S�T�?
e. What does your answer from part d tell you about �RST?
f. Combine your answers from parts c and e to describe �RST as completely as possible.
g. Find m�SRT and m�STR.
h. Find m�OSR and m�OST.
Helping You Remember
3. Many students find it easier to remember mathematical formulas if they can put theminto words in a compact way. How can you use this approach to remember the slope andmidpoint formulas easily?
x
y
S(0, a)
R(–a, 0) T(a, 0)O(0, 0)
x
y
F(?, ?)E(?, a)
D(?, ?)x
y
T(a, ?)
R(?, b)
S(?, ?)
© Glencoe/McGraw-Hill 224 Glencoe Geometry
How Many Triangles?Each puzzle below contains many triangles. Count them carefully.Some triangles overlap other triangles.
How many triangles are there in each figure?
1. 2. 3.
4. 5. 6.
How many triangles can you form by joining points on each circle? List the vertices of each triangle.
7. 8.
8. 9. QR
P
U
S
TV
J K
O
L
MN
E F
I
GH
B
C
DA
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
4-74-7
Chapter 4 Test, Form 144
© Glencoe/McGraw-Hill 225 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of eachquestion.
1. How would this triangle be classified by angles? A. acute B. equiangularC. obtuse D. right
2. What is the value of x if �ABC is equilateral?
A. �8 B. ��18�
C. �12� D. 2
Use the figure for Questions 3–4 and write the letter for the correct answer in the blank at the right of each question.
3. What is m�2?A. 50 B. 70 C. 110 D. 120
4. What is m�4?A. 10 B. 60 C. 100 D. 120
5. What are the congruent triangles in the diagram? A. �ABC � �EBD B. �ABE � �CBDC. �AEB � �CBD D. �ABE � �CDB
6. If �CJW � �AGS, m�A � 50, m�J � 45,and m�S � 16x � 5, what is x? A. 17.5 B. 11.875C. 6 D. 5
7. Which postulate can be used to prove the triangles congruent? A. SSS B. SAS C. ASA D. AAS
8. What reason should be given for statement 5 in the proof?Given: D�B� is the perpendicular bisector of A�C�.Prove: �ADB � �CDB
Statements Reasons1. DB is the perpendicular bisector of A�C�. 1. Given2. A�B� � C�B� 2. Midpoint Theorem3. �ABD � �CBD 3. ⊥ line; all right � are �.4. D�B� � D�B� 4. Reflexive Property5. �ADB � �CDB 5.
A. SSS B. AAS C. ASA D. SAS
?
A C
D
B
C W
J
(16x � 5)�
45�
A S
G
50�
A
E
CB
D
A C
B
10x � 5
6x � 37.5x
1.
2.
3.
4.
5.
6.
7.
8.
NAME DATE PERIOD
SCORE
70�
2 13 460�
40�
© Glencoe/McGraw-Hill 226 Glencoe Geometry
Chapter 4 Test, Form 1 (continued)44
9.
10.
11.
12.
13.
Use the proof for Questions 9–10 and write the letter for the correct answer in the blank at theright of each question.
Given: L is the midpoint of J�M�; J�K� || N�M�.Prove: �JKL � �MNLStatements Reasons
1. L is the midpoint of J�M�. 1. Given2. J�L� � M�L� 2. Definition of midpoint3. J�K� || M�N� 3. Given4. �JKL � �MNL 4. Alt. int. � are �.5. �JLK � �MLN 5.6. �JKL � �MNL 6.
9. What is the reason for �JLK � �MLN?A. definition of midpointB. corresponding anglesC. vertical anglesD. alternate interior angles
10. What is the reason for �JKL � �MNL?A. AAS B. ASA C. SAS D. SSS
Use the figure for Questions 11–12 and write the letter for the correct answer in the blank at the right of each question.
11. If �LMN is isosceles and T is the midpoint of L�N�,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?A. CPCTCB. definition of isosceles triangleC. definition of perpendicularD. definition of angle bisector
13. What are the missing coordinates of this triangle?A. (2a, 2c) B. (2a, 0)C. (0, 2a) D. (a, 2c)
Bonus What is the classification by sides of a triangle with coordinates A(5, 0), B(0, 5), and C(�5, 0)?
x
yM(a, c)
N(?, ?)L(0, 0)
1 2
M
TL N
(Question 10)(Question 9)
N
K
L MJ
B:
NAME DATE PERIOD
Chapter 4 Test, Form 2A44
© Glencoe/McGraw-Hill 227 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of eachquestion.
1. What is the length of the sides of this equilateral triangle?A. 42 B. 30C. 15 D. 12
2. What is the classification of �ABC with vertices A(4, 1), B(2, �1), and C(�2, �1) by its sides?A. equilateral B. isosceles C. scalene D. right
Use the figure for Questions 3–4 and write the letter for the correct answer in the blank at the right ofeach question.
3. What is m�1?A. 40 B. 50 C. 70 D. 90
4. What is m�3?A. 40 B. 70 C. 90 D. 110
5. If �DJL � �EGS, which segment in �EGS corresponds to D�L�?A. E�G� B. E�S� C. G�S� D. G�E�
6. Which triangles are congruent in the figure?A. �KLJ � �MNL B. �JLK � �NLMC. �JKL � �LMN D. �JKL � �MNL
Use the proof for Questions 7–8 and write the letter for the correct answer in the blank at the right of each question.
Given: R�J� || E�I�; R�I� bisects J�E�.Prove: �RJN � �IENStatements Reasons
1. R�J� || I�E� 1. Given2. �RJN � �IEN 2.3. R�I� bisects J�E�. 3. Given4. J�N� � E�N� 4. Definition of bisector5. �RNJ � �INE 5. Vert. � are �.6. �RJN � �IEN 6.
7. What is the reason for statement 2 in the proof?A. Isosceles Triangle Theorem B. same side interior anglesC. corresponding angles D. Alternate Interior Angle Theorem
8. What is the reason for statement 6?A. ASA B. AAS C. SAS D. SSS
(Question 8)
(Question 7)
E N
R
J
I
LK
NJ
M
70�
50�1
2
3
6x � 3
9x � 123x � 61.
2.
3.
4.
5.
6.
7.
8.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 228 Glencoe Geometry
Chapter 4 Test, Form 2A (continued)44
9.
10.
11.
12.
13.
9. If �ABC is isosceles and A�E� � F�C�, which theorem or postulate can be used to prove �AEB � �CFB? A. SSS B. SASC. ASA D. AAS
Use the proof for Questions 10–11 and write the letter for the correct answer in the blank at the right of each question.
Given: D�A� || Y�N�; D�A� � Y�N�Prove: �NDY � �DNAStatements Reasons
1. D�A� || Y�N� 1. Given2. �ADN � �YND 2. Alt. int. � are �.3. D�A� � Y�N� 3. Given4. D�N� � D�N� 4. Reflexive Property5. �NDY � �DNA 5.6. �NDY � �DNA 6.
10. What is the reason for statement 5?A. ASA B. AASC. SAS D. SSS
11. What is the reason for statement 6?A. Alt. int. �s are �. B. CPCTCC. Corr. angles are �. D. Isosceles Triangle Theorem
12. What is the classification of a triangle with vertices A(3, 3), B(6, �2), C(0, �2)by its sides?A. isosceles B. scaleneC. equilateral D. right
13. What are the missing coordinates of the triangle?A. (�2b, 0) B. (0, 2b)C. (�c, 0) D. (0, �c)
Bonus Name the coordinates of points A and C in isosceles right �ABC if point Cis in the second quadrant.
x
y
B(0, a)
A(?, ?)
x
y(0, c)
(?, ?) (2b, 0)
(Question 11)(Question 10)
D A
NY
A C
B
E F
B:
NAME DATE PERIOD
Chapter 4 Test, Form 2B44
© Glencoe/McGraw-Hill 229 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of eachquestion.
1. What is the length of the sides of this equilateral triangle? A. 2.5 B. 5C. 15 D. 20
2. What is the classification of �ABC with vertices A(0, 0), B(4, 3), and C(4, �3)by its sides?A. equilateral B. isosceles C. scalene D. right
Use the figure for Questions 3–4 and write the letter for the correct answer in the blank at the right ofeach question.
3. What is m�1?A. 120 B. 90 C. 60 D. 30
4. What is m�2?A. 120 B. 90 C. 60 D. 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?A. �HMN � �HGN B. �HMN � �NGHC. �NMH � �NGH D. �MNH � �HGN
Use the proof for Questions 7–8 and write the letter for the correct answer in the blank at the right of each question.
Given: A�B� || C�D�; A�C� bisects B�D�.Prove: �ABE � �CDEStatements Reasons1. A�C� bisects B�D�. 1. Given2. B�E� � D�E� 2.3. A�B� || C�D� 3. Given4. �ABE � �CDE 4. Alt. int. � are �.5. 5.Vert. � are �.6. �ABE � �CDE 5. ASA
7. What is the reason for statement 2?A. Definition of bisector B. Midpoint TheoremC. Given D. Alternate Interior Angle Theorem
8. What is the statement for reason 5?A. �BEA � �DEC B. �ABE � �CDEC. �EAB � �ECD D. �BEC � �DEA
(Question 8)
(Question 7)
AB
CD
E
H G
NM
120�12
7x � 15
3x � 54x1.
2.
3.
4.
5.
6.
7.
8.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 230 Glencoe Geometry
Chapter 4 Test, Form 2B (continued)44
9.
10.
11.
12.
13.
9. If A�F� � D�E�, A�B� � F�C� and A�B� || F�C�, which theorem or postulate can be used to prove �ABE � �FCD?A. AAS B. ASAC. SAS D. SSS
Use the proof for Questions 10–11 and write the letter for the correct answer in the blank at the right of each question.
Given: E�G� � I�A�; �EGA � �IAGProve: �GEN � �AINStatements Reasons
1. E�G� � I�A� 1. Given
2. �EGA � �IAG 2. Given
3. G�A� � G�A� 3. Reflexive Property
4. �EGA � �IAG 4.
5. �GEN � �AIN 5.
10. What is the reason for statement 4?A. SSS B. ASA C. SAS D. AAS
11. What is the reason for statement 5?A. Alt. int. � are �. B. Same Side Interior AnglesC. Corr. angles are �. D. CPCTC
12. What is the classification of a triangle with vertices A(�3, �1), B(�2, 2),C(3, 1) by its sides?A. scalene B. isoscelesC. equilateral D. right
13. What are the missing coordinates of the triangle?A. (a, 0) B. (b, 0)C. (c, 0) D. (0, c)
Bonus Find x in the triangle.(5x � 60)� (2x � 51)�
(30 � 10x)�(43 � 2x)�
x
y
(?, ?)
(a, 0)(�a, 0)
(Question 11)
(Question 10)
G A
N
E I
A DF E
CB
B:
NAME DATE PERIOD
Chapter 4 Test, Form 2C44
© Glencoe/McGraw-Hill 231 Glencoe Geometry
Ass
essm
ents
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 thetriangle by its sides.
Find the measure of each angle.
4. m�1
5. m�2
6. m�3
7. Identify the congruent triangles and name their correspondingcongruent angles.
8. Verify that �ABC � �A�B�C�preserves congruence, assuming that corresponding angles arecongruent.
x
y
O
A�
B�
B
C�
AC
A
D F
GB
C
110� 2
1
3
A C
B
8x
7x � 310x � 6
1.
2.
3.
4.
5.
6.
7.
8.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 232 Glencoe Geometry
Chapter 4 Test, Form 2C (continued)44
9. ABCD is a quadrilateral with A�B� � C�D� and A�B� || C�D�. Name the postulate that could be used to prove �BAC � �DCA. Choose from SSS, SAS, ASA, and AAS.
10. �KLM is an isosceles triangle and �1 � �2. Name the theorem that could be used to determine �LKP � �LMN. Then name thepostulate that could be used to prove �LKP � �LMN. Choose from SSS, SAS, ASA, and AAS.
11. Use the figure to find m�1.
12. Find x.
13. Position and label isosceles �ABC with base A�B� b units longon the coordinate plane.
14. C�P� joins point C in isosceles right �ABC to the midpoint P, of A�B�.Name the coordinates of P. Thendetermine the relationship between A�B� and C�P�.
Bonus Without finding any other angles or sides congruent,which pair of triangles can be proved to be congruent bythe HL Theorem?
A
B
C D
E
F X
Y
Z M
N
O
x
yA(0, b)
B(b, 0)C(0, 0)
(10x � 20)�15x �
(18x � 12)�
40�
190�
1
P
1 2
N MK
L
A
D
B
C
NAME DATE PERIOD
9.
10.
11.
12.
13.
14.
B:
C(b–2, c)
B(b, 0)A
Chapter 4 Test, Form 2D44
© Glencoe/McGraw-Hill 233 Glencoe Geometry
Ass
essm
ents
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.
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 thetriangle by its sides.
Find the measure of each angle.
4. m�1
5. m�2
6. m�3
7. Identify the congruent triangles and name their corresponding congruentangles.
8. Verify that �JKL � �J�K�L�preserves congruence, assuming that corresponding angles arecongruent.
9. In quadrilateral JKLM, J�K� � L�K�and M�K� bisects �LKJ. Name thepostulate that could be used to prove �MKL � �MKJ. Choose from SSS, SAS, ASA, and AAS.
M
K
J
L
x
y
O
K�J�
K
L�
J
L
DB
EC
FA
70�1 32
80�
A C
B
2x � 20
9x � 5
5x � 5
1.
2.
3.
4.
5.
6.
7.
8.
9.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 234 Glencoe Geometry
Chapter 4 Test, Form 2D (continued)44
10. �ABC is an isosceles triangle with B�D� ⊥ A�C�. 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. Use the figure to find m�1.
12. Find x.
13. Position and label equilateral �KLM with side lengths 3a units long on the coordinate plane.
14. M�N� joins the midpoint of A�B� and the midpoint of A�C� in �ABC. Findthe coordinates of M and N, and theslopes of M�N� and B�C�.
Bonus Without finding any other angles or sides congruent,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
NAME DATE PERIOD
10.
11.
12.
13.
14.
B:
M(3a, 0)
L(1.5a, b)
K(0, 0)
Chapter 4 Test, Form 344
© Glencoe/McGraw-Hill 235 Glencoe Geometry
Ass
essm
ents
1. If �ABC is isosceles, �B is the vertex angle, AB � 20x � 2,BC � 12x � 30, and AC � 25x, find x and the measure of eachside of the triangle.
2. Given A(0, 4), B(5, 4), and C(�3, �2), find the measure of thesides of the triangle. Then classify the triangle by its sides and angles.
Use the figure to answer Questions 3–5.
3. Find x.
4. m�1, if m�1 � 4x � 10.
5. m�2
6. Verify that the following preserves congruence, assuming thatcorresponding angles are congruent. �ABC is reflected overthe x-axis as follows.A(�1, 1) → A�(�1, �1)B(4, 2) → B�(4, �2)C(1, 5) → C�(1, �5)Verify �ABC � �A�B�C�.
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, A�C� � F�D�, A�B� || D�E�,and A�C� || F�D�. Name the postulate that could be used to prove �ABC � �DEC. 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.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 236 Glencoe Geometry
Chapter 4 Test, Form 3 (continued)44
For Questions 9 and 10, complete this two-column proof.
Given: �ABC is an isosceles triangle with base A�C�.D is the midpoint of A�C�.
Prove: B�D� bisects �ABC.
Statements Reasons1. �ABC is isosceles 1. Given
with base A�C�.
2. A�B� � C�B� 2. Def. of isosceles triangle.
3. �A � �C 3.
4. D is the midpoint of A�C�. 4. Given
5. A�D� � C�D� 5. Midpoint Theorem
6. �ABD � �CBD 6.
7. �1 � �2 7. CPCTC
8. B�D� bisects �ABC. 8. Def. of angle bisector
11. Find x.
12. Position and label isosceles �ABC with base A�B� (a � b) unitslong on a coordinate plane
Bonus In the figure, �ABC is isosceles, �ADC is equilateral,�AEC is isosceles, and the measures of �9, �1, and �3are all equal. Find the measures of the nine numberedangles.
987
1
65
23
4
B
C
DE
A
(15x � 15)�(21x � 3)�
(17x � 9)�
(Question 10)
(Question 9)
21
A C
B
D
NAME DATE PERIOD
9.
10.
11.
12.
B:
B(a � b, 0)
C(a � b, c)2
A(0, 0)
Chapter 4 Open-Ended Assessment44
© Glencoe/McGraw-Hill 237 Glencoe Geometry
Ass
essm
ents
Demonstrate your knowledge by giving a clear, concise solution toeach problem. Be sure to include all relevant drawings and justifyyour answers. You may show your solution in more than one way orinvestigate 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. State the theorem or postulate that can be used to prove that thetriangles are congruent.
b. List their corresponding congruent angles and sides.
5.
Given: A�B� || D�E�, A�D� bisects B�E�.Prove: �ABC � �DEC by 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)�
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 238 Glencoe Geometry
Chapter 4 Vocabulary Test/Review44
Choose from the terms above to complete each sentence.
1. A triangle that is equilateral is also called a(n) .
2. A(n) has at least one obtuse angle.
3. The sum of the is equivalent to the exterior angle of atriangle.
4. The angles of an isosceles triangle are congruent.
5. A triangle with different measures for each side is classified asa(n) .
6. A organizes a series of statements in logical orderwritten in boxes and uses arrows to indicate the order of thestatements.
7. A triangle that is translated, reflected or rotated and preservesits shape, is said to be a(n) .
8. The ASA postulate involves two corresponding angles andtheir corresponding .
9. A uses figures in the coordinate plane and algebra toprove geometric concepts.
10. The is formed by the congruent legs of an isoscelestriangle.
In your own words—
11. corollary
12. congruent triangles
13. acute triangle
?
?
?
?
?
?
?
?
?
? 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
acute trianglebase anglescongruence
transformationscongruent triangles
coordinate proofcorollaryequiangular triangleequilateral triangleexterior angle
flow proofincluded angleincluded sideisosceles triangleobtuse triangle
remote interior anglesright trianglescalene trianglevertex angle
NAME DATE PERIOD
SCORE
Chapter 4 Quiz (Lessons 4–1 and 4–2)
44
© Glencoe/McGraw-Hill 239 Glencoe Geometry
Ass
essm
ents
NAME DATE PERIOD
SCORE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
1. Use a protractor to classify the triangle by its angles and sides.
2. STANDARDIZED TEST PRACTICE What is the best classification of this triangle by its angles and sides?A. acute isosceles B. right isoscelesC. 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 x and themeasures of each side of the triangle.
4. If A(1, 5), B(3, �2), and C(�3, 0), find the measures of thesides of �ABC. Then classify the triangle by its 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
70�65�
1
2 34
56 107�
43�
Chapter 4 Quiz (Lessons 4–3 and 4–4)
44
1.
2.
3.
4.
1. Identify the congruent triangles in the figure.
2. STANDARDIZED TEST PRACTICE If �JGO � �RWI, whichangle corresponds to �I?A. �J B. �R C. �G D. �O
3. Verify that the following preserves congruence assuming that correspondingangles are congruent. �ABC � �A�B�C�
4. In quadrilateral EFGH, F�G� � H�E�, and F�G� || H�E�. Name the postulate that could be used to prove �EHF � �GFH. Choosefrom SSS, SAS, ASA, and AAS.
E H
GF
x
y
OC�
BB�
AC
A�
K
MLN
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 240 Glencoe Geometry
Chapter 4 Quiz (Lessons 4–5 and 4–6)
44
1.
2.
3.
4.
For Questions 1 and 2, complete the two-column proof bysupplying the missing information for each correspondinglocation.
Given: �Z � �C; A�K� bisects �ZKC.Prove: �AKZ � �AKCStatements Reasons
1. �Z � �C; A�K� bisects �ZKC. 1. Given2. �ZKA � �CKA 2.3. A�K� � A�K� 3. Reflexive Property4. �AKZ � �AKC 4.
Refer to the figure for Questions 3 and 4.
3. Find m�1. 4. Find m�2.21
(Question 2)
(Question 1)
A
K
Z C
NAME DATE PERIOD
SCORE
Chapter 4 Quiz (Lesson 4–7)
44
1.
2.
3.
4.
5.
1. Find the missing coordinates.
Position and label each triangle on a coordinate plane.
2. Right �DJL with hypotenuse D�J�; LJ � �12�DL and D�L� is
a units long.
3. isosceles �EGS with base E�S� �12�b units long
For Questions 4 and 5, complete the coordinate proof bysupplying the missing information for each correspondinglocation.
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 asfollows: . Since , �ABC is isosceles.(Question 5)(Question 4)
x
y
C(?, ?)
I(?, ?)
M(�b, 0)
S(1–2b, 0)
G(1–4b, c)
E(0, 0)
J(a–2, 0)
D(0, a)
L(0, 0)
NAME DATE PERIOD
SCORE
Chapter 4 Mid-Chapter Test (Lessons 4–1 through 4–3)
44
© Glencoe/McGraw-Hill 241 Glencoe Geometry
Ass
essm
ents
1. What is the best classification for this triangle?A. acute scaleneB. obtuse equilateralC. acute isoscelesD. obtuse isosceles
Find the missing angle measures.
2. What is m�1?A. 50 B. 60C. 100 D. 105
3. What is m�2?A. 40 B. 50C. 60 D. 100
4. If �SJL � �DMT, which segment in �DMT corresponds to L�S� in �SJL?A. D�T� B. T�D�C. M�D� D. M�T�
50�
50�
60�
2
1
5.
6.
7.
NAME DATE PERIOD
SCORE
1.
2.
3.
4.
Part II
5. Find the measures of the sides of �ABC and classify it by itssides. A(1, 3), B(5, �2), and C(0, �4)
6. In �ABC and �A�B�C�, �A � �A�,�B � �B�, and �C � �C�. Find the lengths needed to prove �ABC � �A�B�C�.
7. What information would you need to know about P�O� and L�N� for �LMPto be congruent to �NMO by SSS? P O
N
L
M
x
y
O
C�
B
B�A
CA�
Part I Write the letter for the correct answer in the blank at the right of each question.
© Glencoe/McGraw-Hill 242 Glencoe Geometry
Chapter 4 Cumulative Review(Chapters 1–4)
44
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
1. Name the geometric figure that is modeled by the second handof a clock. (Lesson 1-1)
2. Find the precision for a measurement of 36 inches. (Lesson 1-2)
For Questions 3–5, use the number line.
3. Find BC. (Lesson 1-3)
4. Find the coordinate of the midpoint of A�D�. (Lesson 1-3)
5. If B is the midpoint of a segment having one endpoint at E,what is the coordinate of its other endpoint? (Lesson 1-3)
For Questions 6 and 7, determine whether each statementis always, sometimes, or never true. Explain your answer.(Lesson 2-5)
6. If D�E� � E�F�, then E is the midpoint of D�F�.
7. If points A and B lie in plane Q , then AB��� lies in Q .
8. Find the slope of a line parallel to x � 2. (Lesson 3-3)
9. Find the distance between y � �9 and y � �5. (Lesson 3-6)
For Questions 10–12, use the figure.
10. Name the segment that represents the distance from F to AD���. (Lesson 3-6)
11. Classify �ADC. (Lesson 4-1)
12. Find m�ACD. (Lesson 4-2)
13. Name the corresponding congruent angles and sides for �PQR � �HGB. (Lesson 4-3)
14. If �QRP � �SRT, and R is the midpoint of P�T�, which theorem or postulate can beused to prove �QRP � �SRT? Choose from SSS, SAS, ASA, and AAS. (Lesson 4-5)
15. Name the missing coordinates of �GEF. (Lesson 4-7)
x
y
F(2b, ?)D(0, 0) G(?, ?)
E(?, ?)
Q S
P TR
50�30�
85�
ED
A CB
F
�5 �4 �3 �2 �1�10 �9 �8 �7 �6 0 1 2 3 4 5 6
B ECA D
NAME DATE PERIOD
SCORE
Standardized Test Practice (Chapters 1–4)
© Glencoe/McGraw-Hill 243 Glencoe Geometry
1. If m�1 � 5x � 4, and m�2 � 52 � 9y, which values for x and ywould make �1 and �2 complementary? (Lesson 1-5)
A. x � 2, y � 12 B. x � 12, y � 2
C. x � 27, y � �13� D. x � �
13�, y � 27
2. Which is not a polygon? (Lesson 1-6)
E. F. G. H.
3. Complete the statement so that its conditional and its converseare true.If �1 � �2, then �1 and �2 . (Lesson 2-3)
A. are supplementary. B. are complementary.C. have the same measure. D. are alternate interior angles.
4. Complete this proof. (Lesson 2-7)
Given: U�V� � V�W�V�W� � W�X�
Prove: UV � WXProof:Statements Reasons
1. U�V� � V�W�; V�W� � W�X� 1. Given
2. UV � VW; VW � WX 2.3. UV � WX 3. Transitive Property
E. Definition of congruent segmentsF. Substitution PropertyG. Segment Addition PostulateH. Symmetric Property
5. Which equation has a slope of �13� and a y-intercept of �2? (Lesson 3-4)
A. y � �13�x � 2 B. y � �
13�x � 2
C. y � 2x � �13� D. y � �2x � �
13�
6. Classify �DEF with vertices D(2, 3), E(5, 7) and F(9, 4). (Lesson 4-1)
E. acute F. equiangular G. obtuse H. right
7. Which postulate or theorem can be used to prove �ABD � �CBD? (Lesson 4-4)
A. SAS B. SSSC. ASA D. AAS
A CB
D
?
U V
XW
?
NAME DATE PERIOD
SCORE 44
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
1.
2.
3.
4.
5.
6.
7. A B C D
E F G H
A B C D
E F G H
A B C D
E F G H
A B C D
Ass
essm
ents
© Glencoe/McGraw-Hill 244 Glencoe Geometry
Standardized Test Practice (continued)
8. What is the y-coordinate of the midpoint ofA(12, 6) and B(�15, �6)? (Lesson 1-3)
9. If m�1 � 112, find m�10.(Lesson 3-2)
10. If J�K� || L�M�, then �4 must be supplementary to� . (Lesson 3-5)
11. Find PR if �PQR is isosceles, �Q is the vertexangle, PQ � 4x � 8, QR � x � 7, and PR � 6x � 12. (Lesson 4-1)
?
6
7
4
5H
L
M
J
K
�k
m
n
1 23
9 1011 12
45 6
7 8
NAME DATE PERIOD
44
Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.
Part 3: Short Response
Instructions: Show your work or explain in words how you found your answer.
8. 9.
10. 11.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
12. The perimeter of a regular pentagon is 14.5 feet. If each sidelength of the pentagon is doubled, what is the new perimeter?(Lesson 1-6)
13. Make a conjecture about the next number in the sequence 5, 7,11, 17, 25. (Lesson 2-1)
14. Find m�PQR. (Lesson 4-2)
15. If PQ � QS, QS � SR, and m�R � 20, find m�PSQ.(Lesson 4-6) P RS
Q
63� 125� 10�P S
Q
RT
12.
13.
14.
15.
0 6 8
6 1 8
Standardized Test PracticeStudent Record Sheet (Use with pages 232–233 of the Student Edition.)
44
© Glencoe/McGraw-Hill A1 Glencoe Geometry
An
swer
s
Select the best answer from the choices given and fill in the corresponding oval.
1 4 7
2 5 8
3 6 DCBADCBA
DCBADCBADCBA
DCBADCBADCBA
NAME DATE PERIOD
Part 1 Multiple ChoicePart 1 Multiple Choice
Part 2 Short Response/Grid InPart 2 Short Response/Grid In
Part 3 Open-EndedPart 3 Open-Ended
Solve the problem and write your answer in the blank.
For Questions 12 and 14, also enter your answer by writing each number orsymbol in a box. Then fill in the corresponding oval for that number or symbol.
9 12 14
10
11
12 (grid in)
13
14 (grid in)
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
Record your answers for Questions 15–16 on the back of this paper.
© Glencoe/McGraw-Hill A2 Glencoe Geometry
Stu
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Cla
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e is
by
the
mea
sure
s of
its
an
gles
.
•If
one
of t
he a
ngle
s of
a t
riang
le is
an
obtu
se a
ngle
, th
en t
he t
riang
le is
an
ob
tuse
tri
ang
le.
•If
one
of t
he a
ngle
s of
a t
riang
le is
a r
ight
ang
le,
then
the
tria
ngle
is a
rig
ht
tria
ng
le.
•If
all t
hree
of t
he a
ngle
s of
a t
riang
le a
re a
cute
ang
les,
the
n th
e tr
iang
le is
an
acu
te t
rian
gle
.
•If
all t
hree
ang
les
of a
n ac
ute
tria
ngle
are
con
grue
nt,
then
the
tria
ngle
is a
n eq
uia
ng
ula
r tr
ian
gle
.
Cla
ssif
y ea
ch t
rian
gle.
a.
All
th
ree
angl
es a
re c
ongr
uen
t,so
all
th
ree
angl
es h
ave
mea
sure
60°
.T
he
tria
ngl
e is
an
equ
ian
gula
r tr
ian
gle.
b. T
he
tria
ngl
e h
as o
ne
angl
e th
at i
s ob
tuse
.It
is a
n o
btu
se t
rian
gle.
c.
Th
e tr
ian
gle
has
on
e ri
ght
angl
e.It
is
a ri
ght
tria
ngl
e.
Cla
ssif
y ea
ch t
rian
gle
as a
cute
,eq
uia
ngu
lar,
obtu
se,o
r ri
ght.
1.2.
3.
rig
ht
ob
tuse
equ
ian
gu
lar
4.5.
6.
acu
teri
gh
to
btu
se
60�
28�
92�
FDB
45�
45�
90�
XY
W
65�
65�
50�
UV
T
60�
60�
60�
Q
RS
120�
30�
30�
NO
P
67�
90�
23�
K LM
90�
60�
30�
G
HJ
25�
35�
120�
DF
E
60�A
BC
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill18
4G
lenc
oe G
eom
etry
Cla
ssif
y Tr
ian
gle
s b
y Si
des
You
can
cla
ssif
y a
tria
ngl
e by
th
e m
easu
res
of i
ts s
ides
.E
qual
nu
mbe
rs o
f h
ash
mar
ks i
ndi
cate
con
gru
ent
side
s.
•If
all t
hree
side
s of
a t
riang
le a
re c
ongr
uent
, th
en t
he t
riang
le is
an
equ
ilate
ral t
rian
gle
.
•If
at le
ast
two
side
s of
a t
riang
le a
re c
ongr
uent
, th
en t
he t
riang
le is
an
iso
scel
es t
rian
gle
.
•If
no t
wo
side
s of
a t
riang
le a
re c
ongr
uent
, th
en t
he t
riang
le is
a s
cale
ne
tria
ng
le.
Cla
ssif
y ea
ch t
rian
gle.
a.b
.c.
Tw
o si
des
are
con
gru
ent.
All
th
ree
side
s ar
e T
he
tria
ngl
e h
as n
o pa
irT
he
tria
ngl
e is
an
co
ngr
uen
t.T
he
tria
ngl
e of
con
gru
ent
side
s.It
is
isos
cele
s tr
ian
gle.
is a
n e
quil
ater
al t
rian
gle.
a sc
alen
e tr
ian
gle.
Cla
ssif
y ea
ch t
rian
gle
as e
qu
ila
tera
l,is
osce
les,
or s
cale
ne.
1.2.
3.
scal
ene
equ
ilate
ral
scal
ene
4.5.
6.
iso
scel
esis
osc
eles
equ
ilate
ral
7.F
ind
the
mea
sure
of
each
sid
e of
equ
ilat
eral
�R
ST
wit
h R
S�
2x�
2,S
T�
3x,
and
TR
�5x
�4.
2
8.F
ind
the
mea
sure
of
each
sid
e of
iso
scel
es �
AB
Cw
ith
AB
�B
Cif
AB
�4y
,B
C�
3y�
2,an
d A
C�
3y.
AB
�B
C�
8,A
C�
6
9.F
ind
the
mea
sure
of
each
sid
e of
�A
BC
wit
h v
erti
ces
A(�
1,5)
,B(6
,1),
and
C(2
,�6)
.C
lass
ify
the
tria
ngl
e.
AB
�B
C�
�65�
,AC
��
130
�;
�A
BC
is is
osc
eles
.
DE
Fx
xx
8x32
x
32x
B CA
UW
S
1217
19Q
O
MG
KI
18
1818
21
3�
�G
CA
2312
15X
V
TN
RP
LJ
H
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Cla
ssif
yin
g T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-1
4-1
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 4-1)
© Glencoe/McGraw-Hill A3 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Cla
ssif
yin
g T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-1
4-1
©G
lenc
oe/M
cGra
w-H
ill18
5G
lenc
oe G
eom
etry
Lesson 4-1
Use
a p
rotr
acto
r to
cla
ssif
y ea
ch t
rian
gle
as a
cute
,eq
uia
ngu
lar,
obtu
se,o
r ri
ght.
1.2.
3.
equ
ian
gu
lar
ob
tuse
rig
ht
4.5.
6.
acu
teo
btu
seac
ute
Iden
tify
th
e in
dic
ated
typ
e of
tri
angl
es.
7.ri
ght
8.is
osce
les
�A
BE
,�B
CE
�B
CD
,�B
DE
9.sc
alen
e10
.obt
use
�A
BE
,�B
CE
�B
DE
ALG
EBR
AF
ind
xan
d t
he
mea
sure
of
each
sid
e of
th
e tr
ian
gle.
11.�
AB
Cis
equ
ilat
eral
wit
h A
B�
3x�
2,B
C�
2x�
4,an
d C
A�
x�
10.
x�
6,A
B�
16,B
C�
16,C
A�
16
12.�
DE
Fis
iso
scel
es,�
Dis
th
e ve
rtex
an
gle,
DE
�x
�7,
DF
�3x
�1,
and
EF
�2x
�5.
x�
4,D
E�
11,D
F�
11,E
F�
13
Fin
d t
he
mea
sure
s of
th
e si
des
of
�R
ST
and
cla
ssif
y ea
ch t
rian
gle
by
its
sid
es.
13.R
(0,2
),S
(2,5
),T
(4,2
)
RS
��
13�,S
T�
�13�
,RT
�4;
iso
scel
es
14.R
(1,3
),S
(4,7
),T
(5,4
)
RS
�5,
ST
��
10�,R
T�
�17�
;sc
alen
e
EC
D
AB
©G
lenc
oe/M
cGra
w-H
ill18
6G
lenc
oe G
eom
etry
Use
a p
rotr
acto
r to
cla
ssif
y ea
ch t
rian
gle
as a
cute
,eq
uia
ngu
lar,
obtu
se,o
r ri
ght.
1.2.
3.
ob
tuse
acu
teri
gh
t
Iden
tify
th
e in
dic
ated
typ
e of
tri
angl
es i
f A �
B��
A�D�
�B�
D��
D�C�
,B�E�
�E�
D�,A�
B�⊥
B�C�
,an
d E�
D�⊥
D�C�
.
4.ri
ght
5.ob
tuse
�A
BC
,�C
DE
�B
ED
,�B
DC
6.sc
alen
e7.
isos
cele
s
�A
BC
,�C
DE
�A
BD
,�B
ED
,�B
DC
AL
GE
BR
AF
ind
xan
d t
he
mea
sure
of
each
sid
e of
th
e tr
ian
gle.
8.�
FG
His
equ
ilat
eral
wit
h F
G�
x�
5,G
H�
3x�
9,an
d F
H�
2x�
2.
x�
7,F
G�
12,G
H�
12,F
H�
12
9.�
LM
Nis
iso
scel
es,�
Lis
the
ver
tex
angl
e,L
M�
3x�
2,L
N�
2x�
1,an
d M
N�
5x�
2.
x�
3,L
M�
7,L
N�
7,M
N�
13
Fin
d t
he
mea
sure
s of
th
e si
des
of
�K
PL
and
cla
ssif
y ea
ch t
rian
gle
by
its
sid
es.
10.K
(�3,
2) P
(2,1
),L
(�2,
�3)
KP
��
26�,P
L�
4�2�,
LK
��
26�;
iso
scel
es
11.K
(5,�
3),P
(3,4
),L
(�1,
1)
KP
��
53�,P
L�
5,L
K�
2�13�
;sc
alen
e
12.K
(�2,
�6)
,P(�
4,0)
,L(3
,�1)
KP
�2�
10�,P
L�
5�2�,
LK
�5�
2�;is
osc
eles
13.D
ESIG
ND
ian
a en
tere
d th
e de
sign
at
the
righ
t in
a l
ogo
con
test
sp
onso
red
by a
wil
dlif
e en
viro
nm
enta
l gr
oup.
Use
a p
rotr
acto
r.H
ow m
any
righ
t an
gles
are
th
ere?
5
AC
DE
B
Pra
ctic
e (
Ave
rag
e)
Cla
ssif
yin
g T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-1
4-1
Answers (Lesson 4-1)
© Glencoe/McGraw-Hill A4 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csC
lass
ifyi
ng
Tri
ang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-1
4-1
©G
lenc
oe/M
cGra
w-H
ill18
7G
lenc
oe G
eom
etry
Lesson 4-1
Pre-
Act
ivit
yW
hy
are
tria
ngl
es i
mp
orta
nt
in c
onst
ruct
ion
?
Rea
d th
e in
trod
uct
ion
to
Les
son
4-1
at
the
top
of p
age
178
in y
our
text
book
.
•W
hy a
re t
rian
gles
use
d fo
r br
aces
in c
onst
ruct
ion
rath
er t
han
othe
r sh
apes
?S
amp
le a
nsw
er:T
rian
gle
s lie
in a
pla
ne
and
are
rig
id s
hap
es.
•W
hy
do y
ou t
hin
k th
at i
sosc
eles
tri
angl
es a
re u
sed
mor
e of
ten
th
ansc
alen
e tr
ian
gles
in
con
stru
ctio
n?
Sam
ple
an
swer
:Is
osc
eles
tria
ng
les
are
sym
met
rica
l.
Rea
din
g t
he
Less
on
1.S
upp
ly t
he
corr
ect
nu
mbe
rs t
o co
mpl
ete
each
sen
ten
ce.
a.In
an
obt
use
tri
angl
e,th
ere
are
acu
te a
ngl
e(s)
,ri
ght
angl
e(s)
,an
d
obtu
se a
ngl
e(s)
.
b.
In a
n a
cute
tri
angl
e,th
ere
are
acu
te a
ngl
e(s)
,ri
ght
angl
e(s)
,an
d
obtu
se a
ngl
e(s)
.
c.In
a r
igh
t tr
ian
gle,
ther
e ar
e ac
ute
an
gle(
s),
righ
t an
gle(
s),a
nd
obtu
se a
ngl
e(s)
.
2.D
eter
min
e w
het
her
eac
h s
tate
men
t is
alw
ays,
som
etim
es,o
r n
ever
tru
e.
a.A
rig
ht
tria
ngl
e is
sca
len
e.so
met
imes
b.
An
obt
use
tri
angl
e is
iso
scel
es.
som
etim
esc.
An
equ
ilat
eral
tri
angl
e is
a r
igh
t tr
ian
gle.
nev
erd
.A
n e
quil
ater
al t
rian
gle
is i
sosc
eles
.al
way
se.
An
acu
te t
rian
gle
is i
sosc
eles
.so
met
imes
f.A
sca
len
e tr
ian
gle
is o
btu
se.
som
etim
es
3.D
escr
ibe
each
tri
angl
e by
as
man
y of
th
e fo
llow
ing
wor
ds a
s ap
ply:
acu
te,o
btu
se,r
igh
t,sc
alen
e,is
osce
les,
or e
quil
ater
al.
a.b
.c.
acu
te,s
cale
ne
ob
tuse
,iso
scel
esri
gh
t,sc
alen
e
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r a
new
mat
hem
atic
al t
erm
is
to r
elat
e it
to
a n
onm
ath
emat
ical
defi
nit
ion
of
the
sam
e w
ord.
How
is
the
use
of
the
wor
d ac
ute
,wh
en u
sed
to d
escr
ibe
acu
te p
ain
,rel
ated
to
the
use
of
the
wor
d ac
ute
wh
en u
sed
to d
escr
ibe
an a
cute
an
gle
oran
acu
te t
rian
gle?
Sam
ple
an
swer
:B
oth
are
rel
ated
to
th
e m
ean
ing
of
acu
teas
sh
arp
.An
acu
te p
ain
is a
sh
arp
pai
n,a
nd
an
acu
te a
ng
leca
n b
eth
ou
gh
t o
f as
an
an
gle
wit
h a
sh
arp
po
int.
In a
n a
cute
tri
ang
leal
l of
the
ang
les
are
acu
te.
5
34
135�
80�70�
30�
01
20
03
10
2
©G
lenc
oe/M
cGra
w-H
ill18
8G
lenc
oe G
eom
etry
Rea
din
g M
ath
emat
ics
Wh
en y
ou r
ead
geom
etry
,you
may
nee
d to
dra
w a
dia
gram
to
mak
e th
e te
xtea
sier
to
un
ders
tan
d.
Con
sid
er t
hre
e p
oin
ts,A
,B,a
nd
Con
a c
oord
inat
e gr
id.
Th
e y-
coor
din
ates
of
Aan
d B
are
the
sam
e.T
he
x-co
ord
inat
e of
Bis
grea
ter
than
th
e x-
coor
din
ate
of A
.Bot
h c
oord
inat
es o
f C
are
grea
ter
than
th
e co
rres
pon
din
g co
ord
inat
es o
f B
.Is
tria
ngl
e A
BC
acu
te,r
igh
t,or
ob
tuse
?
To
answ
er t
his
qu
esti
on,f
irst
dra
w a
sam
ple
tria
ngl
e
that
fit
s th
e de
scri
ptio
n.
Sid
e A
Bm
ust
be
a h
oriz
onta
l se
gmen
t be
cau
se t
he
y-co
ordi
nat
es a
re t
he
sam
e.P
oin
t C
mu
st b
e lo
cate
d to
th
e ri
ght
and
up
from
poi
nt
B.
Fro
m t
he
diag
ram
you
can
see
th
at t
rian
gle
AB
Cm
ust
be
obtu
se.
An
swer
eac
h q
ues
tion
.Dra
w a
sim
ple
tri
angl
e on
th
e gr
id a
bov
e to
hel
p y
ou.
1.C
onsi
der
thre
e po
ints
,R,S
,an
d 2.
Con
side
r th
ree
non
coll
inea
r po
ints
,T
on a
coo
rdin
ate
grid
.Th
e J
,K,a
nd
Lon
a c
oord
inat
e gr
id.T
he
x-co
ordi
nat
es o
f R
and
Sar
e th
ey-
coor
din
ates
of
Jan
d K
are
the
sam
e.T
he
y-co
ordi
nat
e of
Tis
sam
e.T
he
x-co
ordi
nat
es o
f K
and
Lbe
twee
n t
he
y-co
ordi
nat
es o
f R
are
the
sam
e.Is
tri
angl
e J
KL
acu
te,
and
S.T
he
x-co
ordi
nat
e of
Tis
les
sri
ght,
or o
btu
se?
rig
ht
than
th
e x-
coor
din
ate
of R
.Is
angl
eR
of t
rian
gle
RS
T a
cute
,rig
ht,
or
obtu
se?
acu
te
3.C
onsi
der
thre
e n
onco
llin
ear
poin
ts,
4.C
onsi
der
thre
e po
ints
,G,H
,an
d I
D,E
,an
d F
on a
coo
rdin
ate
grid
.on
a c
oord
inat
e gr
id.P
oin
ts G
and
Th
e x-
coor
din
ates
of
Dan
d E
are
Har
e on
th
e po
siti
ve y
-axi
s,an
dop
posi
tes.
Th
e y-
coor
din
ates
of
Dan
dth
e y-
coor
din
ate
of G
is t
wic
e th
e E
are
the
sam
e.T
he
x-co
ordi
nat
e of
y-co
ordi
nat
e of
H.P
oin
t I
is o
n t
he
Fis
0.W
hat
kin
d of
tri
angl
e m
ust
posi
tive
x-a
xis,
and
the
x-co
ordi
nat
e�
DE
F b
e:sc
alen
e,is
osce
les,
orof
Iis
gre
ater
th
an t
he
y-co
ordi
nat
eeq
uil
ater
al?
iso
scel
esof
G.I
s tr
ian
gle
GH
Isc
alen
e,is
osce
les,
or e
quil
ater
al?
scal
eneB
A
Q x
y
O
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-1
4-1
Exam
ple
Exam
ple
Answers (Lesson 4-1)
© Glencoe/McGraw-Hill A5 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
An
gle
s o
f Tri
ang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-2
4-2
©G
lenc
oe/M
cGra
w-H
ill18
9G
lenc
oe G
eom
etry
Lesson 4-2
An
gle
Su
m T
heo
rem
If t
he
mea
sure
s of
tw
o an
gles
of
a tr
ian
gle
are
know
n,
the
mea
sure
of
the
thir
d an
gle
can
alw
ays
be f
oun
d.
An
gle
Su
mT
he s
um o
f th
e m
easu
res
of t
he a
ngle
s of
a t
riang
le is
180
.T
heo
rem
In t
he f
igur
e at
the
rig
ht,
m�
A�
m�
B�
m�
C�
180.
CA
B
Fin
d m
�T
.
m�
R�
m�
S�
m�
T�
180
Ang
le S
um
The
orem
25 �
35 �
m�
T�
180
Sub
stitu
tion
60 �
m�
T�
180
Add
.
m�
T�
120
Sub
trac
t 60
from
eac
h si
de.
35�
25�
RT
S
Fin
d t
he
mis
sin
g an
gle
mea
sure
s.
m�
1 �
m�
A�
m�
B�
180
Ang
le S
um T
heor
em
m�
1 �
58 �
90�
180
Sub
stitu
tion
m�
1 �
148
�18
0A
dd.
m�
1�
32S
ubtr
act
148
from
each
sid
e.
m�
2�
32V
ertic
al a
ngle
s ar
e
cong
ruen
t.
m�
3 �
m�
2 �
m�
E�
180
Ang
le S
um T
heor
em
m�
3 �
32 �
108
�18
0S
ubst
itutio
n
m�
3 �
140
�18
0A
dd.
m�
3�
40S
ubtr
act
140
from
each
sid
e.
58�90
�
108�
12
3
E
DA
C
B
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d t
he
mea
sure
of
each
nu
mb
ered
an
gle.
1.m
�1
�28
2.m
�1
�12
0
3.4.
5.6.
m�
1 �
820
�
152�
DG
A
1
m�
1 �
30,
m�
2 �
6030
�60
�12
S
R
TW
m�
1 �
56,
m�
2 �
56,
m�
3 �
74Q
O
NM P
58�
66�
50�
32
1
m�
1 �
30,
m�
2 �
60V W
T
U
30�
60�
2
1
S
QR
30�
1
90�
62�
1NM
P
©G
lenc
oe/M
cGra
w-H
ill19
0G
lenc
oe G
eom
etry
Exte
rio
r A
ng
le T
heo
rem
At
each
ver
tex
of a
tri
angl
e,th
e an
gle
form
ed b
y on
e si
dean
d an
ext
ensi
on o
f th
e ot
her
sid
e is
cal
led
an e
xter
ior
angl
eof
th
e tr
ian
gle.
For
eac
hex
teri
or a
ngl
e of
a t
rian
gle,
the
rem
ote
inte
rior
an
gles
are
the
inte
rior
an
gles
th
at a
re n
otad
jace
nt
to t
hat
ext
erio
r an
gle.
In t
he
diag
ram
bel
ow,�
Ban
d �
Aar
e th
e re
mot
e in
teri
oran
gles
for
ext
erio
r �
DC
B.
Ext
erio
r A
ng
leT
he m
easu
re o
f an
ext
erio
r an
gle
of a
tria
ngle
is e
qual
to
Th
eore
mth
e su
m o
f th
e m
easu
res
of t
he t
wo
rem
ote
inte
rior
angl
es.
m�
1 �
m�
A�
m�
BA
C
B
D1
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
An
gle
s o
f Tri
ang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-2
4-2
Fin
d m
�1.
m�
1�
m�
R�
m�
SE
xter
ior
Ang
le T
heor
em
�60
�80
Sub
stitu
tion
�14
0A
dd.
RT
S
60�80
�
1
Fin
d x
.
m�
PQ
S�
m�
R�
m�
SE
xter
ior
Ang
le T
heor
em
78 �
55 �
xS
ubst
itutio
n
23 �
xS
ubtr
act
55 f
rom
eac
h si
de.
SR
Q
P
55�
78�
x�
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d t
he
mea
sure
of
each
nu
mb
ered
an
gle.
1.2.
m�
1 �
115
m�
1 �
60,m
�2
�12
0
3.4.
m�
1 �
60,m
�2
�60
,m�
3 �
120
m�
1 �
109,
m�
2 �
29,m
�3
�71
Fin
d x
.
5.25
6.29
E
FG
H58
�x�
x�B
A
DC
95�
2 x�
145�
UT
SR
V
35�
36�
80�
13
2
PO
Q
NM
60� 60
�3
2
1
BC
D
A
25�
35� 1
2Y
ZW
X 65�50
�
1
Answers (Lesson 4-2)
© Glencoe/McGraw-Hill A6 Glencoe Geometry
Skil
ls P
ract
ice
An
gle
s o
f Tri
ang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-2
4-2
©G
lenc
oe/M
cGra
w-H
ill19
1G
lenc
oe G
eom
etry
Lesson 4-2
Fin
d t
he
mis
sin
g an
gle
mea
sure
s.
1.27
2.17
,17
Fin
d t
he
mea
sure
of
each
an
gle.
3.m
�1
55
4.m
�2
55
5.m
�3
70
Fin
d t
he
mea
sure
of
each
an
gle.
6.m
�1
125
7.m
�2
55
8.m
�3
95
Fin
d t
he
mea
sure
of
each
an
gle.
9.m
�1
140
10.m
�2
40
11.m
�3
65
12.m
�4
75
13.m
�5
115
Fin
d t
he
mea
sure
of
each
an
gle.
14.m
�1
27
15.m
�2
2763
�
1
2D
AC
B
80�
60�
40�
105�
14
52
3
150�
55�
70�
12
3
85�
55�
40�
12
3
146�
TIG
ERS
80� 73�
©G
lenc
oe/M
cGra
w-H
ill19
2G
lenc
oe G
eom
etry
Fin
d t
he
mis
sin
g an
gle
mea
sure
s.
1.18
2.85
Fin
d t
he
mea
sure
of
each
an
gle.
3.m
�1
97
4.m
�2
83
5.m
�3
62
Fin
d t
he
mea
sure
of
each
an
gle.
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 t
he
mea
sure
of
each
an
gle
if �
BA
Dan
d
�B
DC
are
righ
t an
gles
an
d m
�A
BC
�84
.
12.m
�1
26
13.m
�2
32
14.C
ON
STR
UC
TIO
NT
he
diag
ram
sh
ows
an
exam
ple
of t
he
Pra
tt T
russ
use
d in
bri
dge
con
stru
ctio
n.U
se t
he
diag
ram
to
fin
d m
�1.
55
145�
1
64�
1
2AB
C
D
118�
36�
68�
70� 65
� 82�
1
2
34
5
6
58�
39�
35�
12
3
40�
55�
72�
?
Pra
ctic
e (
Ave
rag
e)
An
gle
s o
f Tri
ang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-2
4-2
Answers (Lesson 4-2)
© Glencoe/McGraw-Hill A7 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csA
ng
les
of T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-2
4-2
©G
lenc
oe/M
cGra
w-H
ill19
3G
lenc
oe G
eom
etry
Lesson 4-2
Pre-
Act
ivit
yH
ow a
re t
he
angl
es o
f tr
ian
gles
use
d t
o m
ake
kit
es?
Rea
d th
e in
trod
uct
ion
to
Les
son
4-2
at
the
top
of p
age
185
in y
our
text
book
.
Th
e fr
ame
of t
he
sim
ples
t ki
nd
of k
ite
divi
des
the
kite
in
to f
our
tria
ngl
es.
Des
crib
e th
ese
fou
r tr
ian
gles
an
d h
ow t
hey
are
rel
ated
to
each
oth
er.
Sam
ple
an
swer
:Th
ere
are
two
pai
rs o
f ri
gh
t tr
ian
gle
s th
at h
ave
the
sam
e si
ze a
nd
sh
ape.
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.
a.N
ame
the
thre
e in
teri
or a
ngl
es o
f th
e tr
ian
gle.
(Use
th
ree
lett
ers
to n
ame
each
an
gle.
)�
BA
C,�
AB
C,�
BC
Ab
.N
ame
thre
e ex
teri
or a
ngl
es o
f th
e tr
ian
gle.
(Use
th
ree
lett
ers
to n
ame
each
an
gle.
)�
EA
B,�
DB
C,�
FC
Ac.
Nam
e th
e re
mot
e in
teri
or a
ngl
es o
f �
EA
B.
�A
BC
,�B
CA
d.
Fin
d th
e m
easu
re o
f ea
ch a
ngl
e w
ith
out
usi
ng
a pr
otra
ctor
.
i.�
DB
C62
ii.
�A
BC
118
iii.
�A
CF
157
iv.
�E
AB
141
2.In
dica
te w
het
her
eac
h s
tate
men
t is
tru
eor
fal
se.I
f th
e st
atem
ent
is f
alse
,rep
lace
th
eu
nde
rlin
ed w
ord
or n
um
ber
wit
h a
wor
d or
nu
mbe
r th
at w
ill
mak
e th
e st
atem
ent
tru
e.
a.T
he
acu
te a
ngl
es o
f a
righ
t tr
ian
gle
are
.fa
lse;
com
ple
men
tary
b.
Th
e su
m o
f th
e m
easu
res
of t
he
angl
es o
f an
y tr
ian
gle
is
.fa
lse;
180
c.A
tri
angl
e ca
n h
ave
at m
ost
one
righ
t an
gle
or
angl
e.fa
lse;
ob
tuse
d.
If t
wo
angl
es o
f on
e tr
ian
gle
are
con
gru
ent
to t
wo
angl
es o
f an
oth
er t
rian
gle,
then
th
eth
ird
angl
es o
f th
e tr
ian
gles
are
.
tru
ee.
Th
e m
easu
re o
f an
ext
erio
r an
gle
of a
tri
angl
e is
equ
al t
o th
e of
th
em
easu
res
of t
he
two
rem
ote
inte
rior
an
gles
.fa
lse;
sum
f.If
th
e m
easu
res
of t
wo
angl
es o
f a
tria
ngl
e ar
e 62
an
d 93
,th
en t
he
mea
sure
of
the
thir
d an
gle
is
.fa
lse;
25g.
An
an
gle
of a
tri
angl
e fo
rms
a li
nea
r pa
ir w
ith
an
in
teri
or a
ngl
e of
th
etr
ian
gle.
tru
e
Hel
pin
g Y
ou
Rem
emb
er
3.M
any
stu
den
ts r
emem
ber
mat
hem
atic
al i
deas
an
d fa
cts
mor
e ea
sily
if
they
see
th
emde
mon
stra
ted
visu
ally
rat
her
th
an h
avin
g th
em s
tate
d in
wor
ds.D
escr
ibe
a vi
sual
way
to d
emon
stra
te t
he
An
gle
Su
m T
heo
rem
.S
amp
le a
nsw
er:
Cu
t o
ff t
he
ang
les
of
a tr
ian
gle
an
d p
lace
th
em
sid
e-by
-sid
e o
n o
ne
sid
e o
f a
line
so t
hat
th
eir
vert
ices
mee
t at
a c
om
mo
np
oin
t.T
he
resu
lt w
ill s
ho
w t
hre
e an
gle
s w
ho
se m
easu
res
add
up
to
180
.
exte
rior
35
diff
eren
ce
con
gru
ent
acu
te
100
supp
lem
enta
ry
39�
23�
EA
BD
CF
©G
lenc
oe/M
cGra
w-H
ill19
4G
lenc
oe G
eom
etry
Fin
din
g A
ng
le M
easu
res
in T
rian
gle
sYo
u c
an u
se a
lgeb
ra t
o so
lve
prob
lem
s in
volv
ing
tria
ngl
es.
In t
rian
gle
AB
C,m
�A
,is
twic
e m
�B
,an
d m
�C
is 8
mor
e th
an m
�B
.Wh
at i
s th
e m
easu
re o
f ea
ch a
ngl
e?
Wri
te a
nd
solv
e an
equ
atio
n.L
et 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
,an
d m
�C
�43
�8
or51
.
Sol
ve e
ach
pro
ble
m.
1.In
tri
angl
e D
EF
,m�
E i
s th
ree
tim
es2.
In t
rian
gle
RS
T,m
�T
is 5
mor
e th
an
m�
D,a
nd
m�
Fis
9 l
ess
than
m�
E.
m�
R,a
nd
m�
Sis
10
less
th
an m
�T
.W
hat
is
the
mea
sure
of
each
an
gle?
Wh
at i
s th
e m
easu
re o
f ea
ch a
ngl
e?
m�
D�
27,m
�E
�81
,m�
F�
72m
�R
�60
,m�
S�
55,m
�T
�65
3.In
tri
angl
e J
KL
,m�
K i
s fo
ur
tim
es4.
In t
rian
gle
XY
Z,m
�Z
is 2
mor
e th
an t
wic
em
�J
,an
d m
�L
is f
ive
tim
es m
�J
.m
�X
,and
m�
Yis
7 l
ess
than
tw
ice
m�
X.
Wh
at i
s th
e m
easu
re o
f ea
ch a
ngl
e?W
hat
is
the
mea
sure
of
each
an
gle?
m�
J�
18,m
�K
�72
,m�
L�
90m
�X
�37
,m�
Y�
67,m
�Z
�76
5.In
tri
angl
e G
HI,
m�
H i
s 20
mor
e th
an6.
In t
rian
gle
MN
O,m
�M
is e
qual
to
m�
N,
m�
G,a
nd
m�
Gis
8 m
ore
than
m�
I.an
d m
�O
is 5
mor
e th
an t
hre
e ti
mes
W
hat
is
the
mea
sure
of
each
an
gle?
m�
N.W
hat
is
the
mea
sure
of
each
an
gle?
m�
G�
56,m
�H
�76
,m�
I�48
m�
M�
m�
N�
35,m
�O
�11
0
7.In
tri
angl
e S
TU
,m�
U i
s h
alf
m�
T,
8.In
tri
angl
e P
QR
,m�
Pis
equ
al t
o an
d m
�S
is 3
0 m
ore
than
m�
T.W
hat
m�
Q,a
nd
m�
Ris
24
less
th
an m
�P
.is
th
e m
easu
re o
f ea
ch a
ngl
e?W
hat
is
the
mea
sure
of
each
an
gle?
m�
S�
90,m
�T
�60
,m�
U�
30m
�P
�m
�Q
�68
,m�
R�
44
9.W
rite
you
r ow
n p
robl
ems
abou
t m
easu
res
of t
rian
gles
.S
ee s
tud
ents
’wo
rk.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-2
4-2
Exam
ple
Exam
ple
Answers (Lesson 4-2)
© Glencoe/McGraw-Hill A8 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Co
ng
ruen
t Tri
ang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-3
4-3
©G
lenc
oe/M
cGra
w-H
ill19
5G
lenc
oe G
eom
etry
Lesson 4-3
Co
rres
po
nd
ing
Par
ts o
f C
on
gru
ent
Tria
ng
les
Tri
angl
es t
hat
hav
e th
e sa
me
size
an
d sa
me
shap
e ar
e co
ngr
uen
t tr
ian
gles
.Tw
o tr
ian
gles
are
con
gru
ent
if a
nd
only
if
all
thre
e pa
irs
of c
orre
spon
din
g an
gles
are
con
gru
ent
and
all
thre
e pa
irs
of c
orre
spon
din
g si
des
are
con
gru
ent.
In
the
figu
re,�
AB
C�
�R
ST
.
If �
XY
Z�
�R
ST
,nam
e th
e p
airs
of
con
gru
ent
angl
es a
nd
con
gru
ent
sid
es.
�X
��
R,�
Y�
�S
,�Z
��
TX �
Y��
R�S�
,X�Z�
�R�
T�,Y�
Z��
S�T�
Iden
tify
th
e co
ngr
uen
t tr
ian
gles
in
eac
h f
igu
re.
1.2.
3.
�A
BC
��
JKL
�A
BC
��
DC
B�
JKM
��
LM
K
Nam
e th
e co
rres
pon
din
g co
ngr
uen
t an
gles
an
d s
ides
for
th
e co
ngr
uen
t tr
ian
gles
.
4.5.
6.
�E
��
J;�
F�
�K
;�
A�
�D
;�
R�
�T
;�
G�
�L
;E�
F��
J�K�;
�A
BC
��
DC
B;
�R
SU
��
TS
U;
E�G�
�J�L�
;F�G�
�K�
L��
AC
B�
�D
BC
;�
RU
S�
�T
US
;A�
B��
D�C�
;A�
C��
D�B�
;R�
U��
T�U�;
R�S�
�T�S�
;B�
C��
C�B�
S�U�
�S�
U�
R TUS
BD
CA
FG
LK J
E
K J
L MC
D
A
B
CA
B
LJK
Y
XZ
T
SR
AC
B
R
TS
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill19
6G
lenc
oe G
eom
etry
Iden
tify
Co
ng
ruen
ce T
ran
sfo
rmat
ion
sIf
tw
o tr
ian
gles
are
con
gru
ent,
you
can
slid
e,fl
ip,o
r tu
rn o
ne
of t
he
tria
ngl
es a
nd
they
wil
l st
ill
be c
ongr
uen
t.T
hes
e ar
e ca
lled
con
gru
ence
tra
nsf
orm
atio
ns
beca
use
th
ey d
o n
ot c
han
ge t
he
size
or
shap
e of
th
e fi
gure
.It
is
com
mon
to
use
pri
me
sym
bols
to
dist
ingu
ish
bet
wee
n a
n o
rigi
nal
�A
BC
and
atr
ansf
orm
ed �
A�B
�C�.
Nam
e th
e co
ngr
uen
ce t
ran
sfor
mat
ion
th
at p
rod
uce
s �
A�B
�C�
from
�A
BC
.T
he
con
gru
ence
tra
nsf
orm
atio
n i
s a
slid
e.�
A�
�A
�;�
B�
�B
�;�
C�
�C
�;A �
B��
A���B�
��;A�
C��
A���C�
��;B�
C��
B���C�
��
Des
crib
e th
e co
ngr
uen
ce t
ran
sfor
mat
ion
bet
wee
n t
he
two
tria
ngl
es a
s a
slid
e,a
flip
,or
a tu
rn.T
hen
nam
e th
e co
ngr
uen
t tr
ian
gles
.
1.2.
flip
;�
RS
T�
�R
S�T
�sl
ide;
�M
NP
��
M�N
�P�
3.4.
turn
;�
OP
Q�
�O
P�Q
�fl
ip;
�A
BC
��
AB
�C
5.6.
slid
e;�
AB
C�
�A
�B�C
�tu
rn;
�M
NP
��
MN
�P�
x
y
OM
N
P
N�
P�
x
y
OA
�B
�
C�
ABC
x
y
OB
�B
A Cx
y
O
Q�
P�
P Q
x
y
ON
�
M�
P�
N MP
x
y
OR
S�
T�
S
T
x
y
O
A�
B�
B
C�
AC
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Co
ng
ruen
t Tri
ang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-3
4-3
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 4-3)
© Glencoe/McGraw-Hill A9 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Co
ng
ruen
t Tri
ang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-3
4-3
©G
lenc
oe/M
cGra
w-H
ill19
7G
lenc
oe G
eom
etry
Lesson 4-3
Iden
tify
th
e co
ngr
uen
t tr
ian
gles
in
eac
h f
igu
re.
1.2.
�JP
L�
�T
VS
�A
BC
��
WX
Y
3.4.
�P
QR
��
PS
R�
DE
F�
�D
GF
Nam
e th
e co
ngr
uen
t an
gles
an
d s
ides
for
eac
h p
air
of c
ongr
uen
t tr
ian
gles
.
5.�
AB
C�
�F
GH
�A
��
F,�
B�
�G
,�C
��
H;
A�B�
�F�G�
,B�C�
�G�
H�,A�
C��
F�H�
6.�
PQ
R�
�S
TU
�P
��
S,�
Q�
�T
,�R
��
U;
P�Q�
�S�
T�,Q�
R��
T�U�,P�
R��
S�U�
Ver
ify
that
eac
h o
f th
e fo
llow
ing
tran
sfor
mat
ion
s p
rese
rves
con
gru
ence
,an
d n
ame
the
con
gru
ence
tra
nsf
orm
atio
n.
7.�
AB
C�
�A
�B�C
�8.
�D
EF
��
D�E
�F�
AB
�2�
2�,A
�B�
�2�
2�,D
E�
4,D
�E�
�4,
EF
�5,
BC
�2�
2�,B
�C�
�2�
2�,E
�F�
�5,
DF
� 3
,D�F
��
3,
AC
�4,
A�C
��
4,�
A�
�A
�,�
D�
�D
�,�
E�
�E
�,
�B
��
B�,
�C
��
C�;
slid
e�
F�
�F
�;fl
ip
x
y
OD
�
E�
F�
DE
F
x
y
OA
�
B�
C�
A
B
C
D
E G
FR
P
Q S
WY
XC
AB
L
P
J
S
V
T
©G
lenc
oe/M
cGra
w-H
ill19
8G
lenc
oe G
eom
etry
Iden
tify
th
e co
ngr
uen
t tr
ian
gles
in
eac
h f
igu
re.
1.2.
�A
BC
��
DR
S�
LM
N�
�Q
PN
Nam
e th
e co
ngr
uen
t an
gles
an
d s
ides
for
eac
h p
air
of c
ongr
uen
t tr
ian
gles
.
3.�
GK
P�
�L
MN
�G
��
L,�
K�
�M
,�P
��
N;
G�K�
�L�M�
,K�P�
�M�
N�,G�
P��
L�N�
4.�
AN
C�
�R
BV
�A
��
R,�
N�
�B
,�C
��
V;
A�N�
�R�
B�,N�
C��
B�V�
,A�C�
�R�
V�
Ver
ify
that
eac
h o
f th
e fo
llow
ing
tran
sfor
mat
ion
s p
rese
rves
con
gru
ence
,an
d n
ame
the
con
gru
ence
tra
nsf
orm
atio
n.
5.�
PS
T�
�P
�S�T
�6.
�L
MN
��
L�M
�N�
PS
��
13�,P
�S�
��
13�,
LM
�2�
2�,L
�M�
�2�
2�,S
T�
�5�,
S�T
��
�5�,
PT
��
10�,
MN
��
29�,M
�N�
��
29�,
P�T
��
�10�
,�P
��
P�,
LN
�7,
L�N
��
7,�
L�
�L
�,
�S
��
S�,
�T
��
T�;
flip
�M
��
M�,
�N
��
N�;
flip
QU
ILTI
NG
For
Exe
rcis
es 7
an
d 8
,ref
er t
o th
e q
uil
t d
esig
n.
7.In
dica
te t
he
tria
ngl
es t
hat
app
ear
to b
e co
ngr
uen
t.
�A
BI�
�E
BF
,�C
BD
��
HB
G
8.N
ame
the
con
gru
ent
angl
es a
nd
con
gru
ent
side
s of
a p
air
of
con
gru
ent
tria
ngl
es.
Sam
ple
an
swer
:�
A�
�E
,�A
BI�
�E
BF
,�I�
�F
;A�
B��
E�B�
,B�I��
B�F�,
A�I��
E�F�
B
A I
E FH
G
CD
x
y
O
M�
N�
L�
M
NL
x
y
O
S�
T�
P�
S
TP
MN
L
P
QD
R
SC
A
BPra
ctic
e (
Ave
rag
e)
Co
ng
ruen
t Tri
ang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-3
4-3
Answers (Lesson 4-3)
©G
lenc
oe/M
cGra
w-H
ill20
0G
lenc
oe G
eom
etry
Tran
sfo
rmat
ion
s in
Th
e C
oo
rdin
ate
Pla
ne
Th
e fo
llow
ing
stat
emen
t te
lls
one
way
to
map
pre
imag
e po
ints
to
imag
e po
ints
in
th
e co
ordi
nat
e pl
ane.
(x,y
) →
(x�
6,y
�3)
Th
is c
an b
e re
ad,“
Th
e po
int
wit
h c
oord
inat
es (
x,y)
is
map
ped
to t
he
poin
t w
ith
coo
rdin
ates
(x
�6,
y�
3).”
Wit
h t
his
tra
nsf
orm
atio
n,f
or e
xam
ple,
(3,5
) is
map
ped
to
(3 �
6,5
�3)
or
(9,2
).T
he
figu
re s
how
s h
ow t
he
tria
ngl
e A
BC
is m
appe
d to
tri
angl
e X
YZ
.
1.D
oes
the
tran
sfor
mat
ion
abo
ve a
ppea
r to
be
a co
ngr
uen
ce t
ran
sfor
mat
ion
? E
xpla
in y
our
answ
er.
Yes;
the
tran
sfo
rmat
ion
slid
es t
he
fig
ure
to
th
e lo
wer
rig
ht
wit
ho
ut
chan
gin
g it
s si
ze o
r sh
ape.
Dra
w t
he
tran
sfor
mat
ion
im
age
for
each
fig
ure
.Th
en t
ell
wh
eth
er t
he
tran
sfor
mat
ion
is
or i
s n
ot a
con
gru
ence
tra
nsf
orm
atio
n.
2.(x
,y)
→(x
�4,
y)ye
s3.
(x,y
) →
(x�
8,y
�7)
yes
4.(x
,y)
→(�
x,�
y)ye
s5.
(x,y
) →
���1 2� x
,y�
no
x
y
Ox
y
O
x
y
Ox
y
O
x
y
B
A
CX
ZY
O
(x, y
) → (x
� 6
, y �
3)
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-3
4-3
© Glencoe/McGraw-Hill A10 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csC
on
gru
ent T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-3
4-3
©G
lenc
oe/M
cGra
w-H
ill19
9G
lenc
oe G
eom
etry
Lesson 4-3
Pre-
Act
ivit
yW
hy
are
tria
ngl
es u
sed
in
bri
dge
s?
Rea
d th
e in
trod
uct
ion
to
Les
son
4-3
at
the
top
of p
age
192
in y
our
text
book
.
In t
he
brid
ge s
how
n i
n t
he
phot
ogra
ph i
n y
our
text
book
,dia
gon
al b
race
sw
ere
use
d to
div
ide
squ
ares
in
to t
wo
isos
cele
s ri
ght
tria
ngl
es.W
hy
do y
outh
ink
thes
e br
aces
are
use
d on
th
e br
idge
?S
amp
le a
nsw
er:T
he
dia
go
nal
bra
ces
mak
e th
e st
ruct
ure
str
on
ger
an
d p
reve
nt
itfr
om
bei
ng
def
orm
ed w
hen
it h
as t
o w
ith
stan
d a
hea
vy lo
ad.
Rea
din
g t
he
Less
on
1.If
�R
ST
��
UW
V,c
ompl
ete
each
pai
r of
con
gru
ent
part
s.
�R
��
�W
�T
�
R�T�
��
U�W�
�W�
V�
2.Id
enti
fy t
he
con
gru
ent
tria
ngl
es i
n e
ach
dia
gram
.
a.�
AB
C�
�A
DC
b.
�P
QS
��
RQ
S
c.d
.
�M
NO
��
QP
O�
RT
V�
�U
SV
3.D
eter
min
e w
het
her
eac
h s
tate
men
t sa
ys t
hat
con
gru
ence
of
tria
ngl
es i
s re
flex
ive,
sym
met
ric,
or t
ran
siti
ve.
a.If
th
e fi
rst
of t
wo
tria
ngl
es i
s co
ngr
uen
t to
th
e se
con
d tr
ian
gle,
then
th
e se
con
dtr
ian
gle
is c
ongr
uen
t to
th
e fi
rst.
sym
met
ric
b.
If t
here
are
thr
ee t
rian
gles
for
whi
ch t
he f
irst
is c
ongr
uent
to
the
seco
nd a
nd t
he s
econ
dis
con
gru
ent
to t
he
thir
d,th
en t
he
firs
t tr
ian
gle
is c
ongr
uen
t to
th
e th
ird.
tran
siti
vec.
Eve
ry t
rian
gle
is c
ongr
uen
t to
its
elf.
refl
exiv
e
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r so
met
hing
is t
o ex
plai
n it
to
som
eone
els
e.Yo
ur c
lass
mat
e B
en is
hav
ing
trou
ble
wri
tin
g co
ngr
uen
ce s
tate
men
ts f
or t
rian
gles
bec
ause
he
thin
ks h
e h
as t
om
atch
up
thre
e pa
irs
of s
ides
and
thr
ee p
airs
of
angl
es.H
ow c
an y
ou h
elp
him
und
erst
and
how
to
wri
te c
orre
ct c
ongr
uen
ce s
tate
men
ts m
ore
easi
ly?
Sam
ple
an
swer
:Wri
te t
he
thre
e ve
rtic
es o
f o
ne
tria
ng
le in
any
ord
er.T
hen
wri
te t
he
corr
esp
on
din
gve
rtic
es o
f th
e se
con
d t
rian
gle
in t
he
sam
e o
rder
.If
the
ang
les
are
wri
tten
in t
he
corr
ect
corr
esp
on
den
ce,t
he
sid
es w
ill a
uto
mat
ical
ly b
e in
th
eco
rrec
t co
rres
po
nd
ence
als
o.
RT
US
V
NO
P
QM
S
PR
Q
CA
B D
S�T�
R�S�
U�V�
�V
�S
�U
Answers (Lesson 4-3)
© Glencoe/McGraw-Hill A11 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Pro
vin
g C
on
gru
ence
—S
SS
,SA
S
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-4
4-4
©G
lenc
oe/M
cGra
w-H
ill20
1G
lenc
oe G
eom
etry
Lesson 4-4
SSS
Post
ula
teYo
u k
now
th
at t
wo
tria
ngl
es a
re c
ongr
uen
t if
cor
resp
ondi
ng
side
s ar
eco
ngr
uen
t an
d co
rres
pon
din
g an
gles
are
con
gru
ent.
Th
e S
ide-
Sid
e-S
ide
(SS
S)
Pos
tula
te l
ets
you
sh
ow t
hat
tw
o tr
ian
gles
are
con
gru
ent
if y
ou k
now
on
ly t
hat
th
e si
des
of o
ne
tria
ngl
ear
e co
ngr
uen
t to
th
e si
des
of t
he
seco
nd
tria
ngl
e.
SS
S P
ost
ula
teIf
the
side
s of
one
tria
ngle
are
con
grue
nt t
o th
e si
des
of a
sec
ond
tria
ngle
, th
en t
he t
riang
les
are
cong
ruen
t.
Wri
te a
tw
o-co
lum
n p
roof
.G
iven
:A �
B��
D�B�
and
Cis
th
e m
idpo
int
of A�
D�.
Pro
ve:�
AB
C�
�D
BC
Sta
tem
ents
Rea
son
s
1.A�
B��
D�B�
1.G
iven
2.C
is t
he
mid
poin
t of
A�D�
.2.
Giv
en
3.A�
C��
D�C�
3.D
efin
itio
n o
f m
idpo
int
4.B�
C��
B�C�
4.R
efle
xive
Pro
pert
y of
�5.
�A
BC
��
DB
C5.
SS
S P
ostu
late
Wri
te a
tw
o-co
lum
n p
roof
.
B CD
A
Exam
ple
Exam
ple
Exer
cises
Exer
cises
1.
Giv
en:A�
B��
X�Y�
,A�C�
�X�
Z�,B�
C��
Y�Z�
Pro
ve:�
AB
C�
�X
YZ
Sta
tem
ents
Rea
son
s
1.A�
B��
X�Y�
1.G
iven
A�C�
�X�
Z�B�
C��
Y�Z�
2.�
AB
C�
�X
YZ
2.S
SS
Po
st.
2.
Giv
en:R�
S��
U�T�
,R�T�
�U�
S�P
rove
:�R
ST
��
UT
S
Sta
tem
ents
Rea
son
s
1.R�
S��
U�T�
1.G
iven
R�T�
�U�
S�2.
S�T�
�T�S�
2.R
efl.
Pro
p.
3.�
RS
T�
�U
TS
3.S
SS
Po
st.
TU
RS
BY
CA
XZ
©G
lenc
oe/M
cGra
w-H
ill20
2G
lenc
oe G
eom
etry
SAS
Post
ula
teA
not
her
way
to
show
th
at t
wo
tria
ngl
es a
re c
ongr
uen
t is
to
use
th
e S
ide-
An
gle-
Sid
e (S
AS
) P
ostu
late
.
SA
S P
ost
ula
teIf
two
side
s an
d th
e in
clud
ed a
ngle
of
one
tria
ngle
are
con
grue
nt t
o tw
o si
des
and
the
incl
uded
ang
le o
f an
othe
r tr
iang
le,
then
the
tria
ngle
s ar
e co
ngru
ent.
For
eac
h d
iagr
am,d
eter
min
e w
hic
h p
airs
of
tria
ngl
es c
an b
ep
rove
d c
ongr
uen
t b
y th
e S
AS
Pos
tula
te.
a.b
.c.
In �
AB
C,t
he
angl
e is
not
T
he
righ
t an
gles
are
T
he
incl
ude
d an
gles
,�1
“in
clu
ded”
by t
he
side
s A�
B�co
ngr
uen
t an
d th
ey a
re t
he
and
�2,
are
con
gru
ent
an
d A �
C�.S
o th
e tr
ian
gles
in
clu
ded
angl
es f
or t
he
beca
use
th
ey a
re
can
not
be
prov
ed c
ongr
uen
t co
ngr
uen
t si
des.
alte
rnat
e in
teri
or a
ngl
es
by t
he
SA
S P
ostu
late
.�
DE
F�
�J
GH
by t
he
for
two
para
llel
lin
es.
SA
S P
ostu
late
.�
PS
R�
�R
QP
by t
he
SA
S P
ostu
late
.
For
eac
h f
igu
re,d
eter
min
e w
hic
h p
airs
of
tria
ngl
es c
an b
e p
rove
d c
ongr
uen
t b
yth
e S
AS
Pos
tula
te.
1.2.
3.
�T
RU
��
PM
Nby
th
e �
XQ
Yan
d �
WQ
Zar
e �
MP
L�
�N
PL
SA
S P
ost
ula
te.
no
t th
e in
clu
ded
an
gle
s b
ecau
se b
oth
are
fo
r th
e co
ng
ruen
t ri
gh
t an
gle
s.se
gm
ents
.Th
e tr
ian
gle
s �
MP
L�
�N
PL
by
are
no
t co
ng
ruen
t by
th
e S
AS
Po
stu
late
.th
e S
AS
Po
stu
late
.
4.5.
6.
Th
e tr
ian
gle
s ca
nn
ot
�D
��
Bb
ecau
se
Th
e co
ng
ruen
t b
e p
rove
d c
on
gru
ent
bo
th a
re r
igh
t an
gle
s.an
gle
s ar
e th
e by
th
e S
AS
Po
stu
late
.T
he
two
tri
ang
les
are
incl
ud
ed a
ng
les
for
con
gru
ent
by t
he
SA
S
the
con
gru
ent
sid
es.
Po
stu
late
.�
FJH
��
GH
Jby
the
SA
S P
ost
ula
te.
JH
GF
K
CBA D
V
T
W
M
PL
N M
X
WZY
QT
P
UN
MR
PQ
S
1
2R
DH
FE
G
J
A BC
X YZ
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Pro
vin
g C
on
gru
ence
—S
SS
,SA
S
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-4
4-4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 4-4)
© Glencoe/McGraw-Hill A12 Glencoe Geometry
Skil
ls P
ract
ice
Pro
vin
g C
on
gru
ence
—S
SS
,SA
S
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-4
4-4
©G
lenc
oe/M
cGra
w-H
ill20
3G
lenc
oe G
eom
etry
Lesson 4-4
Det
erm
ine
wh
eth
er �
AB
C�
�K
LM
give
n t
he
coor
din
ates
of
the
vert
ices
.Exp
lain
.
1.A
(�3,
3),B
(�1,
3),C
(�3,
1),K
(1,4
),L
(3,4
),M
(1,6
)
AB
�2,
KL
�2,
BC
�2�
2�,L
M�
2�2�,
AC
�2,
KM
�2.
Th
e co
rres
po
nd
ing
sid
es h
ave
the
sam
e m
easu
re a
nd
are
co
ng
ruen
t,so
�A
BC
��
KL
Mby
SS
S.
2.A
(�4,
�2)
,B(�
4,1)
,C(�
1,�
1),K
(0,�
2),L
(0,1
),M
(4,1
)
AB
�3,
KL
�3,
BC
��
13�,L
M�
4,A
C�
�10�
,KM
�5.
Th
e co
rres
po
nd
ing
sid
es a
re n
ot
con
gru
ent,
so �
AB
Cis
no
t co
ng
ruen
t to
�K
LM
.
3.W
rite
a f
low
pro
of.
Giv
en:
P �R�
�D�
E�,P�
T��
D�F�
�R
��
E,�
T�
�F
Pro
ve:
�P
RT
��
DE
F
Pro
of:
Det
erm
ine
wh
ich
pos
tula
te c
an b
e u
sed
to
pro
ve t
hat
th
e tr
ian
gles
are
con
gru
ent.
If i
t is
not
pos
sib
le t
o p
rove
th
at t
hey
are
con
gru
ent,
wri
te n
ot p
ossi
ble.
4.5.
6.
SS
SS
AS
no
t p
oss
ible
PR �
DE
Give
n
PT �
DF
Give
n
�R
� �
E Gi
ven
�P
� �
D
Third
Ang
leTh
eore
m
�PR
T �
�D
EF
SAS
�T
� �
F Gi
ven
T R
P
F E
D
©G
lenc
oe/M
cGra
w-H
ill20
4G
lenc
oe G
eom
etry
Det
erm
ine
wh
eth
er �
DE
F�
�P
QR
give
n t
he
coor
din
ates
of
the
vert
ices
.Exp
lain
.
1.D
(�6,
1),E
(1,2
),F
(�1,
�4)
,P(0
,5),
Q(7
,6),
R(5
,0)
DE
�5�
2�,P
Q�
5�2�,
EF
�2�
10�,Q
R�
2�10�
,DF
�5�
2�,P
R�
5�2�.
�D
EF
��
PQ
Rby
SS
S s
ince
co
rres
po
nd
ing
sid
es h
ave
the
sam
em
easu
re a
nd
are
co
ng
ruen
t.
2.D
(�7,
�3)
,E(�
4,�
1),F
(�2,
�5)
,P(2
,�2)
,Q(5
,�4)
,R(0
,�5)
DE
��
13�,P
Q�
�13
,�
EF
�2�
5�,Q
R�
�26�
,DF
��
29�,P
R�
�13�
.C
orr
esp
on
din
g s
ides
are
no
t co
ng
ruen
t,so
�D
EF
is n
ot
con
gru
ent
to �
PQ
R.
3.W
rite
a f
low
pro
of.
Giv
en:
R �S�
�T�
S�V
is t
he
mid
poin
t of
R�T�
.P
rove
:�
RS
V�
�T
SV
Pro
of:
Det
erm
ine
wh
ich
pos
tula
te c
an b
e u
sed
to
pro
ve t
hat
th
e tr
ian
gles
are
con
gru
ent.
If i
t is
not
pos
sib
le t
o p
rove
th
at t
hey
are
con
gru
ent,
wri
te n
ot p
ossi
ble.
4.5.
6.
no
t p
oss
ible
SA
S o
r S
SS
SS
S
7.IN
DIR
ECT
MEA
SUR
EMEN
TT
o m
easu
re t
he
wid
th o
f a
sin
khol
e on
h
is p
rope
rty,
Har
mon
mar
ked
off
con
gru
ent
tria
ngl
es a
s sh
own
in
th
edi
agra
m.H
ow d
oes
he
know
th
at t
he
len
gth
s A�
B�
and
AB
are
equ
al?
Sin
ce �
AC
Ban
d �
A�C
B�
are
vert
ical
an
gle
s,th
ey a
re
con
gru
ent.
In t
he
fig
ure
,A�C�
�A�
��C�an
d B�
C��
B���C�
.So
�
AB
C�
�A
�B�C
by S
AS
.By
CP
CT
C,t
he
len
gth
s A
�B�
and
AB
are
equ
al.A
�B
�
AB
C
RS
� T
SGi
ven
SV �
SV
Refle
xive
Prop
erty
RV
� V
TDe
finiti
onof
mid
poin
t
V is
th
em
idp
oin
t o
f R
T.
Give
n
�R
SV �
�TS
V
SSS
S
R V T
Pra
ctic
e (
Ave
rag
e)
Pro
vin
g C
on
gru
ence
—S
SS
,SA
S
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-4
4-4
Answers (Lesson 4-4)
© Glencoe/McGraw-Hill A13 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csP
rovi
ng
Co
ng
ruen
ce—
SS
S,S
AS
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-4
4-4
©G
lenc
oe/M
cGra
w-H
ill20
5G
lenc
oe G
eom
etry
Lesson 4-4
Pre-
Act
ivit
yH
ow d
o la
nd
su
rvey
ors
use
con
gru
ent
tria
ngl
es?
Rea
d th
e in
trod
uct
ion
to
Les
son
4-4
at
the
top
of p
age
200
in y
our
text
book
.
Wh
y do
you
th
ink
that
lan
d su
rvey
ors
wou
ld u
se c
ongr
uen
t ri
ght
tria
ngl
esra
ther
th
an o
ther
con
gru
ent
tria
ngl
es t
o es
tabl
ish
pro
pert
y bo
un
dari
es?
Sam
ple
an
swer
:L
and
is u
sual
ly d
ivid
ed in
to r
ecta
ng
ula
r lo
ts,
so t
hei
r b
ou
nd
arie
s m
eet
at r
igh
t an
gle
s.
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.
a.N
ame
the
side
s of
�L
MN
for
wh
ich
�L
is t
he
incl
ude
d an
gle.
L�M�,L�
N�b
.N
ame
the
side
s of
�L
MN
for
wh
ich
�N
is t
he
incl
ude
d an
gle.
N�L�,
N�M�
c.N
ame
the
side
s of
�L
MN
for
wh
ich
�M
is t
he
incl
ude
d an
gle.
M�L�,
M�N�
2.D
eter
min
e w
het
her
you
hav
e en
ough
in
form
atio
n t
o pr
ove
that
th
e tw
o tr
ian
gles
in
eac
hfi
gure
are
con
gru
ent.
If s
o,w
rite
a c
ongr
uen
ce s
tate
men
t an
d n
ame
the
con
gru
ence
post
ula
te t
hat
you
wou
ld u
se.I
f n
ot,w
rite
not
pos
sibl
e.
a.b
.
�A
BD
��
CB
D;
SA
Sn
ot
po
ssib
le
c.E�
H�an
d D�
G�bi
sect
eac
h o
ther
.d
.
�D
EF
��
GH
F;
SA
S�
RS
U�
�T
SU
;S
SS
Hel
pin
g Y
ou
Rem
emb
er
3.F
ind
thre
e w
ords
th
at e
xpla
in w
hat
it
mea
ns
to s
ay t
hat
tw
o tr
ian
gles
are
con
gru
ent
and
that
can
hel
p yo
u r
ecal
l th
e m
ean
ing
of t
he
SS
S P
ostu
late
.S
amp
le a
nsw
er:
Co
ng
ruen
t tr
ian
gle
s ar
e tr
ian
gle
s th
at a
re t
he
sam
e si
zean
d s
hap
e,an
d t
he
SS
S P
ost
ula
te e
nsu
res
that
tw
o t
rian
gle
s w
ith
th
ree
corr
esp
on
din
g s
ides
co
ng
ruen
t w
ill b
e th
e sa
me
size
an
d s
hap
e.
GE
F
HD
R T
SU
G
FD
E
CA
DB
L
N
M
©G
lenc
oe/M
cGra
w-H
ill20
6G
lenc
oe G
eom
etry
Co
ng
ruen
t P
arts
of
Reg
ula
r P
oly
go
nal
Reg
ion
sC
ongr
uen
t fi
gure
s ar
e fi
gure
s th
at h
ave
exac
tly
the
sam
e si
ze a
nd
shap
e.T
her
e ar
e m
any
way
s to
div
ide
regu
lar
poly
gon
al r
egio
ns
into
con
gru
ent
part
s.T
hre
e w
ays
to d
ivid
e an
equ
ilat
eral
tri
angu
lar
regi
on a
re s
how
n.Y
ou c
an v
erif
y th
at t
he
part
s ar
e co
ngr
uen
t by
trac
ing
one
part
,th
en r
otat
ing,
slid
ing,
or r
efle
ctin
g th
at p
art
on t
op o
f th
e ot
her
par
ts.
1.D
ivid
e ea
ch s
quar
e in
to f
our
con
gru
ent
part
s.U
se t
hre
e di
ffer
ent
way
s.S
amp
le a
nsw
ers
are
sho
wn
.
2.D
ivid
e ea
ch p
enta
gon
in
to f
ive
con
gru
ent
part
s.U
se t
hre
e di
ffer
ent
way
s.S
amp
le a
nsw
ers
are
sho
wn
.
3.D
ivid
e ea
ch h
exag
on i
nto
six
con
gru
ent
part
s.U
se t
hre
e di
ffer
ent
way
s.S
amp
le a
nsw
ers
are
sho
wn
.
4.W
hat
hin
ts m
igh
t yo
u g
ive
anot
her
stu
den
t w
ho
is t
ryin
g to
div
ide
figu
res
like
th
ose
into
con
gru
ent
part
s?S
ee s
tud
ents
’wo
rk.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-4
4-4
Answers (Lesson 4-4)
© Glencoe/McGraw-Hill A14 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Pro
vin
g C
on
gru
ence
—A
SA
,AA
S
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-5
4-5
©G
lenc
oe/M
cGra
w-H
ill20
7G
lenc
oe G
eom
etry
Lesson 4-5
ASA
Po
stu
late
Th
e A
ngl
e-S
ide-
An
gle
(AS
A)
Pos
tula
te l
ets
you
sh
ow t
hat
tw
o tr
ian
gles
are
con
gru
ent.
AS
A P
ost
ula
teIf
two
angl
es a
nd t
he in
clud
ed s
ide
of o
ne t
riang
le a
re c
ongr
uent
to
two
angl
es
and
the
incl
uded
sid
e of
ano
ther
tria
ngle
, th
en t
he t
riang
les
are
cong
ruen
t.
Fin
d t
he
mis
sin
g co
ngr
uen
t p
arts
so
that
th
e tr
ian
gles
can
be
pro
ved
con
gru
ent
by
the
AS
A P
ostu
late
.Th
en w
rite
th
e tr
ian
gle
con
gru
ence
.
a.
Tw
o pa
irs
of c
orre
spon
din
g an
gles
are
con
gru
ent,
�A
��
Dan
d �
C�
�F
.If
the
incl
ude
d si
des
A�C�
and
D�F�
are
con
gru
ent,
then
�A
BC
��
DE
Fby
th
e A
SA
Pos
tula
te.
b.
�R
��
Yan
d S�
R��
X�Y�
.If
�S
��
X,t
hen
�R
ST
��
YX
Wby
th
e A
SA
Pos
tula
te.
Wh
at c
orre
spon
din
g p
arts
mu
st b
e co
ngr
uen
t in
ord
er t
o p
rove
th
at t
he
tria
ngl
esar
e co
ngr
uen
t b
y th
e A
SA
Pos
tula
te?
Wri
te t
he
tria
ngl
e co
ngr
uen
ce s
tate
men
t.
1.2.
3.
D�C�
�B�
C�;
W�Y�
�W�
Y�;
�A
BE
��
CB
D;
�C
DE
��
CB
A�
XY
W�
�Z
YW
;�
AB
E�
�C
BD
�W
XY
��
WZ
Y
4.5.
6.
B�D�
�D�
B�;
S�T�
�V�
T�;�
AC
B�
�E
;�
AD
B �
�C
BD
;�
RS
T�
�U
VT
�A
BC
��
CD
E�
AB
D�
�C
DB
AC
B
E
D
S
V
U
RT
D
AB
C
DC
EA
B
YW
X ZE
A
BD
C
RT
WY
SX
AC
B
DF
E
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill20
8G
lenc
oe G
eom
etry
AA
S Th
eore
mA
not
her
way
to
show
th
at t
wo
tria
ngl
es a
re c
ongr
uen
t is
th
e A
ngl
e-A
ngl
e-S
ide
(AA
S) T
heo
rem
.
AA
S T
heo
rem
If tw
o an
gles
and
a n
onin
clud
ed s
ide
of o
ne t
riang
le a
re c
ongr
uent
to
the
corr
espo
ndin
g tw
oan
gles
and
sid
e of
a s
econ
d tr
iang
le,
then
the
tw
o tr
iang
les
are
cong
ruen
t.
You
now
hav
e fi
ve w
ays
to s
how
th
at t
wo
tria
ngl
es a
re c
ongr
uen
t.•
defi
nit
ion
of
tria
ngl
e co
ngr
uen
ce•
AS
A P
ostu
late
•S
SS
Pos
tula
te•
AA
S T
heo
rem
•S
AS
Pos
tula
te
In t
he
dia
gram
,�B
CA
��
DC
A.W
hic
h s
ides
ar
e co
ngr
uen
t? W
hic
h a
dd
itio
nal
pai
r of
cor
resp
ond
ing
par
ts
nee
ds
to b
e co
ngr
uen
t fo
r th
e tr
ian
gles
to
be
con
gru
ent
by
the
AA
S P
ostu
late
?A �
C��
A�C�
by t
he
Ref
lexi
ve P
rope
rty
of c
ongr
uen
ce.T
he
con
gru
ent
angl
es c
ann
ot b
e �
1 an
d �
2,be
cau
se A �
C�w
ould
be
the
incl
ude
d si
de.
If �
B�
�D
,th
en �
AB
C�
�A
DC
by t
he
AA
S T
heo
rem
.
In E
xerc
ises
1 a
nd
2,d
raw
an
d l
abel
�A
BC
and
�D
EF
.In
dic
ate
wh
ich
ad
dit
ion
alp
air
of c
orre
spon
din
g p
arts
nee
ds
to b
e co
ngr
uen
t fo
r th
e tr
ian
gles
to
be
con
gru
ent
by
the
AA
S T
heo
rem
.
1.�
A�
�D
;�B
��
E2.
BC
�E
F;�
A�
�D
If B�
C��
E�F�
(or
if A�
C��
D�F�
),If
�C
��
F (
or
if �
B�
�E
),th
en �
AB
C�
�D
EF
by
the
then
�A
BC
��
DE
F by
the
A
AS
Th
eore
m.
AA
S T
heor
em.
3.W
rite
a f
low
pro
of.
Giv
en:�
S�
�U
;T �R�
bise
cts
�S
TU
.P
rove
:�S
RT
��
UR
T
Give
n
Give
n
RT
� R
T
Refl.
Pro
p. o
f �
Def.o
f � b
isec
tor
TR b
isec
ts �
STU
.
�SR
T �
�U
RT
�ST
R �
�U
TR
AAS
�SR
T �
�U
RT
CPCT
C�
S �
�U
S
RT
U
B
A
C
E
D
F
CA
B
FD
E
D
C1 2
A
B
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Pro
vin
g C
on
gru
ence
—A
SA
,AA
S
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-5
4-5
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 4-5)
© Glencoe/McGraw-Hill A15 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Pro
vin
g C
on
gru
ence
—A
SA
,AA
S
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-5
4-5
©G
lenc
oe/M
cGra
w-H
ill20
9G
lenc
oe G
eom
etry
Lesson 4-5
Wri
te a
flo
w p
roof
.
1.G
iven
:�
N�
�L
J �K��
M�K�
Pro
ve:
�J
KN
��
MK
L
Pro
of:
2.G
iven
:A�
B��
C�B�
�A
��
CD �
B�bi
sect
s �
AB
C.
Pro
ve:
A �D�
�C�
D�
Pro
of:
3.W
rite
a p
arag
raph
pro
of.
Giv
en:
D�E�
|| F�G�
�E
��
GP
rove
:�
DF
G�
�F
DE
Pro
of:
Sin
ce it
is g
iven
th
at D�
E�|| F�
G�,i
t fo
llow
s th
at �
ED
F�
�G
FD
,b
ecau
se a
lt.i
nt.
�ar
e �
.It
is g
iven
th
at �
E�
�G
.By
the
Ref
lexi
ve
Pro
per
ty, D�
F��
F�D�.S
o �
DF
G�
�F
DE
by A
AS
.
FG
DE
�A
� �
C
Give
nA
B �
CB
Give
nCP
CTC
AD
� C
D
DB
bis
ects
�A
BC
. Gi
ven
�A
BD
� �
CB
DAS
A
�A
BD
� �
CB
DDe
f. of
� b
isec
tor
AC
B D
�N
� �
LGi
ven
JK �
MK
Gi
ven
�JK
N �
�M
KL
Verti
cal �
are
�.
�JK
N �
�M
KL
AAS
N
J
M
KL
©G
lenc
oe/M
cGra
w-H
ill21
0G
lenc
oe G
eom
etry
1.W
rite
a f
low
pro
of.
Giv
en:
Sis
th
e m
idpo
int
of Q �
T�.
Q �R�
|| T�U�
Pro
ve:
�Q
SR
��
TS
US
amp
le p
roo
f:
2.W
rite
a p
arag
raph
pro
of.
Giv
en:
�D
��
FG �
E�bi
sect
s �
DE
F.
Pro
ve:
D �G�
�F�
G�
Pro
of:
Sin
ce it
is g
iven
th
at G�
E�b
isec
ts �
DE
F,�
DE
G�
�F
EG
by t
he
def
init
ion
of
an a
ng
le b
isec
tor.
It is
giv
en t
hat
�D
��
F.B
y th
e R
efle
xive
Pro
per
ty, G�
E��
G�E�
.So
�D
EG
��
FE
Gby
AA
S.T
her
efo
re
D�G�
�F�G�
by C
PC
TC
.
AR
CH
ITEC
TUR
EF
or E
xerc
ises
3 a
nd
4,u
se t
he
foll
owin
g in
form
atio
n.
An
arc
hit
ect
use
d th
e w
indo
w d
esig
n i
n t
he
diag
ram
wh
en r
emod
elin
g an
art
stu
dio.
A �B�
and
C�B�
each
mea
sure
3 f
eet.
3.S
upp
ose
Dis
th
e m
idpo
int
of A�
C�.D
eter
min
e w
het
her
�A
BD
��
CB
D.
Just
ify
you
r an
swer
.
Sin
ce D
is t
he
mid
po
int
of
A�C�
,A�D�
�C�
D�by
th
e d
efin
itio
n o
f m
idp
oin
t.A�
B��
C�B�
by t
he
def
init
ion
of
con
gru
ent
seg
men
ts.B
y th
e R
efle
xive
P
rop
erty
, B�D�
�B�
D�.S
o �
AB
D�
�C
BD
by S
SS
.
4.S
upp
ose
�A
��
C.D
eter
min
e w
het
her
�A
BD
��
CB
D.J
ust
ify
you
r an
swer
.
We
are
giv
en A�
B��
C�B�
and
�A
��
C.B�
D��
B�D�
by t
he
Ref
lexi
ve
Pro
per
ty.S
ince
SS
A c
ann
ot
be
use
d t
o p
rove
th
at t
rian
gle
s ar
e co
ng
ruen
t,w
e ca
nn
ot
say
wh
eth
er �
AB
D�
�C
BD
.
DB
AC
D
G
F
E
�Q
� �
T
Give
n
QR
|| TU
Gi
ven
Def.o
f mid
poin
t
Alt.
Int.
� a
re �
.
QS
� T
S S
is t
he
mid
po
int
of
QT.
�Q
SR �
�TS
UAS
A
�Q
SR �
�TS
UVe
rtica
l � a
re �
.
UQ
S
RT
Pra
ctic
e (
Ave
rag
e)
Pro
vin
g C
on
gru
ence
—A
SA
,AA
S
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-5
4-5
Answers (Lesson 4-5)
© Glencoe/McGraw-Hill A16 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csP
rovi
ng
Co
ng
ruen
ce—
AS
A,A
AS
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-5
4-5
©G
lenc
oe/M
cGra
w-H
ill21
1G
lenc
oe G
eom
etry
Lesson 4-5
Pre-
Act
ivit
yH
ow a
re c
ongr
uen
t tr
ian
gles
use
d i
n c
onst
ruct
ion
?R
ead
the
intr
odu
ctio
n t
o L
esso
n 4
-5 a
t th
e to
p of
pag
e 20
7 in
you
r te
xtbo
ok.
Wh
ich
of
the
tria
ngl
es i
n t
he
phot
ogra
ph i
n y
our
text
book
app
ear
to b
eco
ngru
ent?
Sam
ple
an
swer
:Th
e fo
ur
rig
ht
tria
ng
les
are
con
gru
ent
to e
ach
oth
er.T
he
two
ob
tuse
iso
scel
es t
rian
gle
s ar
e co
ng
ruen
tto
eac
h o
ther
.R
ead
ing
th
e Le
sso
n1.
Exp
lain
in
you
r ow
n w
ords
th
e di
ffer
ence
bet
wee
n h
ow t
he
AS
A P
ostu
late
an
d th
e A
AS
Th
eore
m a
re u
sed
to p
rove
th
at t
wo
tria
ngl
es a
re c
ongr
uen
t.S
amp
le a
nsw
er:
In A
SA
,yo
u u
se t
wo
pai
rs o
f co
ng
ruen
t an
gle
s an
d t
he
incl
ud
edco
ng
ruen
t si
des
.In
AA
S,y
ou
use
tw
o p
airs
of
con
gru
ent
ang
les
and
a p
air
of
no
nin
clu
ded
con
gru
ent
sid
es.
B,D
,E,G
,H2.
Whi
ch o
f th
e fo
llow
ing
cond
itio
ns a
re s
uffi
cien
t to
pro
ve t
hat
two
tria
ngle
s ar
e co
ngru
ent?
A.
Tw
o si
des
of o
ne
tria
ngl
e ar
e co
ngr
uen
t to
tw
o si
des
of t
he
oth
er t
rian
gle.
B.T
he
thre
e si
des
of o
ne
tria
ngl
es a
re c
ongr
uen
t to
th
e th
ree
side
s of
th
e ot
her
tri
angl
e.C
.T
he t
hree
ang
les
of o
ne t
rian
gle
are
cong
ruen
t to
the
thr
ee a
ngle
s of
the
oth
er t
rian
gle.
D.A
ll s
ix c
orre
spon
din
g pa
rts
of t
wo
tria
ngl
es a
re c
ongr
uen
t.E
.T
wo
angl
es a
nd
the
incl
ude
d si
de o
f on
e tr
ian
gle
are
con
gru
ent
to t
wo
side
s an
d th
ein
clu
ded
angl
e of
th
e ot
her
tri
angl
e.F.
Tw
o si
des
and
a n
onin
clu
ded
angl
e of
on
e tr
ian
gle
are
con
gru
ent
to t
wo
side
s an
d a
non
incl
ude
d an
gle
of t
he
oth
er t
rian
gle.
G.T
wo
angl
es a
nd
a n
onin
clu
ded
side
of
one
tria
ngl
e ar
e co
ngr
uen
t to
tw
o an
gles
an
dth
e co
rres
pon
din
g n
onin
clu
ded
side
of
the
oth
er t
rian
gle.
H.T
wo
side
s an
d th
e in
clu
ded
angl
e of
on
e tr
ian
gle
are
con
gru
ent
to t
wo
side
s an
d th
ein
clu
ded
angl
e of
th
e ot
her
tri
angl
e.I.
Tw
o an
gles
an
d a
non
incl
ude
d si
de o
f on
e tr
ian
gle
are
con
gru
ent
to t
wo
angl
es a
nd
an
onin
clu
ded
side
of
the
oth
er t
rian
gle.
3.D
eter
min
e w
het
her
you
hav
e en
ough
in
form
atio
n t
o pr
ove
that
th
e tw
o tr
ian
gles
in
eac
hfi
gure
are
con
gru
ent.
If s
o,w
rite
a c
ongr
uen
ce s
tate
men
t an
d n
ame
the
con
gru
ence
post
ula
te o
r th
eore
m t
hat
you
wou
ld u
se.I
f n
ot,w
rite
not
pos
sibl
e.
a.�
AE
B�
�D
EC
;A
AS
b.
Tis
th
e m
idpo
int
of R�
U�.
�R
ST
��
UV
T;
AS
A
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r m
athe
mat
ical
idea
s is
to
sum
mar
ize
them
in a
gen
eral
sta
tem
ent.
If y
ou w
ant
to p
rove
tri
angl
es c
ongr
uen
t by
usi
ng
thre
e pa
irs
of c
orre
spon
din
g pa
rts,
wh
at i
s a
good
way
to
rem
embe
r w
hic
h c
ombi
nat
ion
s of
par
ts w
ill
wor
k?S
amp
le a
nsw
er:A
t le
ast
on
e p
air
of
corr
esp
on
din
g p
arts
mu
st b
e si
des
.If
you
use
tw
o p
airs
of
sid
es a
nd
on
e p
air
of
ang
les,
the
ang
les
mu
st b
e th
ein
clu
ded
an
gle
s.If
yo
u u
se t
wo
pai
rs o
f an
gle
s an
d o
ne
pai
r o
f si
des
,th
en t
he
sid
es m
ust
bo
th b
e in
clu
ded
by
the
ang
les
or
mu
st b
oth
be
corr
esp
on
din
g n
on
incl
ud
ed s
ides
.
RS
T
U V
AD
CB
E
©G
lenc
oe/M
cGra
w-H
ill21
2G
lenc
oe G
eom
etry
Co
ng
ruen
t Tri
ang
les
in t
he
Co
ord
inat
e P
lan
eIf
you
kn
ow t
he
coor
din
ates
of
the
vert
ices
of
two
tria
ngl
es i
n t
he
coor
din
ate
plan
e,yo
u c
an o
ften
dec
ide
wh
eth
er t
he
two
tria
ngl
es a
re c
ongr
uen
t.T
her
em
ay b
e m
ore
than
on
e w
ay t
o do
th
is.
1.C
onsi
der
�A
BD
and
�C
DB
wh
ose
vert
ices
hav
e co
ordi
nat
es A
(0,0
),B
(2,5
),C
(9,5
),an
d D
(7,0
).B
rief
ly d
escr
ibe
how
you
can
use
wh
at y
oukn
ow a
bou
t co
ngr
uen
t tr
ian
gles
an
d th
e co
ordi
nat
e pl
ane
to s
how
th
at
�A
BD
��
CD
B.Y
ou m
ay w
ish
to
mak
e a
sket
ch t
o h
elp
get
you
sta
rted
.
Sam
ple
an
swer
:S
ho
w t
hat
th
e sl
op
es o
f A �
B �an
d C �
D �ar
eeq
ual
an
d t
hat
th
e sl
op
es o
f A �
D �an
d B �
C �ar
e eq
ual
.Co
ncl
ud
eth
at A �
B � �
C �D �
and
B �C �
�A �
D �.U
se t
he
ang
le r
elat
ion
ship
s fo
rp
aral
lel l
ines
an
d a
tra
nsv
ersa
l an
d t
he
fact
th
at B �
D �is
a c
om
-m
on
sid
e fo
r th
e tr
ian
gle
s to
co
ncl
ud
e th
at
�A
BD
��
CD
Bby
AS
A.
2.C
onsi
der
�P
QR
and
�K
LM
wh
ose
vert
ices
are
th
e fo
llow
ing
poin
ts.
P(1
,2)
Q(3
,6)
R(6
,5)
K(�
2,1)
L(�
6,3)
M(�
5,6)
Bri
efly
des
crib
e h
ow y
ou c
an s
how
th
at �
PQ
R�
�K
LM
.
Use
th
e D
ista
nce
Fo
rmu
la t
o f
ind
th
e le
ng
ths
of
the
sid
es o
fb
oth
tri
ang
les.
Co
ncl
ud
e th
at �
PQ
R�
�K
LM
by S
SS
.
3.If
you
kn
ow t
he
coor
din
ates
of
all
the
vert
ices
of
two
tria
ngl
es,i
s it
al
way
spo
ssib
le t
o te
ll w
het
her
th
e tr
ian
gles
are
con
gru
ent?
Exp
lain
.
Yes;
you
can
use
th
e D
ista
nce
Fo
rmu
la a
nd
SS
S.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-5
4-5
Answers (Lesson 4-5)
© Glencoe/McGraw-Hill A17 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Iso
scel
es T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-6
4-6
©G
lenc
oe/M
cGra
w-H
ill21
3G
lenc
oe G
eom
etry
Lesson 4-6
Pro
per
ties
of
Iso
scel
es T
rian
gle
sA
n i
sosc
eles
tri
angl
eh
as t
wo
con
gru
ent
side
s.T
he
angl
e fo
rmed
by
thes
e si
des
is c
alle
d th
e ve
rtex
an
gle.
Th
e ot
her
tw
o an
gles
are
cal
led
bas
e an
gles
.You
can
pro
ve a
th
eore
m a
nd
its
con
vers
e ab
out
isos
cele
s tr
ian
gles
.
•If
tw
o si
des
of a
tri
angl
e ar
e co
ngr
uen
t,th
en t
he
angl
es o
ppos
ite
thos
e si
des
are
con
gru
ent.
(Iso
scel
es T
rian
gle
Th
eore
m)
•If
tw
o an
gles
of
a tr
ian
gle
are
con
gru
ent,
then
th
e si
des
oppo
site
th
ose
angl
es a
re c
ongr
uen
t.If
A�B�
�C�
B�,
then
�A
��
C.
If �
A�
�C
, th
en A�
B��
C�B�
.
A
B
C
Fin
d x
.
BC
�B
A,s
o m
�A
�m
�C
.Is
os. T
riang
le T
heor
em
5x�
10 �
4x�
5S
ubst
itutio
n
x�
10 �
5S
ubtr
act
4xfr
om e
ach
side
.
x�
15A
dd 1
0 to
eac
h si
de.
B
AC( 4
x �
5) �
( 5x
� 1
0)�
Fin
d x
.
m�
S�
m�
T,s
oS
R�
TR
.C
onve
rse
of I
sos.
�T
hm.
3x�
13 �
2xS
ubst
itutio
n
3x�
2x�
13A
dd 1
3 to
eac
h si
de.
x�
13S
ubtr
act
2xfr
om e
ach
side
.
RT
S 3x �
13
2x
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d x
.
1.35
2.12
3.15
4.12
5.20
6.36
7.W
rite
a t
wo-
colu
mn
pro
of.
Giv
en:�
1 �
�2
Pro
ve:A �
B��
C�B�
Sta
tem
ents
Rea
son
s
1.�
1 �
�2
1.G
iven
2.�
2 �
�3
2.V
erti
cal a
ng
les
are
con
gru
ent.
3.�
1 �
�3
3.Tr
ansi
tive
Pro
per
ty o
f �
4.A�
B��
C�B�
4.If
tw
o a
ng
les
of
a tr
ian
gle
are
�,t
hen
th
e si
des
op
po
site
th
e an
gle
s ar
e �
.
B
AC
D
E
13
2
RS
T 3x�
x�D
BG
L3x
�
30�
D TQP
K
( 6x
� 6
) �2x
�
W
YZ
3x�
S
V
T3x
� 6
2x �
6R
P
Q2x
�40
�
©G
lenc
oe/M
cGra
w-H
ill21
4G
lenc
oe G
eom
etry
Pro
per
ties
of
Equ
ilate
ral T
rian
gle
sA
n e
qu
ilat
eral
tri
angl
eh
as t
hre
e co
ngr
uen
tsi
des.
Th
e Is
osce
les
Tri
angl
e T
heo
rem
can
be
use
d to
pro
ve t
wo
prop
erti
es o
f eq
uil
ater
altr
ian
gles
.
1.A
tria
ngle
is e
quila
tera
l if
and
only
if it
is e
quia
ngul
ar.
2.E
ach
angl
e of
an
equi
late
ral t
riang
le m
easu
res
60°.
Pro
ve t
hat
if
a li
ne
is p
aral
lel
to o
ne
sid
e of
an
eq
uil
ater
al t
rian
gle,
then
it
form
s an
oth
er e
qu
ilat
eral
tr
ian
gle.
Pro
of:
Sta
tem
ents
Rea
son
s
1.�
AB
Cis
equ
ilat
eral
;P�Q�
|| B�C�
.1.
Giv
en2.
m�
A�
m�
B�
m�
C�
602.
Eac
h �
of a
n e
quil
ater
al �
mea
sure
s 60
°.3.
�1
��
B,�
2 �
�C
3.If
||li
nes
,th
en c
orre
s.�
s ar
e �
.4.
m�
1 �
60,m
�2
�60
4.S
ubs
titu
tion
5.�
AP
Qis
equ
ilat
eral
.5.
If a
�is
equ
ian
gula
r,th
en i
t is
equ
ilat
eral
.
Fin
d x
.
1.10
2.5
3.10
4.10
5.12
6.15
7.W
rite
a t
wo-
colu
mn
pro
of.
Giv
en:�
AB
Cis
equ
ilat
eral
;�1
��
2.P
rove
:�A
DB
��
CD
B
Pro
of:
Sta
tem
ents
Rea
son
s
1.�
AB
Cis
eq
uila
tera
l.1.
Giv
en2.
A�B�
�C�
B�;
�A
��
C2.
An
eq
uila
tera
l �h
as �
sid
es a
nd
�an
gle
s.3.
�1
��
23.
Giv
en4.
�A
BD
��
CB
D4.
AS
A P
ost
ula
te5.
�A
DB
��
CD
B5.
CP
CT
CA D C
B1 2
R O
HM
60�
4x�
X
ZY
4x �
4
3x �
860
�
PQ
LV
R60
�
4x40
L N M
K
�K
LM is
equ
ilate
ral.
3x�
G
JH
6x �
55 x
D
FE
6x�
A
B
PQ
C
12
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Iso
scel
es T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-6
4-6
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 4-6)
© Glencoe/McGraw-Hill A18 Glencoe Geometry
Skil
ls P
ract
ice
Iso
scel
es T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-6
4-6
©G
lenc
oe/M
cGra
w-H
ill21
5G
lenc
oe G
eom
etry
Lesson 4-6
Ref
er t
o th
e fi
gure
.
1.If
A�C�
�A�
D�,n
ame
two
con
gru
ent
angl
es.
�A
CD
��
CD
A
2.If
B�E�
�B�
C�,n
ame
two
con
gru
ent
angl
es.
�B
EC
��
BC
E
3.If
�E
BA
��
EA
B,n
ame
two
con
gru
ent
segm
ents
.
E�B�
�E�
A�
4.If
�C
ED
��
CD
E,n
ame
two
con
gru
ent
segm
ents
.
C�E�
�C�
D�
�A
BF
is i
sosc
eles
,�C
DF
is e
qu
ilat
eral
,an
d m
�A
FD
�15
0.F
ind
eac
h m
easu
re.
5.m
�C
FD
606.
m�
AF
B55
7.m
�A
BF
708.
m�
A55
In t
he
figu
re,P�
L��
R�L�
and
L�R�
�B�
R�.
9.If
m�
RL
P�
100,
fin
d m
�B
RL
.20
10.I
f m
�L
PR
�34
,fin
d m
�B
.68
11.W
rite
a t
wo-
colu
mn
pro
of.
Giv
en:
C�D�
�C �
G�D�
E��
G�F�
Pro
ve:
C�E�
�C�
F�
Pro
of:
Sta
tem
ents
Rea
son
s
1.C�
D��
C�G�
1.G
iven
2.�
D�
�G
2.If
2 s
ides
of
a �
are
�,t
hen
th
e �
op
po
site
tho
se s
ides
are
�.
3.D�
E��
G�F�
3.G
iven
4.�
CD
E�
�C
GF
4.S
AS
5.C�
E��
C�F�
5.C
PC
TC
D E F G
CRP
BL
D
C F
B
35�
AE
D
C
B
AE
©G
lenc
oe/M
cGra
w-H
ill21
6G
lenc
oe G
eom
etry
Ref
er t
o th
e fi
gure
.
1.If
R�V�
�R�
T�,n
ame
two
con
gru
ent
angl
es.
�R
TV
��
RV
T
2.If
R�S�
�S�
V�,n
ame
two
con
gru
ent
angl
es.
�S
VR
��
SR
V
3.If
�S
RT
��
ST
R,n
ame
two
con
gru
ent
segm
ents
.S�
T��
S�R�
4.If
�S
TV
��
SV
T,n
ame
two
con
gru
ent
segm
ents
.S�
T��
S�V�
Tri
angl
es G
HM
and
HJ
Mar
e is
osce
les,
wit
h G�
H��
M�H�
and
H�J�
�M�
J�.T
rian
gle
KL
Mis
eq
uil
ater
al,a
nd
m�
HM
K�
50.
Fin
d e
ach
mea
sure
.
5.m
�K
ML
606.
m�
HM
G70
7.m
�G
HM
40
8.If
m�
HJ
M�
145,
fin
d m
�M
HJ
.17
.5
9.If
m�
G�
67,f
ind
m�
GH
M.
46
10.W
rite
a t
wo-
colu
mn
pro
of.
Giv
en:
D�E�
|| B�C�
�1
��
2P
rove
:A �
B��
A�C�
Pro
of:
Sta
tem
ents
Rea
son
s
1.D�
E�|| B�
C�1.
Giv
en
2.�
1 �
�4
2.C
orr
.�ar
e �
.�
2 �
�3
3.�
1 �
�2
3.G
iven
4.�
3 �
�4
4.C
on
gru
ence
of
�is
tra
nsi
tive
.
5.A�
B��
A�C�
5.If
2 �
of
a �
are
�,t
hen
th
e si
des
op
po
site
th
ose
�ar
e �
.
11.S
POR
TSA
pen
nan
t fo
r th
e sp
orts
tea
ms
at L
inco
ln H
igh
S
choo
l is
in
th
e sh
ape
of a
n i
sosc
eles
tri
angl
e.If
th
e m
easu
re
of t
he
vert
ex a
ngl
e is
18,
fin
d th
e m
easu
re o
f ea
ch b
ase
angl
e.81
,81
Linc
oln
Haw
ks
E DBC
A12
3 4
GMLK
J
H
UR
TV
S
Pra
ctic
e (
Ave
rag
e)
Iso
scel
es T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-6
4-6
Answers (Lesson 4-6)
© Glencoe/McGraw-Hill A19 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csIs
osc
eles
Tri
ang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-6
4-6
©G
lenc
oe/M
cGra
w-H
ill21
7G
lenc
oe G
eom
etry
Lesson 4-6
Pre-
Act
ivit
yH
ow a
re t
rian
gles
use
d i
n a
rt?
Rea
d th
e in
trod
uct
ion
to
Les
son
4-6
at
the
top
of p
age
216
in y
our
text
book
.
•W
hy
do y
ou t
hin
k th
at i
sosc
eles
an
d eq
uil
ater
al t
rian
gles
are
use
d m
ore
ofte
n t
han
sca
len
e tr
ian
gles
in
art
?S
amp
le a
nsw
er:T
hei
rsy
mm
etry
is p
leas
ing
to
th
e ey
e.•
Wh
y m
igh
t is
osce
les
righ
t tr
ian
gles
be
use
d in
art
?S
amp
le a
nsw
er:
Two
co
ng
ruen
t is
osc
eles
rig
ht
tria
ng
les
can
be
pla
ced
tog
eth
er t
o f
orm
a s
qu
are.
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.
a.W
hat
kin
d of
tri
angl
e is
�Q
RS
?is
osc
eles
b.
Nam
e th
e le
gs o
f �
QR
S.
Q�S�
,R�S�
c.N
ame
the
base
of
�Q
RS
.Q�
R�d
.N
ame
the
vert
ex a
ngl
e of
�Q
RS
.�
Se.
Nam
e th
e ba
se a
ngl
es o
f �
QR
S.
�Q
,�R
2.D
eter
min
e w
het
her
eac
h s
tate
men
t is
alw
ays,
som
etim
es,o
r n
ever
tru
e.
a.If
a t
rian
gle
has
th
ree
con
gru
ent
side
s,th
en i
t h
as t
hre
e co
ngr
uen
t an
gles
.al
way
sb
.If
a t
rian
gle
is i
sosc
eles
,th
en i
t is
equ
ilat
eral
.so
met
imes
c.If
a r
igh
t tr
ian
gle
is i
sosc
eles
,th
en i
t is
equ
ilat
eral
.n
ever
d.
Th
e la
rges
t an
gle
of a
n i
sosc
eles
tri
angl
e is
obt
use
.so
met
imes
e.If
a r
igh
t tr
ian
gle
has
a 4
5°an
gle,
then
it
is i
sosc
eles
.al
way
sf.
If a
n i
sosc
eles
tri
angl
e h
as t
hre
e ac
ute
an
gles
,th
en i
t is
equ
ilat
eral
.so
met
imes
g.T
he
vert
ex a
ngl
e of
an
iso
scel
es t
rian
gle
is t
he
larg
est
angl
e of
th
e tr
ian
gle.
som
etim
es3.
Giv
e th
e m
easu
res
of t
he
thre
e an
gles
of
each
tri
angl
e.
a.an
equ
ilat
eral
tri
angl
e60
,60,
60b
.an
iso
scel
es r
igh
t tr
ian
gle
45,4
5,90
c.an
iso
scel
es t
rian
gle
in w
hic
h t
he
mea
sure
of
the
vert
ex a
ngl
e is
70
70,5
5,55
d.
an i
sosc
eles
tri
angl
e in
wh
ich
th
e m
easu
re o
f a
base
an
gle
is 7
070
,70,
40e.
an i
sosc
eles
tri
angl
e in
wh
ich
th
e m
easu
re o
f th
e ve
rtex
an
gle
is t
wic
e th
e m
easu
re o
fon
e of
th
e ba
se a
ngl
es90
,45,
45
Hel
pin
g Y
ou
Rem
emb
er4.
If a
th
eore
m a
nd
its
con
vers
e ar
e bo
th t
rue,
you
can
oft
en r
emem
ber
them
mos
t ea
sily
by
com
bini
ng t
hem
into
an
“if-
and-
only
-if”
stat
emen
t.W
rite
suc
h a
stat
emen
t fo
r th
e Is
osce
les
Tri
angl
e T
heo
rem
an
d it
s co
nve
rse.
Sam
ple
an
swer
:Tw
o s
ides
of
a tr
ian
gle
are
con
gru
ent
if a
nd
on
ly if
th
e an
gle
s o
pp
osi
te t
ho
se s
ides
are
co
ng
ruen
t.
R Q
S
©G
lenc
oe/M
cGra
w-H
ill21
8G
lenc
oe G
eom
etry
Tria
ng
le C
hal
len
ges
Som
e pr
oble
ms
incl
ude
dia
gram
s.If
you
are
not
su
re h
ow t
o so
lve
the
prob
lem
,beg
in b
y u
sin
g th
e gi
ven
in
form
atio
n.F
ind
the
mea
sure
s of
as
man
yan
gles
as
you
can
,wri
tin
g ea
ch m
easu
re o
n t
he
diag
ram
.Th
is m
ay g
ive
you
mor
e cl
ues
to
the
solu
tion
.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-6
4-6
1.G
iven
:B
E�
BF
, �B
FG
� �
BE
F �
�B
ED
, m�
BF
E �
82 a
nd
AB
FG
and
BC
DE
each
hav
eop
posi
te s
ides
par
alle
l an
dco
ngr
uen
t.F
ind
m�
AB
C.
148
3.G
iven
:m
�U
ZY
�90
,m�
ZW
X�
45,
�Y
ZU
��
VW
X,U
VX
Yis
asq
uar
e (a
ll s
ides
con
gru
ent,
all
angl
es r
igh
t an
gles
).F
ind
m�
WZ
Y.
45
2.G
iven
:A
C�
AD
,an
d A �
B ��
B �D �
,m
�D
AC
�44
an
dC �
E �bi
sect
s �
AC
D.
Fin
d m
�D
EC
.78
4.G
iven
:m
�N
�12
0,J �N �
�M �
N �,
�J
NM
��
KL
M.
Fin
d m
�J
KM
.15
J
K
L
MN
A
DC
BE
UV
W
XY
Z
A
GD
FE
CB
Answers (Lesson 4-6)
© Glencoe/McGraw-Hill A20 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Tria
ng
les
and
Co
ord
inat
e P
roo
f
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-7
4-7
©G
lenc
oe/M
cGra
w-H
ill21
9G
lenc
oe G
eom
etry
Lesson 4-7
Posi
tio
n a
nd
Lab
el T
rian
gle
sA
coo
rdin
ate
proo
f u
ses
poin
ts,d
ista
nce
s,an
d sl
opes
to
prov
e ge
omet
ric
prop
erti
es.T
he
firs
t st
ep i
n w
riti
ng
a co
ordi
nat
e pr
oof
is t
o pl
ace
a fi
gure
on
the
coor
din
ate
plan
e an
d la
bel
the
vert
ices
.Use
th
e fo
llow
ing
guid
elin
es.
1.U
se t
he o
rigin
as
a ve
rtex
or
cent
er o
f th
e fig
ure.
2.P
lace
at
leas
t on
e si
de o
f th
e po
lygo
n on
an
axis
.3.
Kee
p th
e fig
ure
in t
he f
irst
quad
rant
if p
ossi
ble.
4.U
se c
oord
inat
es t
hat
mak
e th
e co
mpu
tatio
ns a
s si
mpl
e as
pos
sibl
e.
Pos
itio
n a
n e
qu
ilat
eral
tri
angl
e on
th
e co
ord
inat
e p
lan
e so
th
at i
ts s
ides
are
au
nit
s lo
ng
and
on
e si
de
is o
n t
he
pos
itiv
e x-
axis
.S
tart
wit
h R
(0,0
).If
RT
is a
,th
en a
not
her
ver
tex
is T
(a,0
).
For
ver
tex
S,t
he
x-co
ordi
nat
e is
�a 2� .U
se b
for
the
y-co
ordi
nat
e,
so t
he
vert
ex i
s S��a 2� ,
b �.
Fin
d t
he
mis
sin
g co
ord
inat
es o
f ea
ch t
rian
gle.
1.2.
3.
C(p
,q)
T(2
a,2a
)E
(�2g
,0);
F(0
,b)
Pos
itio
n a
nd
lab
el e
ach
tri
angl
e on
th
e co
ord
inat
e p
lan
e.
4.is
osce
les
tria
ngl
e 5.
isos
cele
s ri
ght
�D
EF
6.eq
uila
tera
l tr
iang
le �
EQ
I�
RS
T w
ith
bas
e R �
S�w
ith
leg
s e
un
its
lon
gw
ith
ver
tex
Q(0
,a)
and
4au
nit
s lo
ng
side
s 2b
un
its
lon
g
x
y
I(b,
0)
E( –
b, 0
)
Q( 0
, a)
x
y
E( e
, 0)
F( e
, e)
D( 0
, 0)
x
yT
( 2a,
b)
R( 0
, 0)
S( 4
a, 0
)
Sam
ple
an
swer
sar
e g
iven
.
x
y
G( 2
g, 0
)
F( ?
, b)
E( ?
, ?)
x
y
S( 2
a, 0
)
T( ?
, ?)
R( 0
, 0)
x
y
B( 2
p, 0
)
C( ?
, q)
A( 0
, 0)
x
y
T( a
, 0)
R( 0
, 0)S
�a – 2, b�
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill22
0G
lenc
oe G
eom
etry
Wri
te C
oo
rdin
ate
Pro
ofs
Coo
rdin
ate
proo
fs c
an b
e u
sed
to p
rove
th
eore
ms
and
tove
rify
pro
pert
ies.
Man
y co
ordi
nat
e pr
oofs
use
th
e D
ista
nce
For
mu
la,S
lope
For
mu
la,o
rM
idpo
int
Th
eore
m.
Pro
ve t
hat
a s
egm
ent
from
th
e ve
rtex
an
gle
of a
n i
sosc
eles
tri
angl
e to
th
e m
idp
oin
t of
th
e b
ase
is p
erp
end
icu
lar
to t
he
bas
e.F
irst
,pos
itio
n a
nd
labe
l an
iso
scel
es t
rian
gle
on t
he
coor
din
ate
plan
e.O
ne
way
is
to u
se T
(a,0
),R
(�a,
0),a
nd
S(0
,c).
Th
en U
(0,0
) is
th
e m
idpo
int
of R �
T�.
Giv
en:I
sosc
eles
�R
ST
;Uis
th
e m
idpo
int
of b
ase
R �T�
.P
rove
:S �U�
⊥R�
T�
Pro
of:
Uis
th
e m
idpo
int
of R �
T�so
th
e co
ordi
nat
es o
f U
are ���
a 2�a
�,�
0� 2
0�
��(0
,0).
Th
us
S�U�
lies
on
the
y-ax
is,a
nd
�R
ST
was
pla
ced
so R�
T�li
es o
n t
he
x-ax
is.T
he
axes
are
per
pen
dicu
lar,
so
S �U�
⊥R�
T�.
Pro
ve t
hat
th
e se
gmen
ts j
oin
ing
the
mid
poi
nts
of
the
sid
es o
f a
righ
t tr
ian
gle
form
a ri
ght
tria
ngl
e.
Sam
ple
an
swer
:P
osi
tio
n a
nd
lab
el r
igh
t �
AB
Cw
ith
th
e co
ord
inat
es
A(0
,0),
B(0
,2b
),an
d C
(2a,
0).
Th
e m
idp
oin
t P
of
BC
is ��0
� 22a �
,�2b
2�0
���
(a,b
).
Th
e m
idp
oin
t Q
of
AC
is ��0
� 22a �
,�0
� 20
���
(a,0
).
Th
e m
idp
oin
t R
of
AB
is ��0
� 20
�,�
0� 2
2b ���
(0,b
).
Th
e sl
op
e o
f R�
P�is
�b a� �
b 0�
��0 a�
�0,
so t
he
seg
men
t is
ho
rizo
nta
l.
Th
e sl
op
e o
f P�
Q�is
�b a��
a0�
��b 0� ,
wh
ich
is u
nd
efin
ed,s
o t
he
seg
men
t is
ver
tica
l.
�R
PQ
is a
rig
ht
ang
le b
ecau
se a
ny h
ori
zon
tal l
ine
is p
erp
end
icu
lar
to a
nyve
rtic
al li
ne.
�P
RQ
has
a r
igh
t an
gle
,so
�P
RQ
is a
rig
ht
tria
ng
le.
x
y
C( 2
a, 0
)
B( 0
, 2b)
P Q
R
A( 0
, 0)
x
y
T( a
, 0)
U( 0
, 0)
R( –
a, 0
)
S( 0
, c)
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Tria
ng
les
and
Co
ord
inat
e P
roo
f
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-7
4-7
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 4-7)
© Glencoe/McGraw-Hill A21 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Tria
ng
les
and
Co
ord
inat
e P
roo
f
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-7
4-7
©G
lenc
oe/M
cGra
w-H
ill22
1G
lenc
oe G
eom
etry
Lesson 4-7
Pos
itio
n a
nd
lab
el e
ach
tri
angl
e on
th
e co
ord
inat
e p
lan
e.
1.ri
ght
�F
GH
wit
h l
egs
2.is
osce
les
�K
LP
wit
h
3.is
osce
les
�A
ND
wit
ha
un
its
and
bu
nit
sba
se K �
P�6b
un
its
lon
gba
se A�
D�5a
lon
g
Fin
d t
he
mis
sin
g co
ord
inat
es o
f ea
ch t
rian
gle.
4.5.
6.
A(0
,2a)
Z(b
,c)
M(0
,c)
7.8.
9.
Q(4
a,0)
R��7 2� b
,c�
T(0
,b)
10.W
rite
a c
oord
inat
e pr
oof
to p
rove
th
at i
n a
n i
sosc
eles
rig
ht
tria
ngl
e,th
e se
gmen
t fr
omth
e ve
rtex
of
the
righ
t an
gle
to t
he
mid
poin
t of
th
e h
ypot
enu
se i
s pe
rpen
dicu
lar
to t
he
hyp
oten
use
.
Giv
en:
isos
cele
s ri
ght
�A
BC
wit
h �
AB
Cth
e ri
ght
angl
e an
d M
the
mid
poin
t of
A �C�
Pro
ve:
B�M�
⊥A�
C�
Pro
of:
Th
e M
idp
oin
t F
orm
ula
sh
ow
s th
at t
he
coo
rdin
ates
of
Mar
e ��0
� 22a �
,�2a
2�0
��o
r (a
,a).
Th
e sl
op
e o
f A�
C�is
�2 0a ��
20 a�
��
1.T
he
slo
pe
of
B�M�
is �a a
� �0 0
��
1.T
he
pro
du
ct
of
the
slo
pes
is �
1,so
B�M�
⊥A�
C�.
x
y
C( 2
a, 0
)
A( 0
, 2a)
M
B( 0
, 0)
x
y
U( a
, 0)
T( ?
, ?)
S( –
a, 0
)x
y
P( 7
b, 0
)
R( ?
, ?)
N( 0
, 0)
x
y
Q( ?
, ?)
R( 2
a, b
)
P( 0
, 0)
x
y
N( 3
b, 0
)
M( ?
, ?)
O( 0
, 0)
x
y
Y( 2
b, 0
)
Z( ?
, ?)
X( 0
, 0)
x
y
B( 2
a, 0
)
A( 0
, ?)
C( 0
, 0)
x
yN
�5 – 2a, b
�
A( 0
, 0)
D( 5
a, 0
)x
yL(
3b, c
)
K( 0
, 0)
P( 6
b, 0
)x
y F( 0
, a)
G( 0
, 0)
H( b
, 0)
Sam
ple
an
swer
sar
e g
iven
.
©G
lenc
oe/M
cGra
w-H
ill22
2G
lenc
oe G
eom
etry
Pos
itio
n a
nd
lab
el e
ach
tri
angl
e on
th
e co
ord
inat
e p
lan
e.
1.eq
uil
ater
al �
SW
Yw
ith
2.
isos
cele
s �
BL
Pw
ith
3.
isos
cele
s ri
ght
�D
GJ
side
s �1 4� a
lon
gba
se B�
L�3b
un
its
lon
gw
ith
hyp
oten
use
D�J�
and
legs
2a
un
its
lon
g
Fin
d t
he
mis
sin
g co
ord
inat
es o
f ea
ch t
rian
gle.
4.5.
6.
S��1 6� b
,c�
C(3
a,0)
,E(0
,c)
M(0
,c),
N(�
2b,0
)
NEI
GH
BO
RH
OO
DS
For
Exe
rcis
es 7
an
d 8
,use
th
e fo
llow
ing
info
rmat
ion
.K
arin
a li
ves
6 m
iles
eas
t an
d 4
mil
es n
orth
of
her
hig
h s
choo
l.A
fter
sch
ool
she
wor
ks p
art
tim
e at
the
mal
l in
a m
usic
sto
re.T
he m
all
is 2
mil
es w
est
and
3 m
iles
nor
th o
f th
e sc
hool
.
7.W
rite
a c
oord
inat
e pr
oof
to p
rove
th
at K
arin
a’s
hig
h s
choo
l,h
er h
ome,
and
the
mal
l ar
eat
th
e ve
rtic
es o
f a
righ
t tr
ian
gle.
Giv
en:
�S
KM
Pro
ve:
�S
KM
is a
rig
ht
tria
ngl
e.
Pro
of:
Slo
pe
of
SK
��4 6
� �0 0
�o
r �2 3�
Slo
pe
of
SM
�� �3 2� �
0 0�
or
��3 2�
Sin
ce t
he
slo
pe
of
S�M�
is t
he
neg
ativ
e re
cip
roca
l of
the
slo
pe
of
S�K�
,S�M�
⊥S�
K�.T
her
efo
re,�
SK
Mis
rig
ht
tria
ng
le.
8.F
ind
the
dist
ance
bet
wee
n t
he
mal
l an
d K
arin
a’s
hom
e.
KM
��
(�2
��
6)2
��
(3 �
�4)
2�
��
64 �
�1�
��
65�o
r �
8.1
mile
s
x
y S( 0
, 0)
K( 6
, 4)
M( –
2, 3
)
x
y
P( 2
b, 0
)
M( 0
, ?)
N( ?
, 0)
x
y
C( ?
, 0)
E( 0
, ?)
B( –
3a, 0
)x
yS
( ?, ?
)
J(0,
0)
R�1 – 3b,
0�
x
y D( 0
, 2a)
G( 0
, 0)
J(2a
, 0)
x
yP
�3 – 2b, c
�
B( 0
, 0)
L(3b
, 0)
x
yY
�1 – 8a, b
�
W�1 – 4a,
0�
S( 0
, 0)
Sam
ple
an
swer
sar
e g
iven
.
Pra
ctic
e (
Ave
rag
e)
Tria
ng
les
and
Co
ord
inat
e P
roo
f
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-7
4-7
Answers (Lesson 4-7)
© Glencoe/McGraw-Hill A22 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csTr
ian
gle
s an
d C
oo
rdin
ate
Pro
of
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-7
4-7
©G
lenc
oe/M
cGra
w-H
ill22
3G
lenc
oe G
eom
etry
Lesson 4-7
Pre-
Act
ivit
yH
ow c
an t
he
coor
din
ate
pla
ne
be
use
ful
in p
roof
s?
Rea
d th
e in
trod
uct
ion
to
Les
son
4-7
at
the
top
of p
age
222
in y
our
text
book
.
Fro
m t
he
coor
din
ates
of
A,B
,an
d C
in t
he
draw
ing
in y
our
text
book
,wh
atdo
you
kn
ow a
bou
t �
AB
C?
Sam
ple
an
swer
:�
AB
Cis
iso
scel
esw
ith
�C
as t
he
vert
ex a
ng
le.
Rea
din
g t
he
Less
on
1.F
ind
the
mis
sin
g co
ordi
nat
es o
f ea
ch t
rian
gle.
a.b
.
R(0
,b),
S(0
,0),
T�a,
�b 2� �D
(0,0
),E
(0,a
),F
(a,a
)
2.R
efer
to
the
figu
re.
a.F
ind
the
slop
e of
S�R�
and
the
slop
e of
S�T�
.1;
�1
b.
Fin
d th
e pr
odu
ct o
f th
e sl
opes
of
S�R�
and
S�T�
.Wh
at
does
th
is t
ell
you
abo
ut
S �R�
and
S�T�
?�
1;S�
R�⊥
S�T�
c.W
hat
doe
s yo
ur
answ
er f
rom
par
t b
tell
you
abo
ut
�R
ST
?S
amp
le a
nsw
er:
�R
ST
is a
rig
ht
tria
ng
le w
ith
�S
as t
he
rig
ht
ang
le.
d.
Fin
d S
Ran
d S
T.W
hat
doe
s th
is t
ell
you
abo
ut
S�R�
and
S�T�
?
SR
��
2a2
�o
r a�
2�;S
T�
�2a
2�
or
a�2�;
S�R�
�S�
T�e.
Wh
at d
oes
you
r an
swer
fro
m p
art
d te
ll y
ou a
bou
t �
RS
T?
Sam
ple
an
swer
:�
RS
Tis
iso
scel
es w
ith
�R
ST
as t
he
vert
ex a
ng
le.
f.C
ombi
ne y
our
answ
ers
from
par
ts c
and
e t
o de
scri
be �
RS
Tas
com
plet
ely
as p
ossi
ble.
Sam
ple
an
swer
:�
RS
Tis
an
iso
scel
es r
igh
t tr
ian
gle
.�R
ST
is t
he
rig
ht
ang
le a
nd
is a
lso
th
e ve
rtex
an
gle
.g.
Fin
d m
�S
RT
and
m�
ST
R.
45;
45h
.F
ind
m�
OS
Ran
d m
�O
ST
.45
;45
Hel
pin
g Y
ou
Rem
emb
er
3.M
any
stu
den
ts f
ind
it e
asie
r to
rem
embe
r m
ath
emat
ical
for
mu
las
if t
hey
can
pu
t th
emin
to w
ords
in
a c
ompa
ct w
ay.H
ow c
an y
ou u
se t
his
app
roac
h t
o re
mem
ber
the
slop
e an
dm
idpo
int
form
ula
s ea
sily
?S
amp
le a
nsw
er:
Slo
pe
Fo
rmu
la:
chan
ge
in y
over
ch
ang
e in
x;
Mid
po
int
Fo
rmu
la:
aver
age
of
x-co
ord
inat
es,a
vera
ge
of
y-co
ord
inat
es
x
y S( 0
, a)
R( –
a, 0
)T
( a, 0
)O
( 0, 0
)
x
y
F( ?
, ?)
E( ?
, a)
D( ?
, ?)
x
y
T( a
, ?)
R( ?
, b)
S( ?
, ?)
©G
lenc
oe/M
cGra
w-H
ill22
4G
lenc
oe G
eom
etry
How
Man
y Tr
ian
gle
s?E
ach
pu
zzle
bel
ow c
onta
ins
man
y tr
ian
gles
.Cou
nt
them
car
efu
lly.
Som
e tr
ian
gles
ove
rlap
oth
er t
rian
gles
.
How
man
y tr
ian
gles
are
th
ere
in e
ach
fig
ure
?
1.8
2.40
3.35
4.5
5.13
6.27
How
man
y tr
ian
gles
can
you
for
m b
y jo
inin
g p
oin
ts o
n e
ach
cir
cle?
L
ist
the
vert
ices
of
each
tri
angl
e.
7.8.
4;A
BC
,AB
D,A
CD
,BC
D10
;E
FG
,EF
H,E
FI,
EG
H,E
HI,
FG
H,
FG
I,F
HI,
EG
I,G
HI
8.9.
20;
JKL
,JK
M,J
KN
,JK
O,J
LM
,JL
N,J
LO
,JM
N,J
MO
,JN
O,K
LM
,K
LN
,KL
O,K
MN
,KM
O,K
NO
,L
MN
,LM
O,L
NO
,MN
O
35;
PQ
R,P
QS
,PQ
T,P
QU
,PQ
V,
PR
S,P
RT
,PR
U,P
RV
,PS
T,P
SU
,P
SV
,PT
U,P
TV
,PU
V,Q
RS
,QR
T,
QR
U,Q
RV
,QS
T,Q
SU
,QS
V,Q
TU
,Q
TV
,QU
V,R
ST
,RS
U,R
SV
,RT
U,
RT
V,R
UV,
ST
U,S
TV,
SU
V,T
UV
QR
P
U
S
TV
JK
O
L
MN
EF
I
GH
B
C
DA
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
4-7
4-7
Answers (Lesson 4-7)
© Glencoe/McGraw-Hill A23 Glencoe Geometry
Chapter 4 Assessment Answer Key Form 1 Form 2APage 225 Page 226 Page 227
(continued on the next page)
An
swer
s
1.
2.
3.
4.
5.
6.
7.
8.
C
D
A
C
B
D
B
D
9.
10.
11.
12.
13.
B:
C
A
C
A
B
isosceles
1.
2.
3.
4.
5.
6.
7.
8.
C
C
A
D
B
D
D
A
© Glencoe/McGraw-Hill A24 Glencoe Geometry
Chapter 4 Assessment Answer KeyForm 2A (continued) Form 2BPage 228 Page 229 Page 230
9.
10.
11.
12.
13.
B:
B
C
B
A
A
A(0, 0), C(�a, a)
1.
2.
3.
4.
5.
6.
7.
8.
D
B
D
A
C
B
A
A
9.
10.
11.
12.
13.
B:
C
C
D
A
D
�2
© Glencoe/McGraw-Hill A25 Glencoe Geometry
Chapter 4 Assessment Answer KeyForm 2CPage 231 Page 232
An
swer
s
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
AB � A�B� ��29�,
BC � B�C� � �10�,AC � A�C� � �17�
9.
10.
11.
12.
13.
14.
B:
SAS
Isosceles �Theorem, AAS
45
x � 4
P��b2
�, �b2
��, C�P� ⊥ A�B�
�XYZ � �MNO
x
yC(b–
2, c)
B(b, 0)A
© Glencoe/McGraw-Hill A26 Glencoe Geometry
Chapter 4 Assessment Answer KeyForm 2DPage 233 Page 234
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
JK � J�K � � �10�,
JL � J �L� � �29�,KL � K �L� � �37�
SAS
10.
11.
12.
13.
14.
B:
Isosceles �Theorem, AAS
50
x � 6
M ��a2
�, 0�, N�0, �b2
��,slopes: M�N� � ��
ba
�
and B�C� � ��ba
�
�ABC � �DEF
x
y
M(3a, 0)
L(1.5a, b)
K(0, 0)
© Glencoe/McGraw-Hill A27 Glencoe Geometry
Chapter 4 Assessment Answer KeyForm 3Page 235 Page 236
An
swer
s
1.
2.
3.
4.
5.
6.
7.
8.
x � 4, AB � 78,BC � 78,AC � 100
AB � 5, BC � 10,
AC � 3�5�;scalene obtuse
20
90
40
AB � A�B�� �26�,
BC � B�C� � 3�2�,AC � A�C� � 2�5�
GH � JK � 2�5�,
IG � LJ � 2�5�,IH � LK � 2�2�;
�GHI � �JKL bySSS.
AAS
9.
10.
11.
12.
B:
IsoscelesTriangle Theorem
SAS
x � 3
m�1, m�3, m�4,m�6, and m�9each equal 20,
m�2 � 40,m�5 � 40,
m�8 � 60, andm�7 � 140
x
y
B(a � b, 0)
C(a � b, c)2
A(0, 0)
© Glencoe/McGraw-Hill A28 Glencoe Geometry
Chapter 4 Assessment Answer KeyPage 237, Open-Ended Assessment
Scoring Rubric
Score General Description Specific Criteria
• Shows thorough understanding of the concepts of usingthe Distance Formula to classify triangles and verifycongruence, finding missing angles, solving algebraicequations in isosceles and equilateral triangles, provingtriangles congruent, verifying congruence transformations,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.
• Shows understanding of the concepts of using the DistanceFormula to classify triangles and verify congruence, findingmissing angles, solving algebraic equations in isosceles andequilateral triangles, proving triangles congruent, verifyingcongruence transformations, and writing coordinate 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.
• Shows understanding of most of the concepts of using theDistance Formula to classify triangles and verifycongruence, finding missing angles, solving algebraicequations in isosceles and equilateral triangles, provingtriangles congruent, verifying congruence transformations,and writing coordinate 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.
• 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.
• Shows little or no understanding of most of the conceptsof using the Distance Formula to classify triangles andverify congruence, finding missing angles, solvingalgebraic equations in isosceles and equilateral triangles,proving triangles congruent, verifying congruencetransformations, and writing coordinate 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.
0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given
1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation
2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem
3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation
4 SuperiorA correct solution that is supported by well-developed, accurateexplanations
Chapter 4 Assessment Answer KeyPage 237, Open-Ended Assessment
Sample Answers
© Glencoe/McGraw-Hill A29 Glencoe Geometry
1. a. The figure is an acute isosceles triangle.
b. 9x � 4 � 2(20x � 10) � 1809x � 4 � 40x � 20 � 180
49x � 16 � 18049x � 196
x � 4
2. a. To determine whether a triangle is equilateral, the coordinates shouldbe graphed first to see if they form a triangle. Then list the segmentswhich form the triangle and use the Distance Formula to find theirlengths. A triangle is equilateral only if all three sides have equalmeasures.
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°,�1 � 180 � 78 � 58 or 44. Lastly, since �2 is an exterior angle, itsmeasure is equivalent to the sum of the measures of the two remoteinterior angles, 58 � 78, which equals 136.
4. a. SSS postulate
b. �J � �G, �D � �E, �L � �SD�J� � E�G�, D�L� � E�S�, J�L� � G�S�
5. Statements Reasons
1. A�B� || D�E� 1. Given
2. �ABC � �DEC 2. Alt. int. � are �.
3. A�D� bisects B�E�. 3. Given
4. B�C� � E�C� 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 A28, the following sample answers may be used as guidance in evaluating open-ended assessment items.
An
swer
s
© Glencoe/McGraw-Hill A30 Glencoe Geometry
Chapter 4 Assessment Answer KeyVocabulary Test/Review Quiz 1 Quiz 3Page 238 Page 239 Page 240
Quiz 2Page 239
Quiz 4Page 240
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
equiangulartriangle
obtuse triangle
remote interiorangles
base
scalene triangle
flow proof
congruencetransformation
included side
coordinate proof
vertex angle
a statement that caneasily be provedusing a theorem
two triangles in whichall corresponding
parts are congruent
a triangle in which allangle measures arebetween 0 and 90
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
right scalene
C
x � 2, AB � 15,BC � 15, AC � 10
AB � �53�, BC �
2�10�, AC � �41�;scalene
4045
135737030
1.
2.
3.
4.
�KMN � �KML
D
AB � A�B� � �13�,
BC � B�C� � 2�5�,AC � A�C� � �17�
SAS
1.
2.
3.
4.
Def. of anglebisector
AAS
60
45
1.
2.
3.
4.
5.
I(0, c) and C(b, 0)
AC � �34�, AB � 6,
and CB � �34�
A�C� � C�B�
x
y
S(1–2b, 0)
G(1–4b, c)
E(0, 0)
x
y
J(a–2, 0)
D(0, a)
L(0, 0)
© Glencoe/McGraw-Hill A31 Glencoe Geometry
Chapter 4 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 241 Page 242
An
swer
s
Part I
Part II
5.
6.
7.
AB � �41�,
BC � �29�,AC � 5�2�; scalene
AB � A�B� � �26�,
BC � B�C � � 2�5�,AC � A�C � � 3�2�
P�O� and L�N� bisecteach other.
1.
2.
3.
4.
D
C
A
B
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
a ray
35�12
� in. to 36 �12
� in.
5
�4
�17
Sometimes; D, E,and F can benoncollinear.
always
undefined
4
F�D�
right triangle
15
�P � �H, �Q � �G,
�R � �B, P�Q� � H�G�,Q�R� � G�B�, P�R� � H�B�
ASA
E(b, b); F(2b, 0);G(b, 0)
© Glencoe/McGraw-Hill A32 Glencoe Geometry
Chapter 4 Assessment Answer KeyStandardized Test Practice
Page 243 Page 244
1.
2.
3.
4.
5.
6.
7. A B C D
E F G H
A B C D
E F G H
A B C D
E F G H
A B C D8. 9.
10. 11.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
12.
13.
14.
15.
29 ft
35
62
40
0 6 8
6 1 8