Chapter 5 Resource Masters - Math Class · PDF fileThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 5.As ... ! 6z" 4 " 11 3x! 15 36 ! 21 " 3y 9z"

Chapter 5Resource Masters

Geometry

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

55

© 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 5.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

altitude

centroid

circumcenter

SUHR·kuhm·SEN·tuhr

concurrent lines

incenter

indirect proof

(continued on the next page)

⎧ ⎪ ⎨ ⎪ ⎩

© Glencoe/McGraw-Hill viii Glencoe Geometry

Vocabulary Term Found on Page Definition/Description/Example

indirect reasoning

median

orthocenter

OHR·thoh·CEN·tuhr

perpendicular bisector

point of concurrency

proof by contradiction

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

55

Learning to Read MathematicsProof Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

55

© Glencoe/McGraw-Hill ix Glencoe Geometry

Proo

f Bu

ilderThis is a list of key theorems and postulates you will learn in Chapter 5. 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 5.1

Theorem 5.2

Theorem 5.3Circumcenter Theorem

Theorem 5.4

Theorem 5.5

Theorem 5.6Incenter Theorem

Theorem 5.7Centroid Theorem

(continued on the next page)

© Glencoe/McGraw-Hill x Glencoe Geometry

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 5.8Exterior Angle Inequality Theorem

Theorem 5.9

Theorem 5.10

Theorem 5.11Triangle Inequality Theorem

Theorem 5.12

Theorem 5.13SAS Inequality/Hinge Theorem

Theorem 5.14SSS Inequality

Learning to Read MathematicsProof Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

55

Study Guide and InterventionBisectors, Medians, and Altitudes

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1

© Glencoe/McGraw-Hill 245 Glencoe Geometry

Less

on

5-1

Perpendicular Bisectors and Angle Bisectors A perpendicular bisector of aside of a triangle is a line, segment, or ray that is perpendicular to the side and passesthrough its midpoint. Another special segment, ray, or line is an angle bisector, whichdivides an angle into two congruent angles.

Two properties of perpendicular bisectors are:(1) a point is on the perpendicular bisector of a segment if and only if it is equidistant from

the endpoints of the segment, and(2) the three perpendicular bisectors of the sides of a triangle meet at a point, called the

circumcenter of the triangle, that is equidistant from the three vertices of the triangle.

Two properties of angle bisectors are:(1) a point is on the angle bisector of an angle if and only if it is equidistant from the sides

of the angle, and(2) the three angle bisectors of a triangle meet at a point, called the incenter of the

triangle, that is equidistant from the three sides of the triangle.

BD!!" is the perpendicularbisector of A!C!. Find x.

BD!!" is the perpendicular bisector of A!C!, soAD ! DC.3x " 8 ! 5x # 6

14 ! 2x7 ! x

3x ! 8

5x " 6B

C

D

A

MR!!" is the angle bisectorof !NMP. Find x if m!1 # 5x ! 8 andm!2 # 8x " 16.

MR!!" is the angle bisector of !NMP, so m!1 ! m!2.5x " 8 ! 8x # 16

24 ! 3x8 ! x

12

N R

PM

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find the value of each variable.

1. 2. 3.

DE#!" is the perpendicular "CDF is equilateral. DF!!" bisects !CDE.bisector of A!C!.

4. For what kinds of triangle(s) can the perpendicular bisector of a side also be an anglebisector of the angle opposite the side?

5. For what kind of triangle do the perpendicular bisectors intersect in a point outside thetriangle?

FE

DC(4x ! 30)$

8x $D

F

C

E10y " 46x $

3x $

8y

CE

DA

B

7x " 96x " 2


Medians and Altitudes A median is a line segment that connects the vertex of atriangle to the midpoint of the opposite side. The three medians of a triangle intersect at thecentroid of the triangle.

Centroid The centroid of a triangle is located two thirds of the distance from aTheorem vertex to the midpoint of the side opposite the vertex on a median.

AL ! $23$AE, BL ! $

23$BF, CL ! $

23$CD

Points R, S, and T are the midpoints of A!B!, B!C! and A!C!, respectively. Find x, y, and z.

CU ! $23$CR BU ! $

23$BT AU ! $

23$AS

6x ! $23$(6x " 15) 24 ! $

23$(24 " 3y # 3) 6z " 4 ! $

23$(6z " 4 " 11)

9x ! 6x " 15 36 ! 24 " 3y # 3 $32$(6z " 4) ! 6z " 4 " 11

3x ! 15 36 ! 21 " 3y 9z " 6 ! 6z " 15x ! 5 15 ! 3y 3z ! 9

5 ! y z ! 3

Find the value of each variable.

1. 2.

B!D! is a median. AB ! CB; D, E, and F are midpoints.

3. 4.

EH ! FH ! HG

5. 6.

V is the centroid of "RST;D is the centroid of "ABC. TP ! 18; MS ! 15; RN ! 24

7. For what kind of triangle are the medians and angle bisectors the same segments?

8. For what kind of triangle is the centroid outside the triangle?

P

M

V

T

N

R S

y

x

z

G

FE

B

A C

24

329z ! 6 6z

6x

8y

MJ

PN

O

L

K3y ! 5

2x6z

122410

H GF

E

7x ! 4

9x " 2

5y

DB

E

F

A

C

9x ! 6

10x

3y

15D

BA

C

6x ! 3

7x " 1

A CT

SR U

B

3y " 3

6x

1524

11

6z ! 4

A CF

EDL

Bcentroid

Study Guide and Intervention (continued)

Bisectors, Medians, and Altitudes

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1

ExampleExample

ExercisesExercises

Skills PracticeBisectors, Medians, and Altitudes

NAME ______________________________________________ DATE ____________ PERIOD _____

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ALGEBRA For Exercises 1–4, use the given information to find each value.

1. Find x if E!G! is a median of "DEF. 2. Find x and RT if S!U! is a median of "RST.

3. Find x and EF if B!D! is an angle bisector. 4. Find x and IJ if H!K! is an altitude of "HIJ.

ALGEBRA For Exercises 5–7, use the following information.In "LMN, P, Q, and R are the midpoints of L!M!, M!N!, and L!N!,respectively.

5. Find x.

6. Find y.

7. Find z.

ALGEBRA Lines a, b, and c are perpendicular bisectors of "PQR and meet at A.

8. Find x.

9. Find y.

10. Find z.

COORDINATE GEOMETRY The vertices of "HIJ are G(1, 0), H(6, 0), and I(3, 6). Findthe coordinates of the points of concurrency of "HIJ.

11. orthocenter 12. centroid 13. circumcenter

5y " 6

8x ! 16

7z ! 4

24

18

R QA

ab c

P

y ! 1

2z2.8

23.6

x

L

NQ

RB

P

M

(3x ! 3)$

x ! 8

x " 9

I

JH

K

A

D4x " 1

2x ! 6B

G

E

F

C

R

U5x " 30

2x ! 24

S

T

D

G3x ! 1

5x " 17E

F


ALGEBRA In "ABC, B!F! is the angle bisector of !ABC, A!E!, B!F!,and C!D! are medians, and P is the centroid.

1. Find x if DP ! 4x # 3 and CP ! 30.

2. Find y if AP ! y and EP ! 18.

3. Find z if FP ! 5z " 10 and BP ! 42.

4. If m!ABC ! x and m!BAC ! m!BCA ! 2x # 10, is B!F! an altitude? Explain.

ALGEBRA In "PRS, P!T! is an altitude and P!X! is a median.

5. Find RS if RX ! x " 7 and SX ! 3x # 11.

6. Find RT if RT ! x # 6 and m!PTR ! 8x # 6.

ALGEBRA In "DEF, G!I! is a perpendicular bisector.

7. Find x if EH ! 16 and FH ! 6x # 5.

8. Find y if EG ! 3.2y # 1 and FG ! 2y " 5.

9. Find z if m!EGH ! 12z.

COORDINATE GEOMETRY The vertices of "STU are S(0, 1), T(4, 7), and U(8, "3).Find the coordinates of the points of concurrency of "STU.

10. orthocenter 11. centroid 12. circumcenter

13. MOBILES Nabuko wants to construct a mobile out of flat triangles so that the surfacesof the triangles hang parallel to the floor when the mobile is suspended. How canNabuko be certain that she hangs the triangles to achieve this effect?

D I

HF

G

E

S R

P

TX

A

C

F

E

DP

B

Practice Bisectors, Medians, and Altitudes

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1

Reading to Learn MathematicsBisectors, Medians, and Altitudes

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1


Less

on

5-1

Pre-Activity How can you balance a paper triangle on a pencil point?

Read the introduction to Lesson 5-1 at the top of page 238 in your textbook.

Draw any triangle and connect each vertex to the midpoint of the oppositeside to form the three medians of the triangle. Is the point where the threemedians intersect the midpoint of each of the medians?

Reading the Lesson

1. Underline the correct word or phrase to complete each sentence.

a. Three or more lines that intersect at a common point are called(parallel/perpendicular/concurrent) lines.

b. Any point on the perpendicular bisector of a segment is (parallel to/congruent to/equidistant from) the endpoints of the segment.

c. A(n) (altitude/angle bisector/median/perpendicular bisector) of a triangle is a segment drawn from a vertex of the triangle perpendicular to the line containing the opposite side.

d. The point of concurrency of the three perpendicular bisectors of a triangle is called the(orthocenter/circumcenter/centroid/incenter).

e. Any point in the interior of an angle that is equidistant from the sides of that angle lies on the (median/angle bisector/altitude).

f. The point of concurrency of the three angle bisectors of a triangle is called the(orthocenter/circumcenter/centroid/incenter).

2. In the figure, E is the midpoint of A!B!, F is the midpoint of B!C!,and G is the midpoint of A!C!.

a. Name the altitudes of "ABC.b. Name the medians of "ABC.c. Name the centroid of "ABC.d. Name the orthocenter of "ABC.e. If AF ! 12 and CE ! 9, find AH and HE.

Helping You Remember

3. A good way to remember something is to explain it to someone else. Suppose that aclassmate is having trouble remembering whether the center of gravity of a triangle isthe orthocenter, the centroid, the incenter, or the circumcenter of the triangle. Suggest away to remember which point it is.

A B

C

FG

E D

H


Inscribed and Circumscribed CirclesThe three angle bisectors of a triangle intersect in a single point called the incenter. Thispoint is the center of a circle that just touches the three sides of the triangle. Except for thethree points where the circle touches the sides, the circle is inside the triangle. The circle issaid to be inscribed in the triangle.

1. With a compass and a straightedge, construct the inscribed circle for "PQR by following the steps below.Step 1 Construct the bisectors of ! P and ! Q. Label the point

where the bisectors meet A.Step 2 Construct a perpendicular segment from A to R!Q!. Use

the letter B to label the point where the perpendicularsegment intersects R!Q!.

Step 3 Use a compass to draw the circle with center at A andradius A!B!.

Construct the inscribed circle in each triangle.

2. 3.

The three perpendicular bisectors of the sides of a triangle also meet in a single point. Thispoint is the center of the circumscribed circle, which passes through each vertex of thetriangle. Except for the three points where the circle touches the triangle, the circle isoutside the triangle.

4. Follow the steps below to construct the circumscribed circle for "FGH.Step 1 Construct the perpendicular bisectors of F!G! and F!H!.

Use the letter A to label the point where theperpendicular bisectors meet.

Step 2 Draw the circle that has center A and radius A!F!.

Construct the circumscribed circle for each triangle.

5. 6.

F H

G

P

QR

A

B

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1

Study Guide and InterventionInequalities and Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2


Less

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5-2

Angle Inequalities Properties of inequalities, including the Transitive, Addition,Subtraction, Multiplication, and Division Properties of Inequality, can be used withmeasures of angles and segments. There is also a Comparison Property of Inequality.

For any real numbers a and b, either a % b, a ! b, or a & b.

The Exterior Angle Theorem can be used to prove this inequality involving an exterior angle.

If an angle is an exterior angle of aExterior Angle triangle, then its measure is greater than Inequality Theorem the measure of either of its corresponding

remote interior angles.

m!1 & m!A, m!1 & m!B

List all angles of "EFG whose measures are less than m!1.The measure of an exterior angle is greater than the measure of either remote interior angle. So m!3 % m!1 and m!4 % m!1.

List all angles that satisfy the stated condition.

1. all angles whose measures are less than m!1

2. all angles whose measures are greater than m!3









R O

Q

N

P3 456

Exercises 9–10

78

21

S

X T W V

3

4

5

67 2 1

U

Exercises 3–8

M J K

3

4 521

L

Exercises 1–2

H E F3

4

21

G

A C D1

B

ExampleExample


Angle-Side Relationships When the sides of triangles are not congruent, there is a relationship between the sides and angles of the triangles.

• If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the If AC & AB, then m!B & m!C.angle opposite the shorter side. If m!A & m!C, then BC & AB.

• If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.

B C

A


Inequalities and Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2

List the angles in orderfrom least to greatest measure.

!T, !R, !S

R T9 cm

6 cm 7 cm

S

List the sides in orderfrom shortest to longest.

C!B!, A!B!, A!C!A B

C

20$

35$

125$

Example 1Example 1 Example 2Example 2

ExercisesExercises

List the angles or sides in order from least to greatest measure.

1. 2. 3.

Determine the relationship between the measures of the given angles.

4. !R, !RUS

5. !T, !UST

6. !UVS, !R

Determine the relationship between the lengths of the given sides.

7. A!C!, B!C!

8. B!C!, D!B!

9. A!C!, D!B!

A B

C

D30$

30$30$

90$

R V S

U T

2513

24 24

22

21.635

A C

B

3.8 4.3

4.0R T

S

60$

80$

40$T S

R48 cm

23.7 cm

35 cm

Skills PracticeInequalities and Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

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Determine which angle has the greatest measure.

1. !1, !3, !4 2. !4, !5, !7

3. !2, !3, !6 4. !5, !6, !8

Use the Exterior Angle Inequality Theorem to list all angles that satisfy the stated condition.






9. m!ABD, m!BAD 10. m!ADB, m!BAD

11. m!BCD, m!CDB 12. m!CBD, m!CDB


13. L!M!, L!P! 14. M!P!, M!N!

15. M!N!, N!P! 16. M!P!, L!P!

83$ 57$79$

44$59$

38$LN

P

M

2334

4139

35A

B C

D

1

2 4

6

7

8 93 5

1 2 4 6 7 8

35


Determine which angle has the greatest measure.

1. !1, !3, !4 2. !4, !8, !9

3. !2, !3, !7 4. !7, !8, !10

Use the Exterior Angle Inequality Theorem to list all angles that satisfy the stated condition.






9. m!QRW, m!RWQ 10. m!RTW, m!TWR

11. m!RST, m!TRS 12. m!WQR, m!QRW


13. D!H!, G!H! 14. D!E!, D!G!

15. E!G!, F!G! 16. D!E!, E!G!

17. SPORTS The figure shows the position of three trees on one part of a Frisbee™ course. At which tree position is the angle between the trees the greatest?

53 ft

40 ft

3

2

1

37.5 ft

120$32$

48$ 113$

17$H

D E F

G

3447

45

44

22

14

35

Q

R

S

TW

12

4 67 8

9

35

12

4 678 9

10

3

5

Practice Inequalities and Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2

Reading to Learn MathematicsInequalities and Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2


Less

on

5-2

Pre-Activity How can you tell which corner is bigger?


• Which side of the patio is opposite the largest corner?

• Which side of the patio is opposite the smallest corner?

Reading the Lesson1. Name the property of inequality that is illustrated by each of the following.

a. If x & 8 and 8 & y, then x & y.b. If x % y, then x # 7.5 % y # 7.5.c. If x & y, then #3x % #3y.d. If x is any real number, x & 0, x ! 0, or x % 0.

2. Use the definition of inequality to write an equation that shows that each inequality is true.a. 20 & 12 b. 101 & 99c. 8 & #2 d. 7 & #7e. #11 & #12 f. #30 & #45

3. In the figure, m!IJK ! 45 and m!H & m!I.a. Arrange the following angles in order from largest to

smallest: !I, !IJK, !H, !IJHb. Arrange the sides of "HIJ in order from shortest to longest.

c. Is "HIJ an acute, right, or obtuse triangle? Explain your reasoning.

d. Is "HIJ scalene, isosceles, or equilateral? Explain your reasoning.

Helping You Remember4. A good way to remember a new geometric theorem is to relate it to a theorem you

learned earlier. Explain how the Exterior Angle Inequality Theorem is related to theExterior Angle Theorem, and why the Exterior Angle Inequality Theorem must be true ifthe Exterior Angle Theorem is true.

KJH

I


Construction ProblemThe diagram below shows segment AB adjacent to a closed region. Theproblem requires that you construct another segment XY to the right of theclosed region such that points A, B, X, and Y are collinear. You are not allowedto touch or cross the closed region with your compass or straightedge.

Follow these instructions to construct a segment XY so that it iscollinear with segment AB.

1. Construct the perpendicular bisector of A!B!. Label the midpoint as point C,and the line as m.

2. Mark two points P and Q on line m that lie well above the closed region.Construct the perpendicular bisector n of P!Q!. Label the intersection oflines m and n as point D.

3. Mark points R and S on line n that lie well to the right of the closedregion. Construct the perpendicular bisector k of R!S!. Label theintersection of lines n and k as point E.

4. Mark point X on line k so that X is below line n and so that E!X! iscongruent to D!C!.

5. Mark points T and V on line k and on opposite sides of X, so that X!T! andX!V! are congruent. Construct the perpendicular bisector ! of T!V!. Call thepoint where the line ! hits the boundary of the closed region point Y. X!Y!corresponds to the new road.

Q

P

m

k

nD

R E

TX

V

BAC

S

ExistingRoad

Closed Region(Lake)

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2

Study Guide and InterventionIndirect Proof

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3


Less

on

5-3

Indirect Proof with Algebra One way to prove that a statement is true is to assumethat its conclusion is false and then show that this assumption leads to a contradiction ofthe hypothesis, a definition, postulate, theorem, or other statement that is accepted as true.That contradiction means that the conclusion cannot be false, so the conclusion must betrue. This is known as indirect proof.

Steps for Writing an Indirect Proof

1. Assume that the conclusion is false.2. Show that this assumption leads to a contradiction.3. Point out that the assumption must be false, and therefore, the conclusion must be true.

Given: 3x ! 5 % 8Prove: x % 1

Step 1 Assume that x is not greater than 1. That is, x ! 1 or x % 1.Step 2 Make a table for several possibilities for x ! 1 or x % 1. The

contradiction is that when x ! 1 or x % 1, then 3x " 5 is notgreater than 8.

Step 3 This contradicts the given information that 3x " 5 & 8. Theassumption that x is not greater than 1 must be false, which means that the statement “x & 1” must be true.

Write the assumption you would make to start an indirect proof of each statement.

1. If 2x & 14, then x & 7.

2. For all real numbers, if a " b & c, then a & c # b.

Complete the proof.Given: n is an integer and n2 is even.Prove: n is even.

3. Assume that

4. Then n can be expressed as 2a " 1 by

5. n2 ! Substitution

6. ! Multiply.

7. ! Simplify.

8. ! 2(2a2 " 2a) " 1

9. 2(2a2 " 2a)" 1 is an odd number. This contradicts the given that n2 is even,

so the assumption must be

10. Therefore,

x 3x " 5

1 8

0 5

#1 2

#2 #1

#3 #4

ExampleExample

ExercisesExercises


Indirect Proof with Geometry To write an indirect proof in geometry, you assumethat the conclusion is false. Then you show that the assumption leads to a contradiction.The contradiction shows that the conclusion cannot be false, so it must be true.

Given: m!C # 100Prove: !A is not a right angle.

Step 1 Assume that !A is a right angle.

Step 2 Show that this leads to a contradiction. If !A is a right angle,then m!A ! 90 and m!C " m!A ! 100 " 90 ! 190. Thus the sum of the measures of the angles of "ABC is greater than 180.

Step 3 The conclusion that the sum of the measures of the angles of "ABC is greater than 180 is a contradiction of a known property.The assumption that !A is a right angle must be false, which means that the statement “!A is not a right angle” must be true.

Write the assumption you would make to start an indirect proof of eachstatement.

1. If m!A ! 90, then m!B ! 45.

2. If A!V! is not congruent to V!E!, then "AVE is not isosceles.

Complete the proof.

Given: !1 " !2 and D!G! is not congruent to F!G!.Prove: D!E! is not congruent to F!E!.

3. Assume that Assume the conclusion is false.

4. E!G! " E!G!

5. "EDG " "EFG

6.

7. This contradicts the given information, so the assumption must

be

8. Therefore,

12

D G

FE

A B

C


Indirect Proof

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3

ExercisesExercises

ExampleExample

Skills PracticeIndirect Proof

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3


Less

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5-3


1. m!ABC % m!CBA

2. "DEF " "RST

3. Line a is perpendicular to line b.

4. !5 is supplementary to !6.

PROOF Write an indirect proof.

5. Given: x2 " 8 ' 12Prove: x ' 2

6. Given: !D # !F.Prove: DE ( EF

D F

E



1. B!D! bisects !ABC.

2. RT ! TS

PROOF Write an indirect proof.

3. Given: #4x " 2 % #10Prove: x & 3

4. Given: m!2 " m!3 ( 180Prove: a ⁄|| b

5. PHYSICS Sound travels through air at about 344 meters per second when thetemperature is 20°C. If Enrique lives 2 kilometers from the fire station and it takes 5 seconds for the sound of the fire station siren to reach him, how can you proveindirectly that it is not 20°C when Enrique hears the siren?

12

3

a

b

Practice Indirect Proof

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3

Reading to Learn MathematicsIndirect Proof

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3


Less

on

5-3

Pre-Activity How is indirect proof used in literature?


How could the author of a murder mystery use indirect reasoning to showthat a particular suspect is not guilty?

Reading the Lesson1. Supply the missing words to complete the list of steps involved in writing an indirect proof.

Step 1 Assume that the conclusion is .

Step 2 Show that this assumption leads to a of the

or some other fact, such as a definition, postulate,

, or corollary.

Step 3 Point out that the assumption must be and, therefore, the

conclusion must be .

2. State the assumption that you would make to start an indirect proof of each statement.

a. If #6x & 30, then x % #5.

b. If n is a multiple of 6, then n is a multiple of 3.

c. If a and b are both odd, then ab is odd.

d. If a is positive and b is negative, then ab is negative.

e. If F is between E and D, then EF " FD ! ED.

f. In a plane, if two lines are perpendicular to the same line, then they are parallel.

g. Refer to the figure. h. Refer to the figure.

If AB ! AC, then m!B ! m!C. In "PQR, PR " QR & QP.

Helping You Remember3. A good way to remember a new concept in mathematics is to relate it to something you have

already learned. How is the process of indirect proof related to the relationship between aconditional statement and its contrapositive?

P

RQ

A C

B


More CounterexamplesSome statements in mathematics can be proven false by counterexamples.Consider the following statement.

For any numbers a and b, a # b ! b # a.

You can prove that this statement is false in general if you can find oneexample for which the statement is false.

Let a ! 7 and b ! 3. Substitute these values in the equation above.

7 # 3 # 3 # 74 ( #4

In general, for any numbers a and b, the statement a # b ! b # a is false.You can make the equivalent verbal statement: subtraction is not acommutative operation.

In each of the following exercises a, b, and c are any numbers. Prove that the statement is false by counterexample.

1. a # (b # c) # (a # b) # c 2. a ) (b ) c) # (a ) b) ) c

3. a ) b # b ) a 4. a ) (b " c) # (a ) b) " (a ) c)

5. a " (bc) # (a " b)(a " c) 6. a2 " a2 # a4

7. Write the verbal equivalents for Exercises 1, 2, and 3.

8. For the Distributive Property a(b " c) ! ab " ac it is said that multiplicationdistributes over addition. Exercises 4 and 5 prove that some operations do notdistribute. Write a statement for each exercise that indicates this.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3

Study Guide and InterventionThe Triangle Inequality

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4


Less

on

5-4

The Triangle Inequality If you take three straws of lengths 8 inches, 5 inches, and 1 inch and try to make a triangle with them, you will find that it is not possible. Thisillustrates the Triangle Inequality Theorem.

Triangle Inequality The sum of the lengths of any two sides of aTheorem triangle is greater than the length of the third side.

The measures of two sides of a triangle are 5 and 8. Find a rangefor the length of the third side.By the Triangle Inequality, all three of the following inequalities must be true.

5 " x & 8 8 " x & 5 5 " 8 & xx & 3 x & #3 13 & x

Therefore x must be between 3 and 13.

Determine whether the given measures can be the lengths of the sides of atriangle. Write yes or no.

1. 3, 4, 6 2. 6, 9, 15

3. 8, 8, 8 4. 2, 4, 5

5. 4, 8, 16 6. 1.5, 2.5, 3

Find the range for the measure of the third side given the measures of two sides.

7. 1 and 6 8. 12 and 18

9. 1.5 and 5.5 10. 82 and 8

11. Suppose you have three different positive numbers arranged in order from least togreatest. What single comparison will let you see if the numbers can be the lengths ofthe sides of a triangle?

BC

A

a

cb

ExercisesExercises

ExampleExample


Distance Between a Point and a Line


The Triangle Inequality

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4

The perpendicular segment from a point toa line is the shortest segment from thepoint to the line.

P!C! is the shortest segment from P to AB#!".

The perpendicular segment from a point toa plane is the shortest segment from thepoint to the plane.

Q!T! is the shortest segment from Q to plane N .

Q

TN

B

P

CA

Given: Point P is equidistant from the sides of an angle.

Prove: B!A! " C!A!Proof:1. Draw B!P! and C!P! ⊥ to 1. Dist. is measured

the sides of !RAS. along a ⊥.2. !PBA and !PCA are right angles. 2. Def. of ⊥ lines3. "ABP and "ACP are right triangles. 3. Def. of rt. "

4. !PBA " !PCA 4. Rt. angles are ".5. P is equidistant from the sides of !RAS. 5. Given6. B!P! " C!P! 6. Def. of equidistant7. A!P! " A!P! 7. Reflexive Property8. "ABP " "ACP 8. HL9. B!A! " C!A! 9. CPCTC

Complete the proof.Given: "ABC " "RST; !D " !UProve: A!D! " R!U!Proof:

1. "ABC " "RST; !D " !U 1.

2. A!C! " R!T! 2.

3. !ACB " !RTS 3.

4. !ACB and !ACD are a linear pair; 4. Def. of !RTS and !RTU are a linear pair.

5. !ACB and !ACD are supplementary; 5.!RTS and !RTU are supplementary.

6. 6. Angles suppl. to " angles are ".

7. "ADC " "RUT 7.

8. 8. CPCTC

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R

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ExampleExample

ExercisesExercises

Skills PracticeThe Triangle Inequality

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4


Less

on

5-4


1. 2, 3, 4 2. 5, 7, 9

3. 4, 8, 11 4. 13, 13, 26

5. 9, 10, 20 6. 15, 17, 19

7. 14, 17, 31 8. 6, 7, 12

Find the range for the measure of the third side of a triangle given the measuresof two sides.

9. 5 and 9 10. 7 and 14

11. 8 and 13 12. 10 and 12

13. 12 and 15 14. 15 and 27

15. 17 and 28 16. 18 and 22

ALGEBRA Determine whether the given coordinates are the vertices of a triangle.Explain.

17. A(3, 5), B(4, 7), C(7, 6) 18. S(6, 5), T(8, 3), U(12, #1)

19. H(#8, 4), I(#4, 2), J(4, #2) 20. D(1, #5), E(#3, 0), F(#1, 0)



1. 9, 12, 18 2. 8, 9, 17

3. 14, 14, 19 4. 23, 26, 50

5. 32, 41, 63 6. 2.7, 3.1, 4.3

7. 0.7, 1.4, 2.1 8. 12.3, 13.9, 25.2

Find the range for the measure of the third side of a triangle given the measuresof two sides.

9. 6 and 19 10. 7 and 29

11. 13 and 27 12. 18 and 23

13. 25 and 38 14. 31 and 39

15. 42 and 6 16. 54 and 7

ALGEBRA Determine whether the given coordinates are the vertices of a triangle.Explain.

17. R(1, 3), S(4, 0), T(10, #6) 18. W(2, 6), X(1, 6), Y(4, 2)

19. P(#3, 2), L(1, 1), M(9, #1) 20. B(1, 1), C(6, 5), D(4, #1)

21. GARDENING Ha Poong has 4 lengths of wood from which he plans to make a border for atriangular-shaped herb garden. The lengths of the wood borders are 8 inches, 10 inches,12 inches, and 18 inches. How many different triangular borders can Ha Poong make?

Practice The Triangle Inequality

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4

Reading to Learn MathematicsThe Triangle Inequality

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4


Less

on

5-4

Pre-Activity How can you use the Triangle Inequality Theorem when traveling?


In addition to the greater distance involved in flying from Chicago toColumbus through Indianapolis rather than flying nonstop, what are twoother reasons that it would take longer to get to Columbus if you take twoflights rather than one?

Reading the Lesson

1. Refer to the figure.

Which statements are true?A. DE & EF " FD B. DE ! EF " FDC. EG ! EF " FG D. ED " DG & EGE. The shortest distance from D to EG#!" is DF.F. The shortest distance from D to EG#!" is DG.

2. Complete each sentence about "XYZ.

a. If XY ! 8 and YZ ! 11, then the range of values for XZ is % XZ % .

b. If XY ! 13 and XZ ! 25, then YZ must be between and .

c. If "XYZ is isosceles with !Z as the vertex angle, and XZ ! 8.5, then the range of

values for XY is % XY % .

d. If XZ ! a and YZ ! b, with b % a, then the range for XY is % XY % .

Helping You Remember

3. A good way to remember a new theorem is to state it informally in different words. Howcould you restate the Triangle Inequality Theorem?

ZX

Y

G

D

EF


Constructing TrianglesThe measurements of the sides of a triangle are given. If a triangle having sideswith these measurements is not possible, then write impossible. If a triangle ispossible, draw it and measure each angle with a protractor.

1. AR ! 5 cm m!A ! 2. PI ! 8 cm m!P !

RT ! 3 cm m!R ! IN ! 3 cm m!I !

AT ! 6 cm m!T ! PN ! 2 cm m!N !

3. ON ! 10 cm m!O ! 4. TW ! 6 cm m!T !

NE ! 5.3 cm m!N ! WO ! 7 cm m!W!

GE ! 4.6 cm m!E ! TO ! 2 cm m!O !

5. BA ! 3.l cm m!B ! 6. AR ! 4 cm m!A !

AT ! 8 cm m!A ! RM ! 5 cm m!R !

BT ! 5 cm m!T ! AM ! 3 cm m!M !

M

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Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4

Study Guide and InterventionInequalities Involving Two Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5


Less

on

5-5

SAS Inequality The following theorem involves the relationship between the sides oftwo triangles and an angle in each triangle.

If two sides of a triangle are congruent to two sides of another triangle and the included angle in one triangle has a

SAS Inequality/Hinge Theorem greater measure than the included angle in the other, then the third side of the If R!S! " A!B!, S!T! " B!C!, andfirst triangle is longer than the third side m!S & m!B, then RT & AC.of the second triangle.

Write an inequality relating the lengths of C!D! and A!D!.Two sides of "BCD are congruent to two sides of "BAD and m!CBD & m!ABD. By the SAS Inequality/Hinge Theorem,CD & AD.

Write an inequality relating the given pair of segment measures.

1. 2.

MR, RP AD, CD

3. 4.

EG, HK MR, PR

Write an inequality to describe the possible values of x.

5. 6.

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ExampleExample

ExercisesExercises


SSS Inequality The converse of the Hinge Theorem is also useful when two triangleshave two pairs of congruent sides.

If two sides of a triangle are congruent to two sidesof another triangle and the third side in one triangle

SSS Inequalityis longer than the third side in the other, then the angle between the pair of congruent sides in the first triangle is greater than the corresponding angle in the second triangle. If NM ! SR, MP ! RT, and NP & ST, then

m!M & m!R.

Write an inequality relating the measures of !ABD and !CBD.Two sides of "ABD are congruent to two sides of "CBD, and AD & CD.By the SSS Inequality, m!ABD & m!CBD.

Write an inequality relating the given pair of angle measures.

1. 2.

m!MPR, m!NPR m!ABD, m!CBD

3. 4.

m!C, m!Z m!XYW, m!WYZ

Write an inequality to describe the possible values of x.

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Inequalities Involving Two Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5

ExampleExample

ExercisesExercises

Skills PracticeInequalities Involving Two Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5


Less

on

5-5

Write an inequality relating the given pair of angles or segment measures.

1. m!BXA, m!DXA

2. BC, DC


3. m!STR, m!TRU 4. PQ, RQ

5. In the figure, B!A!, B!D!, B!C!, and B!E! are congruent and AC % DE.How does m!1 compare with m!3? Explain your thinking.

6. Write a two-column proof.Given: B!A! " D!A!

BC & DCProve: m!1 & m!2

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3. m!CDF, m!EDF 4. m!R, m!T

5. Write a two-column proof.Given: G is the midpoint of D!F!.

m!1 & m!2Prove: ED & EF

6. TOOLS Rebecca used a spring clamp to hold together a chair leg she repaired with wood glue. When she opened the clamp,she noticed that the angle between the handles of the clampdecreased as the distance between the handles of the clampdecreased. At the same time, the distance between the gripping ends of the clamp increased. When she released the handles, the distance between the gripping end of the clamp decreased and the distance between the handles increased.Is the clamp an example of the SAS or SSS Inequality?

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Practice Inequalities Involving Two Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5

Reading to Learn MathematicsInequalities Involving Two Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5


Less

on

5-5

Pre-Activity How does a backhoe work?


What is the main kind of task that backhoes are used to perform?

Reading the Lesson1. Refer to the figure. Write a conclusion that you can draw from the given information.

Then name the theorem that justifies your conclusion.

a. L!M! " O!P!, M!N! " P!Q!, and LN & OQ

b. L!M! " O!P!, M!N! " P!Q!, and m!P % m!M

c. LM ! 8, LN ! 15, OP ! 8, OQ ! 15, m!L ! 22, and m!O ! 21

2. In the figure, "EFG is isosceles with base F!G! and F is the midpoint of D!G!. Determine whether each of the following is a valid conclusion that you can draw based on the given information. (Write valid or invalid.) If the conclusion is valid,identify the definition, property, postulate, or theorem that supports it.

a. !3 " !4

b. DF ! GF

c. "DEF is isosceles.

d. m!3 & m!1

e. m!2 & m!4

f. m!2 & m!3

g. DE & EG

h. DE & FG

Helping You Remember3. A good way to remember something is to think of it in concrete terms. How can you

illustrate the Hinge Theorem with everyday objects?

F GD

E

1 2 3 4

N Q PM

L O


Drawing a DiagramIt is useful and often necessary to draw a diagram of the situationbeing described in a problem. The visualization of the problem ishelpful in the process of problem solving.

The roads connecting the towns of Kings,Chana, and Holcomb form a triangle. Davis Junction islocated in the interior of this triangle. The distances fromDavis Junction to Kings, Chana, and Holcomb are 3 km,4 km, and 5 km, respectively. Jane begins at Holcomb anddrives directly to Chana, then to Kings, and then back toHolcomb. At the end of her trip, she figures she has traveled25 km altogether. Has she figured the distance correctly?

To solve this problem, a diagram can be drawn. Based on this diagram and the Triangle Inequality Theorem, the distance from Holcomb to Chana is less than 9 km. Similarly,the distance from Chana to Kings is less than 7 km, and thedistance from Kings to Holcomb is less than 8 km.

Therefore, Jane must have traveled less than (9 " 7 " 8) km or 24 km versus her calculated distance of 25 km.

Explain why each of the following statements is true.Draw and label a diagram to be used in the explanation.

1. If an altitude is drawn to one side of a triangle, then thelength of the altitude is less than one-half the sum of thelengths of the other two sides.

2. If point Q is in the interior of *ABC and on the angle bisectorof !B, then Q is equidistant from A!B! and C!B!. (Hint: Draw Q!D!and Q!E! such that Q!D! $ A!B! and Q!E! $ C!B!.)

C E B

A

Q

D

A D C

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Kings

DavisJunction

Chana Holcomb

3 km

5 km4 km

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5

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9x !

6

10x

3y15D

BA

C

6x !

3

7x "

1

AC

T

SR

U

B 3y "

3

6x

1524

11

6z !

4

AC

F

ED

L

Bce

ntro

id

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Bis

ecto

rs,M

edia

ns,a

nd A

ltitu

des

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

5-1

5-1

Exam

ple

Exam

ple

Exercis

esExercis

es

Answers (Lesson 5-1)


An

swer

s

Skil

ls P

ract

ice

Bis

ecto

rs,M

edia

ns,a

nd A

ltitu

des

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

5-1

5-1

©G

lenc

oe/M

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ill24

7G

lenc

oe G

eom

etry

Lesson 5-1

ALG

EBR

AF

or E

xerc

ises

1–4

,use

th

e gi

ven

in

form

atio

n t

o fi

nd

eac

h v

alu

e.

1.F

ind

xif

E!G!

is a

med

ian

of "

DE

F.

2.F

ind

xan

d R

Tif

S!U!

is a

med

ian

of "

RS

T.

x#

9x

#18

;RT

#12

0

3.F

ind

xan

d E

Fif

B!D!

is a

n an

gle

bise

ctor

.4.

Fin

d x

and

IJif

H!K!

is a

n al

titu

de o

f "H

IJ.

x#

3.5;

EF

#13

x#

29;I

J#

57

ALG

EBR

AF

or E

xerc

ises

5–7

,use

th

e fo

llow

ing

info

rmat

ion

.In

"L

MN

,P,Q

,and

Rar

e th

e m

idpo

ints

of L !

M!,M!

N!,a

nd L!

N!,

resp

ecti

vely

.

5.F

ind

x.4

6.F

ind

y.0.

87.

Fin

d z.

0.7

ALG

EBR

AL

ines

a,b

,an

d c

are

per

pen

dic

ula

r bi

sect

ors

of "

PQ

Ran

d m

eet

at A

.

8.F

ind

x.1

9.F

ind

y.6

10.F

ind

z.2

CO

OR

DIN

ATE

GEO

MET

RYT

he

vert

ices

of

"H

IJar

e G

(1,0

),H

(6,0

),an

d I

(3,6

).F

ind

the

coor

din

ates

of

the

poi

nts

of

con

curr

ency

of

"H

IJ.

11.o

rtho

cent

er12

.cen

troi

d13

.cir

cum

cent

er

(3,1

)"&1 30 &

,2#

"&7 2& ,&5 2& #5y

" 6

8x !

16

7 z !

4

24

18

RQ

A

ab

c

P

y !

1

2z2.

8

23.

6

x

L

NQ

RB

P

M

( 3x

! 3

) $x !

8 x "

9

I

JH

K

AD4x

" 1

2x !

6B

G EF

C

RU 5x "

30

2x !

24

S

T

DG 3x !

1

5x "

17

E

F

©G

lenc

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ill24

8G

lenc

oe G

eom

etry

ALG

EBR

AIn

"A

BC

,B!F!

is t

he

angl

e bi

sect

or o

f !

AB

C,A!

E!,B!

F!,

and

C !D!

are

med

ian

s,an

d P

is t

he

cen

troi

d.

1.F

ind

xif

DP

!4x

#3

and

CP

!30

.4.

5

2.F

ind

yif

AP

!y

and

EP

!18

.36

3.F

ind

zif

FP

!5z

"10

and

BP

!42

.2.

2

4.If

m!

AB

C!

xan

d m

!B

AC

!m

!B

CA

!2x

#10

,is

B!F!

an a

ltit

ude?

Exp

lain

.Ye

s;si

nce

x#

40 a

nd B!

F!is

an

angl

e bi

sect

or,i

t fo

llow

s th

at m

!B

AF

#70

an

d m

!A

BF

#20

.So

m!

AFB

#90

,and

B!F!

⊥A!

C!.

ALG

EBR

AIn

"P

RS

,P!T!

is a

n a

ltit

ud

e an

d P!

X!is

a m

edia

n.

5.F

ind

RS

if R

X!

x"

7 an

d S

X!

3x#

11.

32

6.F

ind

RT

if R

T!

x#

6 an

d m

!P

TR

!8x

#6.

6

ALG

EBR

AIn

"D

EF

,G!I!

is a

per

pen

dic

ula

r bi

sect

or.

7.F

ind

xif

EH

!16

and

FH

!6x

#5.

3.5

8.F

ind

yif

EG

!3.

2y#

1 an

d F

G!

2y"

5.5

9.F

ind

zif

m!

EG

H!

12z.

7.5

CO

OR

DIN

ATE

GEO

MET

RYT

he

vert

ices

of

"S

TU

are

S(0

,1),

T(4

,7),

and

U(8

,"3)

.F

ind

th

e co

ord

inat

es o

f th

e p

oin

ts o

f co

ncu

rren

cy o

f "

ST

U.

10.o

rtho

cent

er11

.cen

troi

d12

.cir

cum

cent

er

"&5 4& ,&3 2& #

"4,&5 3& #

"&4 83 &,&

7 4& #or

(5.3

75,1

.75)

13.M

OB

ILES

Nab

uko

wan

ts t

o co

nstr

uct

a m

obile

out

of

flat

tri

angl

es s

o th

at t

he s

urfa

ces

of t

he t

rian

gles

han

g pa

ralle

l to

the

floo

r w

hen

the

mob

ile is

sus

pend

ed.H

ow c

anN

abuk

o be

cer

tain

tha

t sh

e ha

ngs

the

tria

ngle

s to

ach

ieve

thi

s ef

fect

?S

he n

eeds

to

hang

eac

h tr

iang

le f

rom

its

cent

er o

f gr

avity

or

cent

roid

,w

hich

is t

he p

oint

at

whi

ch t

he t

hree

med

ians

of

the

tria

ngle

inte

rsec

t.

DI

HF

G

E

SR

P TX

AC F

E DP

B

Pra

ctic

e (A

vera

ge)

Bis

ecto

rs,M

edia

ns,a

nd A

ltitu

des

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

5-1

5-1



Rea

din

g t

o L

earn

Math

emati

csB

isec

tors

,Med

ians

,and

Alti

tude

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

5-1

5-1

©G

lenc

oe/M

cGra

w-H

ill24

9G

lenc

oe G

eom

etry

Lesson 5-1

Pre-

Act

ivit

yH

ow c

an y

ou b

alan

ce a

pap

er t

rian

gle

on a

pen

cil

poi

nt?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 5-

1 at

the

top

of

page

238

in y

our

text

book

.

Dra

w a

ny t

rian

gle

and

conn

ect

each

ver

tex

to t

he m

idpo

int

of t

he o

ppos

ite

side

to

form

the

thr

ee m

edia

ns o

f th

e tr

iang

le.I

s th

e po

int

whe

re t

he t

hree

med

ians

inte

rsec

t th

e m

idpo

int

of e

ach

of t

he m

edia

ns?

Sam

ple

answ

er:

No;

the

inte

rsec

tion

poin

t ap

pear

s to

be

mor

e th

an h

alfw

ayfr

om e

ach

vert

ex t

o th

e m

idpo

int

of t

he o

ppos

ite s

ide.

Rea

din

g t

he

Less

on

1.U

nder

line

the

corr

ect

wor

d or

phr

ase

to c

ompl

ete

each

sen

tenc

e.

a.T

hree

or

mor

e lin

es t

hat

inte

rsec

t at

a c

omm

on p

oint

are

cal

led

(par

alle

l/per

pend

icul

ar/c

oncu

rren

t) li

nes.

b.A

ny p

oint

on

the

perp

endi

cula

r bi

sect

or o

f a

segm

ent

is

(par

alle

l to/

cong

ruen

t to

/equ

idis

tant

fro

m)

the

endp

oint

s of

the

seg

men

t.

c.A

(n)

(alt

itud

e/an

gle

bise

ctor

/med

ian/

perp

endi

cula

r bi

sect

or)

of a

tri

angl

e is

a

segm

ent

draw

n fr

om a

ver

tex

of t

he t

rian

gle

perp

endi

cula

r to

the

line

con

tain

ing

the

oppo

site

sid

e.

d.T

he p

oint

of c

oncu

rren

cy o

f the

thr

ee p

erpe

ndic

ular

bis

ecto

rs o

f a t

rian

gle

is c

alle

d th

e(o

rtho

cent

er/c

ircu

mce

nter

/cen

troi

d/in

cent

er).

e.A

ny p

oint

in t

he in

teri

or o

f an

ang

le t

hat

is e

quid

ista

nt f

rom

the

sid

es o

f th

at a

ngle

lie

s on

the

(m

edia

n/an

gle

bise

ctor

/alt

itud

e).

f.T

he p

oint

of

conc

urre

ncy

of t

he t

hree

ang

le b

isec

tors

of

a tr

iang

le is

cal

led

the

(ort

hoce

nter

/cir

cum

cent

er/c

entr

oid/

ince

nter

).

2.In

the

fig

ure,

Eis

the

mid

poin

t of

A !B!

,Fis

the

mid

poin

t of

B!C!

,an

d G

is t

he m

idpo

int

of A !

C!.

a.N

ame

the

alti

tude

s of

"A

BC

.A!

C!,B!

C!,C!

D!b.

Nam

e th

e m

edia

ns o

f "A

BC

.A!

F!,B!

G!,C!

E!c.

Nam

e th

e ce

ntro

id o

f "A

BC

.H

d.N

ame

the

orth

ocen

ter

of "

AB

C.

Ce.

If A

F!

12 a

nd C

E!

9,fi

nd A

Han

d H

E.

AH

#8,

HE

#3

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

som

ethi

ng is

to

expl

ain

it t

o so

meo

ne e

lse.

Supp

ose

that

acl

assm

ate

is h

avin

g tr

oubl

e re

mem

beri

ng w

heth

er t

he c

ente

r of

gra

vity

of

a tr

iang

le is

the

orth

ocen

ter,

the

cent

roid

,the

ince

nter

,or

the

circ

umce

nter

of

the

tria

ngle

.Sug

gest

aw

ay t

o re

mem

ber

whi

ch p

oint

it is

.S

ampl

e an

swer

:The

ter

ms

cent

roid

and

cent

er o

f gra

vity

mea

n th

e sa

me

thin

g an

d in

bot

h te

rms,

the

lett

ers

“cen

t”co

me

at t

he b

egin

ning

of

the

term

s.

AB

C

FG

ED

H

©G

lenc

oe/M

cGra

w-H

ill25

0G

lenc

oe G

eom

etry

Insc

ribe

d an

d C

ircu

msc

ribe

d C

ircl

esT

he t

hree

ang

le b

isec

tors

of

a tr

iang

le in

ters

ect

in a

sin

gle

poin

t ca

lled

the

ince

nte

r.T

his

poin

t is

the

cen

ter

of a

cir

cle

that

just

tou

ches

the

thr

ee s

ides

of

the

tria

ngle

.Exc

ept

for

the

thre

e po

ints

whe

re t

he c

ircl

e to

uche

s th

e si

des,

the

circ

le is

insi

de t

he t

rian

gle.

The

cir

cle

issa

id t

o be

insc

ribe

d in

the

tri

angl

e.

1.W

ith

a co

mpa

ss a

nd a

str

aigh

tedg

e,co

nstr

uct

the

insc

ribe

d ci

rcle

for

"P

QR

by f

ollo

win

g th

e st

eps

belo

w.

Ste

p 1

Con

stru

ct t

he b

isec

tors

of !

Pan

d !

Q.L

abel

the

poi

nt

whe

re t

he b

isec

tors

mee

t A

.S

tep

2C

onst

ruct

a p

erpe

ndic

ular

seg

men

t fr

om A

to R !

Q!.U

se

the

lett

er B

to la

bel t

he p

oint

whe

re t

he p

erpe

ndic

ular

segm

ent

inte

rsec

ts R !

Q!.

Ste

p 3

Use

a c

ompa

ss t

o dr

aw t

he c

ircl

e w

ith

cent

er a

t A

and

radi

us A !

B!.

Con

stru

ct t

he

insc

ribe

d c

ircl

e in

eac

h t

rian

gle.

2.3.

The

thr

ee p

erpe

ndic

ular

bis

ecto

rs o

f th

e si

des

of a

tri

angl

e al

so m

eet

in a

sin

gle

poin

t.T

his

poin

t is

the

cen

ter

of t

he c

ircu

msc

ribe

d ci

rcle

,whi

ch p

asse

s th

roug

h ea

ch v

erte

x of

the

tria

ngle

.Exc

ept

for

the

thre

e po

ints

whe

re t

he c

ircl

e to

uche

s th

e tr

iang

le,t

he c

ircl

e is

outs

ide

the

tria

ngle

.

4.Fo

llow

the

ste

ps b

elow

to

cons

truc

t th

e ci

rcum

scri

bed

circ

le

for

"F

GH

.S

tep

1C

onst

ruct

the

per

pend

icul

ar b

isec

tors

of F

!G!an

d F!H!

.U

se t

he le

tter

Ato

labe

l the

poi

nt w

here

the

perp

endi

cula

r bi

sect

ors

mee

t.S

tep

2D

raw

the

cir

cle

that

has

cen

ter

Aan

d ra

dius

A !F!.

Con

stru

ct t

he

circ

um

scri

bed

cir

cle

for

each

tri

angl

e.

5.6.

FH

G

AP

QR

A B

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

5-1

5-1



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Ineq

ualit

ies

and

Tria

ngle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

5-2

5-2

©G

lenc

oe/M

cGra

w-H

ill25

1G

lenc

oe G

eom

etry

Lesson 5-2

An

gle

Ineq

ual

itie

sP

rope

rtie

s of

ineq

ualit

ies,

incl

udin

g th

e T

rans

itiv

e,A

ddit

ion,

Subt

ract

ion,

Mul

tipl

icat

ion,

and

Div

isio

n P

rope

rtie

s of

Ine

qual

ity,

can

be u

sed

wit

hm

easu

res

of a

ngle

s an

d se

gmen

ts.T

here

is a

lso

a C

ompa

riso

n P

rope

rty

of I

nequ

alit

y.

For

any

real

num

bers

aan

d b,

eith

er a

%b,

a!

b,or

a&

b.

The

Ext

erio

r A

ngle

The

orem

can

be

used

to

prov

e th

is in

equa

lity

invo

lvin

g an

ext

erio

r an

gle.

If an

ang

le is

an

exte

rior

angl

e of

aE

xter

ior

Ang

letr

iang

le, t

hen

its m

easu

re is

gre

ater

than

In

equa

lity

Theo

rem

the

mea

sure

of e

ither

of i

ts c

orre

spon

ding

re

mot

e in

terio

r an

gles

.

m!

1 &

m!

A, m

!1

&m

!B

Lis

t al

l an

gles

of

"E

FG

wh

ose

mea

sure

s ar

e le

ss t

han

m!

1.T

he m

easu

re o

f an

ext

erio

r an

gle

is g

reat

er t

han

the

mea

sure

of

eith

er r

emot

e in

teri

or a

ngle

.So

m!

3 %

m!

1 an

d m

!4

%m

!1.

Lis

t al

l an

gles

th

at s

atis

fy t

he

stat

ed c

ond

itio

n.

1.al

l ang

les

who

se m

easu

res

are

less

tha

n m

!1

!3,

!4

2.al

l ang

les

who

se m

easu

res

are

grea

ter

than

m!

3!

1,!

5

3.al

l ang

les

who

se m

easu

res

are

less

tha

n m

!1

!5,

!6

4.al

l ang

les

who

se m

easu

res

are

grea

ter

than

m!

1!

7

5.al

l ang

les

who

se m

easu

res

are

less

tha

n m

!7

!1,

!3,

!5,

!6,

!TU

V

6.al

l ang

les

who

se m

easu

res

are

grea

ter

than

m!

2!

4

7.al

l ang

les

who

se m

easu

res

are

grea

ter

than

m!

5!

1,!

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tha

t th

e co

nclu

sion

can

not

be f

alse

,so

the

conc

lusi

on m

ust

betr

ue.T

his

is k

now

n as

in

dir

ect

pro

of.

Ste

ps fo

r Wri

ting

an In

dire

ct P

roof

1.A

ssum

e th

at th

e co

nclu

sion

is fa

lse.

2.S

how

that

this

ass

umpt

ion

lead

s to

a c

ontr

adic

tion.

3.P

oint

out

that

the

assu

mpt

ion

mus

t be

fals

e, a

nd th

eref

ore,

the

conc

lusi

on m

ust b

e tr

ue.

Giv

en:3

x!

5 %

8P

rove

:x%

1St

ep 1

Ass

ume

that

xis

not

gre

ater

tha

n 1.

Tha

t is

,x!

1 or

x%

1.St

ep 2

Mak

e a

tabl

e fo

r se

vera

l pos

sibi

litie

s fo

r x

!1

or x

%1.

The

cont

radi

ctio

n is

tha

t w

hen

x!

1 or

x%

1,th

en 3

x"

5 is

not

grea

ter

than

8.

Step

3T

his

cont

radi

cts

the

give

n in

form

atio

n th

at 3

x"

5 &

8.T

heas

sum

ptio

n th

at x

is n

ot g

reat

er t

han

1 m

ust

be f

alse

,whi

ch

mea

ns t

hat

the

stat

emen

t “x

&1”

mus

t be

tru

e.

Wri

te t

he

assu

mp

tion

you

wou

ld m

ake

to s

tart

an

in

dir

ect

pro

of o

f ea

ch s

tate

men

t.

1.If

2x

&14

,the

n x

&7.

x(

7

2.Fo

r al

l rea

l num

bers

,if a

"b

&c,

then

a&

c#

b.a

(c

"b

Com

ple

te t

he

pro

of.

Giv

en:n

is a

n in

tege

r an

d n2

is e

ven.

Pro

ve:n

is e

ven.

3.A

ssum

e th

at n

is n

ot e

ven.

That

is,a

ssum

e n

is o

dd.

4.T

hen

nca

n be

exp

ress

ed a

s 2a

"1

by th

e m

eani

ng o

f od

d nu

mbe

r.

5.n2

!(2

a!

1)2

Subs

titu

tion

6.!

(2a

!1)

(2a

!1)

Mul

tipl

y.

7.!

4a2

!4a

!1

Sim

plif

y.

8.!

2(2a

2"

2a) "

1D

istr

ibut

ive

Pro

pert

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9.2(

2a2

"2a

)"1

is a

n od

d nu

mbe

r.T

his

cont

radi

cts

the

give

n th

at n

2is

eve

n,

so t

he a

ssum

ptio

n m

ust

be fa

lse.

10.T

here

fore

,nis

eve

n.

x3x

"5

18

05

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2

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in g

eom

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,you

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ume

that

the

con

clus

ion

is f

alse

.The

n yo

u sh

ow t

hat

the

assu

mpt

ion

lead

s to

a c

ontr

adic

tion

.T

he c

ontr

adic

tion

sho

ws

that

the

con

clus

ion

cann

ot b

e fa

lse,

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mus

t be

tru

e.

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en:m

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ume

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.If !

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ight

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le,

then

m!

A!

90 a

nd m

!C

"m

!A

!10

0 "

90 !

190.

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s th

e su

m o

f th

e m

easu

res

of t

he a

ngle

s of

"A

BC

is g

reat

er t

han

180.

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3T

he c

oncl

usio

n th

at t

he s

um o

f th

e m

easu

res

of t

he a

ngle

s of

"

AB

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gre

ater

tha

n 18

0 is

a c

ontr

adic

tion

of

a kn

own

prop

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.T

he a

ssum

ptio

n th

at !

Ais

a r

ight

ang

le m

ust

be f

alse

,whi

ch

mea

ns t

hat

the

stat

emen

t “!

Ais

not

a r

ight

ang

le”

mus

t be

tru

e.

Wri

te t

he

assu

mp

tion

you

wou

ld m

ake

to s

tart

an

in

dir

ect

pro

of o

f ea

chst

atem

ent.

1.If

m!

A!

90,t

hen

m!

B!

45.

m!

B)

45

2.If

A!V!

is n

ot c

ongr

uent

to

V!E!

,the

n "

AVE

is n

ot is

osce

les.

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VE

is is

osce

les.

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ple

te t

he

pro

of.

Giv

en:!

1 "

!2

and

D!G!

is n

ot c

ongr

uent

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F!G!.

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ve:D !

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not

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grue

nt t

o F!E!

.

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ssum

e th

at D!

E!$

F!E!.

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ume

the

conc

lusi

on is

fal

se.

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G!"

E!G!

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lexi

ve P

rope

rty

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ED

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FG

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S

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PC

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7.T

his

cont

radi

cts

the

give

n in

form

atio

n,so

the

ass

umpt

ion

mus

t

be fa

lse.

8.T

here

fore

,D!

E!is

not

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grue

nt t

o F!E!

.

12

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____

____

____

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____

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5-3

5-3

Exercis

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Exam

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ple



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s

Skil

ls P

ract

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Indi

rect

Pro

of

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ME

____

____

____

____

____

____

____

____

____

____

____

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____

____

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RIO

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___

5-3

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Lesson 5-3

Wri

te t

he

assu

mp

tion

you

wou

ld m

ake

to s

tart

an

in

dir

ect

pro

of o

f ea

ch s

tate

men

t.

1.m

!A

BC

%m

!C

BA

m!

AB

C*

m!

CB

A

2."

DE

F"

"R

ST

"D

EF

%"

RS

T

3.L

ine

ais

per

pend

icul

ar t

o lin

e b.

Line

ais

not

per

pend

icul

ar t

o lin

e b.

4.!

5 is

sup

plem

enta

ry t

o !

6.

!5

is n

ot s

uppl

emen

tary

to

!6.

PRO

OF

Wri

te a

n i

nd

irec

t p

roof

.

5.G

iven

:x2

"8

'12

Pro

ve:x

'2

Pro

of:

Ste

p 1:

Ass

ume

x%

2.S

tep

2:If

x%

2,th

en x

2%

4.B

ut if

x2

%4,

it fo

llow

s th

at x

2!

8 %

12.

This

con

trad

icts

the

giv

en f

act

that

x2

!8

(12

.S

tep

3:S

ince

the

ass

umpt

ion

of x

%2

lead

s to

a c

ontr

adic

tion,

it m

ust

be f

alse

.The

refo

re, x

(2

mus

t be

tru

e.

6.G

iven

:!D

#!

F.

Pro

ve:D

E(

EF

Pro

of:

Ste

p 1:

Ass

ume

DE

#E

F.S

tep

2:If

DE

#E

F,th

en D!

E!$

E!F!by

the

defin

ition

of c

ongr

uent

seg

men

ts.

But

if D!

E!$

E!F!,t

hen

!D

$!

Fby

the

Isos

cele

s Tr

iang

le T

heor

em.

This

con

trad

icts

the

giv

en in

form

atio

n th

at !

D%

!F.

Ste

p 3:

Sin

ce t

he a

ssum

ptio

n th

at D

E#

EF

lead

s to

a c

ontr

adic

tion,

itm

ust

be f

alse

.The

refo

re,i

t m

ust

be t

rue

that

DE

)E

F.

DF

E

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you

wou

ld m

ake

to s

tart

an

in

dir

ect

pro

of o

f ea

ch s

tate

men

t.

1.B!

D!bi

sect

s !

AB

C.

B!D!

does

not

bis

ect

!A

BC

.

2.R

T!

TS

RT

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PRO

OF

Wri

te a

n i

nd

irec

t p

roof

.

3.G

iven

:#4x

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10P

rove

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3P

roof

:S

tep

1:A

ssum

e x

(3.

Ste

p 2:

If x

(3,

then

"4x

*"

12.B

ut "

4x*

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impl

ies

that

"

4x!

2 *

"10

,whi

ch c

ontr

adic

ts t

he g

iven

ineq

ualit

y.S

tep

3:S

ince

the

ass

umpt

ion

that

x(

3 le

ads

to a

con

trad

ictio

n,it

mus

t be

tru

e th

at x

%3.

4.G

iven

:m!

2 "

m!

3 (

180

Pro

ve: a

⁄|| bP

roof

:S

tep

1:A

ssum

e a

|| b.

Ste

p 2:

If a

|| b,t

hen

the

cons

ecut

ive

inte

rior

ang

les

!2

and

!3

are

supp

lem

enta

ry.T

hus

m!

2 !

m!

3 #

180.

This

con

trad

icts

the

give

n st

atem

ent

that

m!

2 !

m!

3 )

180.

Ste

p 3:

Sin

ce t

he a

ssum

ptio

n le

ads

to a

con

trad

ictio

n,th

e st

atem

ent

a|| b

mus

t be

fal

se.T

here

fore

,a⁄|| b

mus

t be

tru

e.

5.PH

YSI

CS

Soun

d tr

avel

s th

roug

h ai

r at

abo

ut 3

44 m

eter

s pe

r se

cond

whe

n th

ete

mpe

ratu

re is

20°

C.I

f E

nriq

ue li

ves

2 ki

lom

eter

s fr

om t

he f

ire

stat

ion

and

it t

akes

5

seco

nds

for

the

soun

d of

the

fir

e st

atio

n si

ren

to r

each

him

,how

can

you

pro

vein

dire

ctly

tha

t it

is n

ot 2

0°C

whe

n E

nriq

ue h

ears

the

sir

en?

Ass

ume

that

it is

20°

C w

hen

Enr

ique

hea

rs t

he s

iren

,the

n sh

ow t

hat

atth

is t

empe

ratu

re it

will

tak

e m

ore

than

5 s

econ

ds fo

r th

e so

und

of t

hesi

ren

to r

each

him

.Sin

ce t

he a

ssum

ptio

n is

fal

se,y

ou w

ill h

ave

prov

edth

at it

is n

ot 2

0°C

whe

n E

nriq

ue h

ears

the

sir

en.

1 23

a b

Pra

ctic

e (A

vera

ge)

Indi

rect

Pro

of

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

5-3

5-3



Rea

din

g t

o L

earn

Math

emati

csIn

dire

ct P

roof

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

5-3

5-3

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Lesson 5-3

Pre-

Act

ivit

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ow i

s in

dir

ect

pro

of u

sed

in

lit

erat

ure

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 5-

3 at

the

top

of

page

255

in y

our

text

book

.

How

cou

ld t

he a

utho

r of

a m

urde

r m

yste

ry u

se in

dire

ct r

easo

ning

to

show

that

a p

arti

cula

r su

spec

t is

not

gui

lty?

Sam

ple

answ

er:A

ssum

e th

atth

e pe

rson

is g

uilty

.The

n sh

ow th

at th

is a

ssum

ptio

n co

ntra

dict

sev

iden

ce t

hat

has

been

gat

here

d ab

out

the

crim

e.

Rea

din

g t

he

Less

on

1.Su

pply

the

mis

sing

wor

ds t

o co

mpl

ete

the

list

of s

teps

invo

lved

in w

riti

ng a

n in

dire

ct p

roof

.

Ste

p 1

Ass

ume

that

the

con

clus

ion

is

.

Ste

p 2

Show

tha

t th

is a

ssum

ptio

n le

ads

to a

of

the

or s

ome

othe

r fa

ct,s

uch

as a

def

init

ion,

post

ulat

e,

,or

coro

llary

.

Ste

p 3

Poin

t ou

t th

at t

he a

ssum

ptio

n m

ust

be

and,

ther

efor

e,th

e

conc

lusi

on m

ust

be

.

2.St

ate

the

assu

mpt

ion

that

you

wou

ld m

ake

to s

tart

an

indi

rect

pro

of o

f ea

ch s

tate

men

t.

a.If

#6x

&30

,the

n x

%#

5.x

*"

5b.

If n

is a

mul

tipl

e of

6,t

hen

nis

a m

ulti

ple

of 3

.n

is n

ot a

mul

tiple

of

3.c.

If a

and

bar

e bo

th o

dd,t

hen

abis

odd

.ab

is e

ven.

abis

gre

ater

d.If

ais

pos

itiv

e an

d b

is n

egat

ive,

then

ab

is n

egat

ive.

than

or

equa

l to

0.e.

If F

is b

etw

een

Ean

d D

,the

n E

F"

FD

!E

D.

EF

!FD

)E

Df.

In a

pla

ne,i

f tw

o lin

es a

re p

erpe

ndic

ular

to

the

sam

e lin

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en t

hey

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para

llel.

Two

lines

are

not

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alle

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Ref

er t

o th

e fi

gure

.h

.R

efer

to

the

figu

re.

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B!

AC

,the

n m

!B

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.In

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athe

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the

pro

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the

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wee

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itio

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:The

con

trap

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the

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tate

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→q

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t &

q→

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In a

nin

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ct p

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of

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p→

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u as

sum

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lse

and

show

tha

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s th

at p

is f

alse

,tha

t is

,you

sho

w t

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tru

e.B

ecau

se a

sta

tem

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quiv

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t to

its

cont

rapo

sitiv

e,pr

ovin

g th

e co

ntra

posi

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rue

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the

orig

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con

ditio

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true

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eth

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cou

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les.

Con

side

r th

e fo

llow

ing

stat

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t.

For

any

num

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aan

d b,

a#

b!

b#

a.

You

can

prov

e th

at t

his

stat

emen

t is

fal

se in

gen

eral

if y

ou c

an f

ind

one

exam

ple

for

whi

ch t

he s

tate

men

t is

fal

se.

Let

a!

7 an

d b

!3.

Subs

titu

te t

hese

val

ues

in t

he e

quat

ion

abov

e.

7 #

3 #

3 #

74

( #

4

In g

ener

al,f

or a

ny n

umbe

rs a

and

b,th

e st

atem

ent

a#

b!

b#

ais

fal

se.

You

can

mak

e th

e eq

uiva

lent

ver

bal s

tate

men

t:su

btra

ctio

n is

not

aco

mm

utat

ive

oper

atio

n.

In e

ach

of

the

foll

owin

g ex

erci

ses

a,b

,an

d c

are

any

nu

mbe

rs.P

rove

th

at

the

stat

emen

t is

fal

se b

y co

un

tere

xam

ple

.S

ampl

e an

swer

s ar

e gi

ven.

1.a

#(b

#c)

# (a

#b)

#c

2.a

)(b

)c)

# (a

)b)

)c

6 "

(4 "

2) #

(6 "

4) "

26

+(4

+2)

# (6

+4)

+2

6 "

2 #

2 "

2&6 2&

#&1 2.5 &

4 )

03

) 0

.75

3.a

)b

# b

)a

4.a

)(b

"c)

# (a

)b)

"(a

)c)

6 +

4 #

4 +

66

+(4

!2)

#(6

+4)

!(6

+2)

&3 2&)

&2 3&6

+6

#1.

5 !

31

) 4

.5

5.a

"(b

c) #

(a"

b)(a

"c)

6.a2

"a2

# a

4

6 !

(4 ,

2)

#(6

!4)

(6 !

2)62

!62

#64

6 !

8 #

(10)

(8)

36 !

36 #

1296

14 )

80

72 )

129

6

7.W

rite

the

ver

bal e

quiv

alen

ts f

or E

xerc

ises

1,2

,and

3.

1.S

ubtr

actio

n is

not

an

asso

ciat

ive

oper

atio

n.2.

Div

isio

n is

not

an

asso

ciat

ive

oper

atio

n.3.

Div

isio

n is

not

a c

omm

utat

ive

oper

atio

n.

8.Fo

r th

e D

istr

ibut

ive

Pro

pert

y a(

b"

c) !

ab"

acit

is s

aid

that

mul

tipl

icat

ion

dist

ribu

tes

over

add

itio

n.E

xerc

ises

4 a

nd 5

pro

ve t

hat

som

e op

erat

ions

do

not

dist

ribu

te.W

rite

a s

tate

men

t fo

r ea

ch e

xerc

ise

that

indi

cate

s th

is.

4.D

ivis

ion

does

not

dis

trib

ute

over

add

ition

.5.

Add

ition

doe

s no

t di

stri

bute

ove

r m

ultip

licat

ion.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

5-3

5-3



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

The

Tria

ngle

Ineq

ualit

y

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

5-4

5-4

©G

lenc

oe/M

cGra

w-H

ill26

3G

lenc

oe G

eom

etry

Lesson 5-4

The

Tria

ng

le In

equ

alit

yIf

you

tak

e th

ree

stra

ws

of le

ngth

s 8

inch

es,5

inch

es,a

nd

1 in

ch a

nd t

ry t

o m

ake

a tr

iang

le w

ith

them

,you

will

fin

d th

at it

is n

ot p

ossi

ble.

Thi

sill

ustr

ates

the

Tri

angl

e In

equa

lity

The

orem

.

Tria

ngle

Ineq

ualit

yT

he s

um o

f the

leng

ths

of a

ny tw

o si

des

of a

Theo

rem

tria

ngle

is g

reat

er th

an th

e le

ngth

of t

he th

ird s

ide.

Th

e m

easu

res

of t

wo

sid

es o

f a

tria

ngl

e ar

e 5

and

8.F

ind

a r

ange

for

the

len

gth

of

the

thir

d s

ide.

By

the

Tri

angl

e In

equa

lity,

all t

hree

of

the

follo

win

g in

equa

litie

s m

ust

be t

rue.

5 "

x&

88

"x

&5

5 "

8 &

xx

&3

x&

#3

13 &

x

The

refo

re x

mus

t be

bet

wee

n 3

and

13.

Det

erm

ine

wh

eth

er t

he

give

n m

easu

res

can

be

the

len

gth

s of

th

e si

des

of

atr

ian

gle.

Wri

te y

esor

no.

1.3,

4,6

yes

2.6,

9,15

no

3.8,

8,8

yes

4.2,

4,5

yes

5.4,

8,16

no6.

1.5,

2.5,

3ye

s

Fin

d t

he

ran

ge f

or t

he

mea

sure

of

the

thir

d s

ide

give

n t

he

mea

sure

s of

tw

o si

des

.

7.1

and

6 8.

12 a

nd 1

8

5 '

n'

76

'n

'30

9.1.

5 an

d 5.

5 10

.82

and

8

4 '

n'

774

'n

'90

11.S

uppo

se y

ou h

ave

thre

e di

ffer

ent

posi

tive

num

bers

arr

ange

d in

ord

er f

rom

leas

t to

grea

test

.Wha

t si

ngle

com

pari

son

will

let

you

see

if t

he n

umbe

rs c

an b

e th

e le

ngth

s of

the

side

s of

a t

rian

gle?

Find

the

sum

of

the

two

smal

ler

num

bers

.If

that

sum

is g

reat

er t

han

the

larg

est

num

ber,

then

the

thr

ee n

umbe

rs c

an b

e th

e le

ngth

s of

the

sid

esof

a t

rian

gle.

BC

A

a

cb

Exercis

esExercis

es

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-H

ill26

4G

lenc

oe G

eom

etry

Dis

tan

ce B

etw

een

a P

oin

t an

d a

Lin

e

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

The

Tria

ngle

Ineq

ualit

y

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

5-4

5-4

The

per

pend

icul

ar s

egm

ent

from

a p

oint

to

a lin

e is

the

sho

rtes

t se

gmen

t fr

om t

hepo

int

to t

he li

ne.

P!C!is

the

shor

test

seg

men

t fro

m P

toA

B#!

" .

The

per

pend

icul

ar s

egm

ent

from

a p

oint

to

a pl

ane

is t

he s

hort

est

segm

ent

from

the

poin

t to

the

pla

ne.

Q!T!

is th

e sh

orte

st s

egm

ent f

rom

Qto

pla

ne N

.

Q TN

B

P CA G

iven

:Poi

nt

Pis

equ

idis

tan

t fr

om t

he

sid

es

of a

n a

ngl

e.P

rove

:B !A!

$C!

A!P

roof

:1.

Dra

w B !

P!an

d C!

P!⊥

to

1.D

ist.

is m

easu

red

the

side

s of

!R

AS

.al

ong

a ⊥.

2.!

PB

Aan

d !

PC

Aar

e ri

ght

angl

es.

2.D

ef.o

f ⊥

lines

3."

AB

Pan

d "

AC

Par

e ri

ght

tria

ngle

s.3.

Def

.of

rt."

4.!

PB

A"

!P

CA

4.R

t.an

gles

are

".

5.P

is e

quid

ista

nt f

rom

the

sid

es o

f !R

AS

.5.

Giv

en6.

B !P!

"C!

P!6.

Def

.of

equi

dist

ant

7.A !

P!"

A!P!

7.R

efle

xive

Pro

pert

y8.

"A

BP

""

AC

P8.

HL

9.B !

A!"

C!A!

9.C

PC

TC

Com

ple

te t

he

pro

of.

Giv

en:"

AB

C"

"R

ST

;!D

"!

UP

rove

:A !D!

"R!

U!P

roof

:

1."

AB

C"

"R

ST

;!D

"!

U1.

Giv

en2.

A!C!

"R!

T!2.

CP

CTC

3.!

AC

B"

!R

TS

3.C

PC

TC4.

!A

CB

and

!A

CD

are

a lin

ear

pair

;4.

Def

.of lin

ear

pair

!R

TS

and

!R

TU

are

a lin

ear

pair

.

5.!

AC

Ban

d !

AC

Dar

e su

pple

men

tary

;5.

Line

ar p

airs

are

sup

pl.

!R

TS

and

!R

TU

are

supp

lem

enta

ry.

6.!

AC

D$

!R

TU6.

Ang

les

supp

l.to

"an

gles

are

".

7."

AD

C"

"R

UT

7.A

AS

8.A!

D!$

R!U!

8.C

PC

TC

A DC

B

R UT

S

AS

CPB

R

Exam

ple

Exam

ple

Exercis

esExercis

es



Skil

ls P

ract

ice

The

Tria

ngle

Ineq

ualit

y

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

5-4

5-4

©G

lenc

oe/M

cGra

w-H

ill26

5G

lenc

oe G

eom

etry

Lesson 5-4

Det

erm

ine

wh

eth

er t

he

give

n m

easu

res

can

be

the

len

gth

s of

th

e si

des

of

atr

ian

gle.

Wri

te y

esor

no.

1.2,

3,4

yes

2.5,

7,9

yes

3.4,

8,11

yes

4.13

,13,

26no

5.9,

10,2

0no

6.15

,17,

19ye

s

7.14

,17,

31no

8.6,

7,12

yes

Fin

d t

he

ran

ge f

or t

he

mea

sure

of

the

thir

d s

ide

of a

tri

angl

e gi

ven

th

e m

easu

res

of t

wo

sid

es.

9.5

and

910

.7 a

nd 1

4

4 '

n'

147

'n

'21

11.8

and

13

12.1

0 an

d 12

5 '

n'

212

'n

'22

13.1

2 an

d 15

14.1

5 an

d 27

3 '

n'

2712

'n

'42

15.1

7 an

d 28

16.1

8 an

d 22

11 '

n'

454

'n

'40

ALG

EBR

AD

eter

min

e w

het

her

th

e gi

ven

coo

rdin

ates

are

th

e ve

rtic

es o

f a

tria

ngl

e.E

xpla

in.

17.A

(3,5

),B

(4,7

),C

(7,6

)18

.S(6

,5),

T(8

,3),

U(1

2,#

1)

Yes;

AB

#'

5!,B

C#

'10!

,and

N

o;S

T#

2'2!,

TU#

4'2!,

and

AC

#'

17!,s

o A

B!

BC

%A

C,

SU

#6'

2!,so

ST

!TU

#S

U.

AB

!A

C%

BC

,and

A

C!

BC

%A

B.

19.H

(#8,

4),I

(#4,

2),J

(4,#

2)20

.D(1

,#5)

,E(#

3,0)

,F(#

1,0)

No;

HI#

2'5!,

IJ#

4'5!,

and

Yes;

DE

#'

41!,E

F#

2,an

d H

J#

6'5!,

so H

I!IJ

#H

J.D

F#

'29!

,so

DE

!E

F%

DF,

DE

!D

F%

EF,

and

DF

!E

F%

DE

.

©G

lenc

oe/M

cGra

w-H

ill26

6G

lenc

oe G

eom

etry

Det

erm

ine

wh

eth

er t

he

give

n m

easu

res

can

be

the

len

gth

s of

th

e si

des

of

atr

ian

gle.

Wri

te y

esor

no.

1.9,

12,1

8ye

s2.

8,9,

17no

3.14

,14,

19ye

s4.

23,2

6,50

no

5.32

,41,

63ye

s6.

2.7,

3.1,

4.3

yes

7.0.

7,1.

4,2.

1no

8.12

.3,1

3.9,

25.2

yes

Fin

d t

he

ran

ge f

or t

he

mea

sure

of

the

thir

d s

ide

of a

tri

angl

e gi

ven

th

e m

easu

res

of t

wo

sid

es.

9.6

and

1910

.7 a

nd 2

913

'n

'25

22 '

n'

36

11.1

3 an

d 27

12.1

8 an

d 23

14 '

n'

405

'n

'41

13.2

5 an

d 38

14.3

1 an

d 39

13 '

n'

638

'n

'70

15.4

2 an

d 6

16.5

4 an

d 7

36 '

n'

4847

'n

'61

ALG

EBR

AD

eter

min

e w

het

her

th

e gi

ven

coo

rdin

ates

are

th

e ve

rtic

es o

f a

tria

ngl

e.E

xpla

in.

17.R

(1,3

),S

(4,0

),T

(10,

#6)

18.W

(2,6

),X

(1,6

),Y

(4,2

)

No;

RS

#3'

2!,S

T#

6'2!,

and

Yes;

WX

#1,

XY

#5,

and

RT

#9'

2!,so

RS

!S

T#

RT.

WY

#2'

5!,so

WX

!X

Y%

WY

,W

X!

WY

%X

Y,a

nd

WY

!X

Y%

WX

.

19.P

(#3,

2),L

(1,1

),M

(9,#

1)20

.B(1

,1),

C(6

,5),

D(4

,#1)

No;

PL

#'

17!,L

M#

2 '

17!,a

nd

Yes;

BC

#'

41!,C

D#

2'10!

,and

PM

#3

'17!

,so

PL

!LM

#P

M.

BD

#'

13!,s

o B

C!

CD

%B

D,

BC

!B

D%

CD

,and

BD

!C

D%

BC

.

21.G

AR

DEN

ING

Ha

Poon

g ha

s 4

leng

ths

of w

ood

from

whi

ch h

e pl

ans

to m

ake

a bo

rder

for

atr

iang

ular

-sha

ped

herb

gar

den.

The

leng

ths

of t

he w

ood

bord

ers

are

8 in

ches

,10

inch

es,

12 in

ches

,and

18

inch

es.H

ow m

any

diff

eren

t tr

iang

ular

bor

ders

can

Ha

Poon

g m

ake?

3

Pra

ctic

e (A

vera

ge)

The

Tria

ngle

Ineq

ualit

y

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

5-4

5-4



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csTh

e Tr

iang

le In

equa

lity

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

5-4

5-4

©G

lenc

oe/M

cGra

w-H

ill26

7G

lenc

oe G

eom

etry

Lesson 5-4

Pre-

Act

ivit

yH

ow c

an y

ou u

se t

he

Tri

angl

e In

equ

alit

y T

heo

rem

wh

en t

rave

lin

g?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 5-

4 at

the

top

of

page

261

in y

our

text

book

.

In a

ddit

ion

to t

he g

reat

er d

ista

nce

invo

lved

in f

lyin

g fr

om C

hica

go t

oC

olum

bus

thro

ugh

Indi

anap

olis

rat

her

than

fly

ing

nons

top,

wha

t ar

e tw

oot

her

reas

ons

that

it w

ould

tak

e lo

nger

to

get

to C

olum

bus

if y

ou t

ake

two

flig

hts

rath

er t

han

one?

Sam

ple

answ

er:t

ime

need

ed fo

r an

ext

rata

keof

f an

d la

ndin

g;la

yove

r tim

e in

Indi

anap

olis

bet

wee

n th

etw

o fli

ghts

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.

Whi

ch s

tate

men

ts a

re t

rue?

C,D

,FA

.DE

&E

F"

FD

B.D

E!

EF

"F

DC

.EG

!E

F"

FG

D.E

D"

DG

&E

GE

.The

sho

rtes

t di

stan

ce f

rom

Dto

EG

#!" i

s D

F.

F.T

he s

hort

est

dist

ance

fro

m D

to E

G# !

" is

DG

.

2.C

ompl

ete

each

sen

tenc

e ab

out

"X

YZ

.

a.If

XY

!8

and

YZ

!11

,the

n th

e ra

nge

of v

alue

s fo

r X

Zis

%

XZ

%.

b.If

XY

!13

and

XZ

!25

,the

n Y

Zm

ust

be b

etw

een

and

.

c.If

"X

YZ

is is

osce

les

wit

h !

Zas

the

ver

tex

angl

e,an

d X

Z!

8.5,

then

the

ran

ge o

f

valu

es f

or X

Yis

%

XY

%.

d.If

XZ

!a

and

YZ

!b,

wit

h b

%a,

then

the

ran

ge fo

r X

Yis

%

XY

%.

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

a ne

w t

heor

em is

to

stat

e it

info

rmal

ly in

dif

fere

nt w

ords

.How

coul

d yo

u re

stat

e th

e T

rian

gle

Ineq

ualit

y T

heor

em?

Sam

ple

answ

er:T

he s

ide

that

con

nect

s on

e ve

rtex

of

a tr

iang

le t

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ion

to

Kin

gs,C

han

a,an

d H

olco

mb

are

3 k

m,

4 k

m,a

nd

5 k

m,r

esp

ecti

vely

.Jan

e be

gin

s at

Hol

com

b an

dd

rive

s d

irec

tly

to C

han

a,th

en t

o K

ings

,an

d t

hen

bac

k t

oH

olco

mb.

At

the

end

of

her

tri

p,s

he

figu

res

she

has

tra

vele

d25

km

alt

oget

her

.Has

sh

e fi

gure

d t

he

dis

tan

ce c

orre

ctly

?

To s

olve

thi

s pr

oble

m,a

dia

gram

can

be

draw

n.B

ased

on

this

dia

gram

and

the

Tri

angl

e In

equa

lity

The

orem

,the

di

stan

ce f

rom

Hol

com

b to

Cha

na is

less

tha

n 9

km.S

imila

rly,

the

dist

ance

fro

m C

hana

to

Kin

gs is

less

tha

n 7

km,a

nd t

hedi

stan

ce f

rom

Kin

gs t

o H

olco

mb

is le

ss t

han

8 km

.

The

refo

re,J

ane

mus

t ha

ve t

rave

led

less

tha

n (9

"7

"8)

km

or

24

km v

ersu

s he

r ca

lcul

ated

dis

tanc

e of

25

km.

Exp

lain

wh

y ea

ch o

f th

e fo

llow

ing

stat

emen

ts i

s tr

ue.

Dra

w a

nd

lab

el a

dia

gram

to

be u

sed

in

th

e ex

pla

nat

ion

.

1.If

an

alti

tude

is d

raw

n to

one

sid

e of

a t

rian

gle,

then

the

leng

th o

f th

e al

titu

de is

less

tha

n on

e-ha

lf t

he s

um o

f th

ele

ngth

s of

the

oth

er t

wo

side

s.

If B!

D!is

the

alti

tude

,the

n it

is t

rue

that

B!D!

$A!

C!.

Then

"B

DC

and

"B

DA

are

righ

t tr

iang

les.

By

Theo

rem

6-8

,BD

'B

C a

nd B

D'

BA

.Usi

ngTh

eore

m 6

-2,2

BD

'B

A!

BC

.Thu

s,B

D'

&1 2&(B

A!

BC

).

2.If

poi

nt Q

is in

the

inte

rior

of *

AB

Can

d on

the

ang

le b

isec

tor

of !

B,t

hen

Qis

equ

idis

tant

fro

m A!

B!an

d C!

B!.(

Hin

t:D

raw

Q!D!

and

Q!E!

such

tha

t Q!

D!$

A!B!

and

Q!E!

$C!

B!.)

If Q

is o

n th

e bi

sect

or o

f !

B,Q!

D!$

A!B!

,and

Q!

E!$

C!B!

,the

n "

QE

B$

"Q

DB

by H

A.T

hus,

Q!E!

$Q!

D!by

CP

CTC

,whi

ch m

eans

tha

t Q

iseq

uidi

stan

t fr

om A!

B!an

d C!

B!.

CE

B

A

Q

D

AD

C

B

King

s

Davi

sJu

nctio

n

Chan

aHo

lcom

b

3 km

5 km

4 km

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

5-5

5-5

Exam

ple

Exam

ple


Documents

Chapter 5 Resource Masters - Math Class · PDF fileThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 5.As ... ! 6z" 4 " 11 3x! 15 36 ! 21 " 3y 9z"