chapter 5 resource masters - math problem solving©glencoe/mcgraw-hill iv glencoe geometry...

Chapter 5Resource 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 5 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: 0-07-860182-7 GeometryChapter 5 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 5-1Study Guide and Intervention . . . . . . . . 245–246Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 247Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 248Reading to Learn Mathematics . . . . . . . . . . 249Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 250





Chapter 5 AssessmentChapter 5 Test, Form 1 . . . . . . . . . . . . 275–276Chapter 5 Test, Form 2A . . . . . . . . . . . 277–278Chapter 5 Test, Form 2B . . . . . . . . . . . 279–280Chapter 5 Test, Form 2C . . . . . . . . . . . 281–282Chapter 5 Test, Form 2D . . . . . . . . . . . 283–284Chapter 5 Test, Form 3 . . . . . . . . . . . . 285–286Chapter 5 Open-Ended Assessment . . . . . . 287Chapter 5 Vocabulary Test/Review . . . . . . . 288Chapter 5 Quizzes 1 & 2 . . . . . . . . . . . . . . . 289Chapter 5 Quizzes 3 & 4 . . . . . . . . . . . . . . . 290Chapter 5 Mid-Chapter Test . . . . . . . . . . . . 291Chapter 5 Cumulative Review . . . . . . . . . . . 292Chapter 5 Standardized Test Practice . 293–294

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A26

© Glencoe/McGraw-Hill iv Glencoe Geometry

Teacher’s Guide to Using theChapter 5 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 5 Resource Masters includes the core materials neededfor Chapter 5. 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 5-1.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 5-1.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 5Resources 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 278–279. 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 _____

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


© 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 _____

5-15-1


Less

on

5-1

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?

DI

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

on

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�

RV 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 _____

5-25-2


Less

on

5-2

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 6

7 89

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 � 99

c. 8 � �2 d. 7 � �7

e. �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, �IJH

b. 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

T

X

V

BAC

S

ExistingRoad

Closed Region(Lake)

Enrichment

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Study Guide and InterventionIndirect Proof

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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

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ExercisesExercises

ExampleExample

Skills PracticeIndirect Proof

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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

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Reading to Learn MathematicsIndirect Proof

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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

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Study Guide and InterventionThe Triangle Inequality

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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

A

DC

B

R

UT

S

AS C

PB

R

ExampleExample

ExercisesExercises

Skills PracticeThe Triangle Inequality

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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

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Reading to Learn MathematicsThe Triangle Inequality

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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 � FD

C. EG � EF � FG D. ED � DG � EG

E. 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 Triangles

The 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

RAT

BA

W

T

O

A R

T

Enrichment

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Study Guide and InterventionInequalities Involving Two Triangles

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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.

62�65�

2.7 cm1.8 cm

1.8 cm (3x � 2.1) cm

115�

120� 24 cm

24 cm40 cm

(4x � 10) cm

M R

N P

48�

46�

20 25

20

E G

H K

J

F60�

62�

10

10

42

42

C

A

DB

22�

38�

N

R

P

M

21�

19�

B D

A

28�22�

C

S T80�

R

B C60�

A

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.

5. 6.

33�

60 cm

60 cm

36 cm

30 cm(3x � 3)�

(1–2x � 6)�

52�30

30

28

12

42

28

ZW

XY

30C

A X

B30

5048 24

24Z Y

11 16

2626

B

CDA

13

10

M

R

NP

13

16

C

D

A

B

3838

2323 3336

TR

SN

M P


Inequalities Involving Two Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5

ExampleExample

ExercisesExercises

Skills PracticeInequalities Involving Two Triangles

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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

12

B

A

D

C

12

3

B

AD C

E

95�7 7

85�P RS

Q31

30

22 22

R S

U T

6

98

3

3

B

A C

D

X



1. AB, BK 2. ST, SR

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?

1 2D F

E

G

20 21

R TS

J K

14 14

14

13

12C F

E

D

(x � 3)�(x � 3)�

10 10

R TS

Q

40�

30�

60�A KM

B

Practice Inequalities Involving Two Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5

Reading to Learn MathematicsInequalities Involving Two Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

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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

B

Kings

DavisJunction

Chana Holcomb

3 km

5 km4 km

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5

ExampleExample

Chapter 5 Test, Form 155


Ass

essm

entsWrite the letter for the correct answer in the blank at the right of each

question.

For Questions 1–4, use the figure at the right.

1. Name an altitude.A. D�E� B. A�B�C. GB�� D. CF��

2. Name a perpendicular bisector.A. D�E� B. A�B� C. GB�� D. CF��

3. Name an angle bisector.A. D�E� B. A�B� C. GB�� D. CF��

4. Name a median.A. D�E� B. A�B� C. GB�� D. CF��

For Questions 5–7, use the figure to determine which is a true statement for the given information.

5. A�C� is a median.A. m�ACD � 90 B. �BAC � �DACC. BC � CD D. �B � �D

6. A�C� is an angle bisector.A. m�ACD � 90 B. �BAC � �DAC C. BC � CD D. �B � �D

7. A�C� is an altitude.A. m�ACD � 90 B. �BAC � �DAC C. BC � CD D. �B � �D

8. Name the longest side of �DEF.A. D�E� B. E�F�C. D�F� D. cannot tell

9. Which angle in �ABC has the greatest measure?A. �A B. �BC. �C D. cannot tell

10. Given �ABC with vertices A(2, 6), B(�4, �2), and C(8, �6), find an equationfor the line containing the altitude to AB.

A. y � ��34�x � �

54� B. y � �

43�x � �

130� C. y � ��

34�x D. y � ��

89�x � �

190�

11. Given �ABC with vertices A(2, 6), B(�4, �2), and C(8, �6), find an equationfor the line containing the median to A�B�.

A. y � ��34�x � �

54� B. y � �

43�x � �

130� C. y � ��

34�x D. y � ��

89�x � �

190�

A C

B

9

5 7

D 62�10�

108�

F

E

A D

C

B

A

BC D

E F

G1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 5 Test, Form 1 (continued)55

12.

13.

14.

15.

16.

17.

18.

19.

20.

12. Find the possible values for m�1.A. 180 � m�1 � 62 B. 90 � m�1 � 62C. 0 � m�1 � 62 D. m�1 � 118

13. Find x.A. 5 B. 7C. 10 D. 15

14. If D is the circumcenter of �ABC and AD � 6, find BD.A. 4 B. 6C. 9 D. 12

15. Choose the assumption you would make to start an indirect proof of x � 3.A. x � 3 B. x � 3 C. x � 3 D. x � 3

16. Choose the assumption you would make to start an indirect proof.Given: a ⁄|| b Prove: �1 and �2 are not supplementary.A. a || b B. �1 and �2 are supplementary.C. �1 � �2 D. �1 and �2 are complementary.

17. Which can be the lengths of the sides of a triangle?A. 12, 9, 4 B. 1, 2, 3 C. 5, 5, 10 D. �2�, �5�, �18�

18. Find the shortest distance from B to A�C�.A. BD B. BCC. BF D. BE

For Questions 19 and 20, use the figure.

19. Given: A�C� � D�F�, A�B� � D�E�, m�A � m�DWhich can be concluded by the SAS Inequality Theorem?A. �ABC � �DEF B. BC � EFC. BC � EF D. BC � EF

20. Given: A�B� � D�E�, B�C� � E�F�, AC � DFWhich can be concluded by the SSS Inequality Theorem?A. m�B � m�E B. m�B � m�EC. m�B � m�E D. �BAC � �EDF

Bonus Q�S� is a median of �PQR with point S on P�R�.If PS � x2 � 3x and SR � 2x � 6, find x.

A

CBD

E F

A C

B

D E F

A C

B

D

x � 3

5W X

M

T V

N

15

62� 1

B:

NAME DATE PERIOD

Chapter 5 Test, Form 2A55


Ass

essm


question.

For Questions 1–4, use the figure.1. Name an angle bisector.

A. K�I� B. GL��

C. JM�� D. H�J�

2. Name a median.A. K�I� B. GL�� C. JM�� D. H�J�

3. Name an altitude.A. K�I� B. GL�� C. JM�� D. H�J�

4. Name a perpendicular bisector.A. K�I� B. GL�� C. JM�� D. H�J�


5. Y�W� is an angle bisector.A. �YWZ is a right angle. B. �XYW � �ZYWC. XW � WZ D. XY � ZY

6. Y�W� is an altitude.A. �YWZ is a right angle. B. �XYW � �ZYWC. XW � WZ D. XY � ZY

7. Y�W� is a median.A. �YWZ is a right angle. B. �XYW � �ZYWC. XW � WZ D. XY � ZY

8. Name the longest side of �ABC.A. A�B� B. B�C�C. A�C� D. cannot tell

9. Name the angle with greatest measure in �DEF.A. �D B. �EC. �F D. cannot tell

10. Given �ABC with vertices A(2, 6), B(�4, �2), and C(8, �6), find an equationfor the line containing the median to B�C�.A. y � 3x � 10 B. y � 3x C. y � ��

13�x � �

130� D. x � 2

11. Given �ABC with vertices A(2, 6), B(�4, �2), and C(8, �6), find an equationfor the perpendicular bisector of B�C�.A. y � 3x � 10 B. y � 3x C. y � ��

13�x � �

130� D. x � 2

3

79

F

DE

22� 84�

74�

A C

B

XY

Z

W

H

J

MI G

L K 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 5 Test, Form 2A (continued)55

12.

13.

14.

15.

16.

17.

18.

19.

20.

12. Find the possible values for m�1.A. 90 � m�1 � 74 B. 180 � m�1 � 74C. 0 � m�1 � 74 D. m�1 � 106

13. Find x.A. 9 B. 11C. 27 D. 32

14. Which is another name for an indirect proof?A. proof by deduction B. proof by converseC. proof by inverse D. proof by contradiction


16. Choose the assumption you would make to start an indirect proof.Given: �1 is an exterior angle of �ABC. Prove: m�1 � m�B � m�C

A. �1 is not an exterior angle of �ABE.B. �1 is an interior angle of �ABC.C. m�1 m�B � m�CD. m�1 � m�B

17. Which can be the lengths of the sides of a triangle?A. 6, 6, 12 B. 6, 7, 13 C. �2�, �5�, �15� D. 2.6, 8.1, 10.2

18. Compare QS to RS.A. QS � RS B. QS � RSC. QS � RS D. cannot tell

19. Compare DC to AD.A. DC � AD B. DC � ADC. DC � AD D. cannot tell

20. Compare m�1 to m�2.A. m�1 � m�2 B. m�1 � m�2C. m�1 � m�2 D. cannot tell

Bonus Y�W� bisects �XYZ in �XYZ. Point W is on X�Z�.If m�XYW � 2x � 18 and m�ZYW � x2 � 5x, find x.

8

8

15

131

2

10

10

30�

20�

A

C

D

B

RT

Q S

x � 2

x � 7A

BC

D

EF27

174�

B:

NAME DATE PERIOD

Chapter 5 Test, Form 2B55


Ass

essm


question.

For Questions 1–4, use the figure.

1. Name a median.A. R�W� B. SV��

C. Q�T� D. RU��

2. Name an angle bisector.A. R�W� B. SV�� C. Q�T� D. RU��

3. Name a perpendicular bisector.A. R�W� B. SV�� C. Q�T� D. RU��

4. Name an altitude.A. R�W� B. R�P� C. Q�T� D. RU��


5. F�G� is an altitude.A. �DGF is a right angle. B. DF � EFC. DG � GE D. �DFG � �EFG

6. F�G� is a median.A. �DGF is a right angle. B. DF � EFC. DG � GE D. �DFG � �EFG

7. F�G� is an angle bisector.A. �DGF is a right angle. B. DF � EFC. DG � GE D. �DFG � �EFG

8. Name the longest side of �ABC.A. A�B� B. B�C�C. A�C� D. cannot tell

9. Name the angle with the greatest measure in �GHI.A. �G B. �HC. �I D. cannot tell

10. Given �ABC with vertices A(2, 6), B(�4, �2), and C(8, �6), find an equationfor the perpendicular bisector of A�C�.

A. y � �12�x � �

52� B. y � �

29�x � �

190� C. y � �

12�x D. y � 0

11. Given �ABC with vertices A(2, 6), B(�4, �2), and C(8, �6), find an equationfor the line containing the altitude to A�C�.

A. y � �12�x � �

52� B. y � �

29�x � �

190� C. y � �

12�x D. y � 0

5

7

9H

GI

70� 48�

62�

A C

B

G

D

E F

P

WR

S

TU VQ

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 5 Test, Form 2B (continued)55

12.

13.

14.

15.

16.

17.

18.

19.

20.

12. Find the possible values for m�1.A. m�1 � 124 B. 0 � m�1 � 56C. 90 � m�1 � 56 D. 180 � m�1 � 56

13. Find ST.A. 12 B. 18C. 23 D. 24

14. Which of the following is the last step in an indirect proof ?A. show the assumption true B. show the assumption falseC. show the conclusion false D. contradict the conclusion


16. Choose the assumption you would make to start this indirect proof.Given: A�B� bisects �CAD.Prove: �ACB � �DAB

A. A�B� does not bisect �CAD. B. �ACD is isosceles.C. A�B� is a median. D. �ACB � �DAB

17. Which of the following can be the lengths of the sides of a triangle?A. 12, 9, 2 B. 11, 12, 23 C. 2, 3, 4 D. �3�, �5�, �18�

18. Compare YW to YX.A. YW � YX B. YW � YXC. YW � YX D. cannot tell

19. Compare DG to GF.A. DG � GF B. DG � GFC. DG � GF D. cannot tell

20. Compare m�1 to m�2.A. m�1 � m�2 B. m�1 � m�2C. m�1 � m�2 D. cannot tell

Bonus H�J� is an altitude of �GHI with point J on G�I�.If m�GJH � 5x � 30, GH � 3x � 4, HI � 5x � 3,JI � 4x � 3, and GJ � x � 6, find the perimeter of �GHI.

11

1010

12

12

M

N K

L

8

830�

20�

D

FG

E

W

Y

X

Z

x � 6

5P

QRS

TU

2x � 1

156�

B:

NAME DATE PERIOD

Chapter 5 Test, Form 2C55


Ass

essm

ents1. Name an angle bisector.

2. The perimeter of ABCD is 44. Find x.Then describe the relationship betweenAC�� and B�D�.

3. If point E is the centroid of �ABC,BD � 12, EF � 7, and AG � 15, find ED.

4. If �XYZ has vertices at X(�2, 6), Y(4, 10), and Z(14, 6), findthe coordinates of the centroid of �XYZ.

5. If P�O� is an angle bisector of �MON,find x.

6. Write a compound inequality for the possible measures of �A.

7. List the angles of �GHI in order from least to greatest measure.

8. List the sides of �PQR in order from shortest to longest.

9. Find the shortest segment.

10. Write the assumption you would make to start an indirect proofof the statement If 16 is a factor of n, then 4 is a factor of n.

11. Write the assumption you would make to start an indirectproof of the statement If A�B� is an altitude of equilateraltriangle ABC, then A�B� is a median.

55�55�65�

65�60�

60�

Y

X Z

W

80� 45�

55�

Q

P R

2 in.1.2 in.

3 in.

IG

H

135�

B

C A

(2x � 10)� (x � 15)�O

PN M

DA C

B

F G

E

2x � 3

2x � 7

x � 5

x � 1

D

B

AC

G

B CD

A

E

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 5 Test, Form 2C (continued)55

12. Write the assumption you would make to start an indirectproof for the following.Given: X�Y� � Y�Z�

Y�W� bisects �XYZ.Prove: �X � �Z

13. If two sides of a triangle are 10 meters and 23 meters long, thenthe third side must have a length between what two measures?

14. Find the shortest distance from P to RQ��.

15. If BD�� bisects �ABC, find x.

16. Write an inequality comparing EFand GH.

17. Write an inequality comparing m�1 and m�2.

For Questions 18–20, complete the proof below by supplyingthe missing information for each corresponding location.Given: �ABC, AD � CB, and AC � DBProve: m�ADC � m�DCB

Statements Reasons

1. AD � CB and AC � DB 1. Given

2. A�D� � C�B� 2.

3. C�D� � C�D� 3.

4. m�ADC � m�DCB 4.

Bonus Write an equation in slope-intercept form for the altitude to B�C�.

x

y

C(2c, 0)A(0, 0)

B(2a, 2b)

O

(Question 20)

(Question 19)

(Question 18)

C

BA D

5 ft

5 ft

12 ft

11 ft

12

23�20�

H

G F

E

6

6

2x � 30

3x � 4B C

A

D

R Q

P

S T U

Y

ZX W

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

NAME DATE PERIOD

Chapter 5 Test, Form 2D55


Ass

essm

ents1. Name a perpendicular bisector.

2. The perimeter of PRQS is 34. Find x. Then describe the relationship between RS�� and P�Q�.

3. If point N is the centroid of �HIJ,IM � 18, KN � 4, and HL � 15, find JN.

4. If �DEF has vertices at D(4, 12), E(14, 6), and F(�6, 2), findthe coordinates of the circumcenter of �DEF.

5. If R�U� is an altitude for �RST, find x.

6. Write a compound inequality for the possible measures of �X.

7. List the angles of �TUV in order from least to greatest measure.

8. List the sides of �FGH in order from shortest to longest.

9. Name the longest segment.

10. Write the assumption you would make to start an indirectproof of the statement If n is an even number, then n2 is aneven number.

11. Write the assumption you would make to start an indirectproof of the statement If A�D� is an angle bisector of equilateraltriangle ABC, then A�D� is an altitude.

50�96�

34�

30� 70�

80�

KN

L M

82� 41�

57�

H

F

G

1.8

5

3.9

VT

U

127�

Z

Y X

(5x � 10)�T

U

S

R

MH J

I

K L

N

4x � 10

2x � 3 7

10R

S

QP

G

H

NK

JL

I

M

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 5 Test, Form 2D (continued)55

12. Write the assumption you would make to start an indirectproof for the following.Given: V is not the midpoint of P�Q�;

�P � �Q.Prove: S�V� ⁄⊥ P�Q�

13. If the lengths of two sides of a triangle are 14 feet and 29 feet,then the third side must have a length between what twomeasures?

14. Find the shortest distance from B to AC��.

15. If YW�� bisects �XYZ, find x.

16. Write an inequality comparing m�1 and m�2.

17. Write an inequality comparing BCand ED.

For Questions 18–20, complete the proof below by supplyingthe missing information for each corresponding location.Given: K is the midpoint of A�B�.

m�MKB � m�MKAProve: MB � AM

Statements Reasons

1. K is the midpoint of A�B�; 1. Given m�MKB � m�MKA.

2. B�K� � K�A� 2.

3. M�K� � M�K� 3.

4. MB � AM 4.

Bonus Write an equation in slope-intercept form for the perpendicular bisector of C�E�.

x

y

E(a, 0)C(0, 0)

D(b, c)

O

(Question 20)

(Question 19)

(Question 18)

M

AB K

24�

8

830�

E

D

BC

77

10

9

1 2

3x � 20

2x � 15Y Z

X

W

A D E F C

B

S

QP V

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

NAME DATE PERIOD

Chapter 5 Test, Form 355


Ass

essm

ents1. If point G is the centroid of �ABC,

AE � 24, DG � 5, and CG � 14,find DB.

2. If �EFG has vertices at E(2, 4), F(10, �6), and G(�4, �8), findthe coordinates of the orthocenter of �EFG.

3. If J�L� is a median for �IJK, find x.

4. Write a compound inequality for the possible measures of �L.

5. List the angles of �GHI in order from least to greatest measure.

6. List the sides of �PQR in order from shortest to longest.

7. Name the shortest and the longest segments.

8. Write the assumption you would make to begin an indirectproof of the statement If 2x � 6 � 12, then x � 3.

9. Justify the statement below algebraically.

If BD�� is the perpendicular bisector of A�C�, then point T lies on BD��.

10. Write the assumption you would make to begin an indirectproof of the statement The three angle bisectors of a triangleare concurrent.

11. Write and solve an inequality to find x.(3x � 4)�

6

10

4

4

(12x � 31)�

7

14 x � y

5y � 34y � 3x

4y � x

D B

A

C

T

53� 64�

63� 55�72�

53�

V

Y

WX

45� 55�

80�

P R

Q

9.6

7 8I H

G

146�

L

N M

3x � 10 2x � 42LI K

J

FA B

C

D EG

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 5 Test, Form 3 (continued)55

12. If F�H� is a median of �EFG, find the perimeter of �EFG.

13. Write the assumption you would make to start an indirect proof for the following.

Given: A�B� � D�E� and A�C� � C�D�Prove: �B � �E

14. If the lengths of two sides of a triangle are 24 inches and 29 inches, then the third side must have a length betweenwhat two measures?

15. Name the shortest distance from Y to XZ��.

16. Write and solve an inequality to find x.

For Questions 17–20, complete the proof below by supplyingthe missing information for each corresponding location.Given: XW � YZ, XK � WK, and KZ � KYProve: m�XWZ � m�YZW

Statements Reasons

1. XW � YZ, XK � WK, 1. Givenand KZ � KY

2. XW � YZ 2.

3. XZ � WY 3.

4. WZ � WZ 4.

5. m�XWZ � m�YZW 5.

Bonus Write an equation in slope-intercept form for the line containing the median to D�E�.

x

y

F(2a, 0)D(0, 0)

E(2c, 2d)

O

(Question 20)

(Question 19)

(Question 18)

(Question 17)

Y

Z W

XK

110�84�

8

866

3x � 10

x � 20

XZ W

T

Y

B

AC

D

E

2x � 23

9x � 6x � 18

7x � 2HE G

F 12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

NAME DATE PERIOD

Chapter 5 Open-Ended Assessment55


Ass

essm

entsDemonstrate your knowledge by giving a clear, concise solution to

each 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. Two sticks are bent and connected with a rubber band as shown in thediagram. Describe what happens to the rubber band as the ends of thesticks are pulled farther apart. Name the theorem this situation illustrates.

2. Mary says FG�� and JK�� are six inches apart and Ashley says they are fourinches apart. Who is correct? Explain your answer.

3. Suppose B�D� is drawn on this figure so that point D is on AC�� and has alength of 6 centimeters. If the shortest distance from B to AC�� is 5centimeters, in how many different places on AC�� could point D be located?Explain how you know.

4. Draw a triangle that satisfies each situation.

a. Two of the sides are altitudes.

b. The altitudes intersect outside the triangle.

c. The altitudes intersect inside the triangle.

d. The altitudes are also the medians of the triangle.

5. �ABC is scalene. Explain the difference between an altitude of �ABC anda perpendicular bisector of a side of �ABC.

6. What is the difference between the SAS Inequality Theorem and thetheorem that says the greatest angle of a triangle is opposite the longestside? Draw a figure to illustrate your explanation.

7. Write an algebraic statement, then write the assumption you would maketo start an indirect proof for your statement.

10 cm

A

B

C

6 in. 4 in.F

J

G

KH

ED

NAME DATE PERIOD

SCORE


Chapter 5 Vocabulary Test/Review55

Write whether each sentence is true or false. If false, replacethe underlined word or number to make a true sentence.

1. The of a triangle is a segment whose endpoints are avertex of a triangle and the midpoint of the side opposite thevertex.

2. The of a triangle is the point where the altitudes ofthe triangle intersect.

3. The point of concurency of the perpendicular bisectors of atriangle is called the .

4. The of a triangle is the intersection of the medians ofthe triangle.

5. can be used to prove statements ingeometry and prove theorems.

6. The of a triangle is the intersection of the anglebisectors of the triangle.

7. The of a triangle is a line, segment, orray that passes through the midpoint of a side and isperpendicular to that side.

8. The is the point where three or morelines intersect.

9. Every triangle has altitude.

10. An indirect proof is a proof where you assume that theconclusion is and then show that this assumption leadsto a contradiction of the hypothesis, a definition, postulate,theorem, or some other accepted fact.

In your own words—

11. Write a definition of concurrent lines.

false

only 1

point of concurrency

perpendicular bisector

orthocenter

Indirect reasoning

incenter

circumcenter

centroid

altitude 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

altitudecentroidcircumcenter

concurrent linesincenterindirect proof

indirect reasoningmedianorthocenter

perpendicular bisectorpoint of concurrencyproof by contradiction

NAME DATE PERIOD

SCORE

Chapter 5 Quiz (Lessons 5–1 and 5–2)

55


Ass

essm

ents

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

1. What is the point called where the perpendicular bisectors ofthe sides of a triangle intersect?

2. True or false? m�4 � m�2

3. What is the name of the point that is two-thirds of the way fromeach vertex of a triangle to the midpoint of the opposite side?

4. If C�D� is the perpendicular bisector of A�B�and A�B� is the perpendicular bisector of C�D�, find x.

5. Find the shortest segment.

P

QR

S56�63�

61�54� 51�

75�

2x � 7

2y � 3

y � 1

A B

C

D

2

1 3 4

Chapter 5 Quiz (Lesson 5–3)

55

1.

2.

3.

4.

5.

1. What do you assume in an indirect proof?

For Questions 2 and 3, write the assumption you wouldmake to start an indirect proof of each statement.

2. If 2x � 7 � 19, then x � 6.

3. If �ABC is isosceles with base A�C�, then A�B� � B�C�.

For Questions 4 and 5, write the assumption you wouldmake to start an indirect proof.

4. Given: 3x � 10 � 20Prove: x � 10

5. Given: C�D� is not a median of �ABC.�1 � �2

Prove: C�B� � C�A�1 2

B A

C

D

NAME DATE PERIOD

SCORE



55

1.

2.

3.

4.

5.

1. Write AB, AC, and AD in order from least to greatest measure.

2. Determine whether A(2, 3), B(7, 12), C(�5, �24) are thevertices of �ABC. Explain your answer.

3. Name the shortest distance from A to B�C�.

4. Write an inequality expressing the possible values for x.

5. STANDARDIZED TEST PRACTICE Which of the followingsets of numbers can be the lengths of the sides of a triangle?A. 5, 5, 10 B. �39�, �8�, �5� C. 2.5, 3.4, 4.6 D. 1, 2, 4

7

9

x

F DEB C

A

CB

A D50�

75�

NAME DATE PERIOD

SCORE


55

1.

2.

3.

4.

5.

1. Write an inequality 2. Write an inequality comparingcomparing m�1 to m�2. AB to DE.

3. Write an inequality about the length of G�H�.

For Questions 4 and 5, complete the proof by supplying themissing information for each corresponding location.

Given: �ABC, AB � DE, and BE � ADProve: m�CAE � m�CEAStatements Reasons

1. AB � DE, BE � AD 1. Given2. A�B� � D�E� 2. Def. of � segments3. 3. Reflexive Prop.4. m�CAE � m�CEA 4. (Question 5)

(Question 4)

D E

BC

A

G

I F

H

50�60� 9

6

6 7

A B D E

FC

75�72�9 95 5

5

6 1 2

9

NAME DATE PERIOD

SCORE

Chapter 5 Mid-Chapter Test (Lessons 5–1 through 5–3)

55


Ass

essm

ents

1. Which of the following can intersect outside a triangle?A. angle bisectors B. mediansC. altitudes D. sides

2. What is the name of the point of concurrency of the altitudes of a triangle?A. orthocenter B. circumcenterC. incenter D. centroid

3. What is the name of the point of concurrency of the medians of a triangle?A. orthocenter B. circumcenterC. incenter D. centroid

4. Name the longest segment.A. B�D� B. B�C�C. A�D� D. C�D�

5. P�S� is the perpendicular bisector of Q�R� and Q�R� is the perpendicular bisectorof P�S�. If PQ � 2x � 17 and QS � 5x � 23, find x.A. 7 B. 5 C. 3 D. 2

A

B

CD

55�

85�

40� 50�

66� 64�

6.

7.

8.

9.

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

Part II

6. Write a compound inequality for the possible values of x.

7. What would you assume to start an indirect proof of thestatement If x � 2, then x2 � 4?

8. What would you assume to start an indirect proof of thestatement If AB � BC, then m�C � m�A?

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

Given: B�D� is not a median of �ABC.�1 � �2

Prove: B�D� does not bisect �ABC.1 2

B

A CD

50�

x

Part I Write the letter for the correct answer in the blank at the right of each question.


Chapter 5 Cumulative Review(Chapters 1–5)

55

1.

2.

3.

4.

5. and 6.

7.

8.

9.

10.

11.

12.

13.

14.

1. Find x and RS, if R is between Q and S, QR � 3x � 2,RS � 2x � 2, and QS � 5x. (Lesson 1-2)

2. Find the perimeter of �HJK with vertices H(2, 6), J(�4, 6),and K(�4, �2). (Lesson 1-6)

For Questions 3 and 4, complete this two-column proof.(Lesson 2-8)

Given: �1 � �2Prove: m�ABC � 2(m�1)Proof:Statements Reasons1. �1 � �2 1. Given2. m�1 � m�2 2. Def. of congruent angles3. 3.4. m�ABC � m�1 � m�1 4. Substitution Property5. m�ABC � 2(m�1) 5. Addition Property

For Questions 5 and 6, graph each line on the same grid.(Lesson 3-4)

5. line � perpendicular to y � �x � 3, and contains (�2, �5).

6. line m contains (2, �1) and parallel to the line containing (4, �1) and (5, 1)

7. Find the distance between the parallel lines whose equationsare y � 2x � 8 and y � 2x � 3. (Lesson 3-6)

For Questions 8–10, use the figure at the right.

8. Classify �CEF. (Lesson 4-1)

9. Find m�B. (Lesson 4-2)

10. Identify the congruent triangles. (Lesson 4-3)

11. Find the coordinates of the orthocenter of �HJK if H(2, 0),J(�4, 2), and K(0, 6). (Lesson 5-1)

12. Determine the relationship between m�PMQ and m�PQM. (Lesson 5-2)

13. Can 52, 53, and 54 be the lengths of the sides of a triangle?(Lesson 5-4)

14. Find the range for the measure of the third side of a trianglehaving two sides measuring 3 inches and 9 inches. (Lesson 5-4)

50� 60�

70�10

13

16

17

P Q

M

R

N

27�

103�

A CF

D E

B

(Question 4)(Question 3)

A

CB

12

NAME DATE PERIOD

SCORE

Standardized Test Practice (Chapters 1–5)


1. If �BXY is a right angle, then which statements are true? (Lesson 1–4)

I m�BXY � 90II The measure of an angle vertical to �BXY would be 90.III The measure of an angle supplementary to �BXY would be 90.A. I only B. I and III C. I, II, and III D. I and II

2. Which is the contrapositive of the conditional statement If m�K � 45, then x � 5? (Lesson 2-3)

E. If m�K 45, then x 5 F. If x 5, then m�K 45G. If x � 5, then m�K � 45 H. If m�K 45, then x � 5

3. Find m�HJK. (Lesson 3-2)

A. 33 B. 45C. 78 D. 147

4. The line y � 5 � �x � 3 satisfies which conditions? (Lesson 3-4)

E. m � �1, contains (�5, 3) F. m � 1, contains (�5, �3)G. m � �1, contains (5, 3) H. m � �1, contains (5, �3)

5. Given D(0, 4), E(2, 4), F(2, 1), A(0, 2), and C(�2, �1), whichcoordinates for B would make �ABC � �DEF? (Lesson 4-4)

A. B(�2, 2) B. B(0, 1)C. B(0, 0) D. B(�1, 0)

6. In �XYZ, which type of line is �? (Lesson 5-1)

E. perpendicular bisectorF. angle bisectorG. altitudeH. median

7. Which assumption would you make to start an indirect proof ofthe statement If 2x � 5 � 17, then x � 11? (Lesson 5-3)

A. x � 11 B. x � 11 C. x � 11 D. x 11

8. Which inequality describes the possible values of x? (Lesson 5-5)

E. x � 6 F. x � 6G. x 12 H. 6 � x � 12

35�

35�3x � 7

x � 5

45�

X Z

Y �

45�33�H K

J

NAME DATE PERIOD

SCORE

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.

1.

2.

3.

4.

5.

6.

7.

8. E F G H

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

55


Standardized Test Practice (continued)

9. A store sells both groceries and clothing. A surveyof 963 customers indicated that 543 customersbought groceries during the month of April.What is x in the Venn Diagram? (Lesson 2-2)

10. In 1999, Caitlin had 20,000 subscribers on hermailing list. In 2000, there were 4000 additionalsubscribers. If Caitlin continues to attract newsubscribers at the same rate, in what year willshe have 44,000 subscribers? (Lesson 3-3)

11. Name the y-intercept of ( y � 2) � 3(x � 5).(Lesson 3-4)

12. If B�D� is an altitude of �ABC, find x. (Lesson 5-2)

13. The measures of two sides of �ABC are 19 and15. The range for measure of the third side, n,would be 4 � n � . (Lesson 5-4)?

(2x � 17)�(3x � 2)�

35�A C

B

D E

GroceryItems

x

Clothing420 126

Purchases in April

NAME DATE PERIOD

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.

9. 10.

11. 12.

13.

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

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

14. Find a counterexample for the statement Five is the onlywhole number between 4.5 and 6.1. (Lesson 2-1)

15. What is the length of the side opposite the vertex angle ofisosceles �XYZ with vertices at X(�3, 4), Y(8, 6), and Z(3, �4)? (Lesson 4-1)

14.

15.

4 1 7 2 0 0 5

1 7

3 4

8

55

Standardized Test PracticeStudent Record Sheet (Use with pages 278–279 of the Student Edition.)

55

© 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 Question 9, also enter your answer by writing each number or symbol in abox. Then fill in the corresponding oval for that number or symbol.

9 (grid in) 9

10

11

12

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

Record your answers for Questions 13–14 on the back of this paper.


Stu

dy G

uid

e a

nd I

nte

rven

tion

Bis

ecto

rs,M

edia

ns,

and

Alt

itu

des

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

E__

____

____

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ER

IOD

____

_

5-1

5-1

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lenc

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eom

etry

Lesson 5-1

Perp

end

icu

lar

Bis

ecto

rs a

nd

An

gle

Bis

ecto

rsA

per

pen

dic

ula

r b

isec

tor

of a

side

of

a tr

ian

gle

is a

lin

e,se

gmen

t,or

ray

th

at i

s pe

rpen

dicu

lar

to t

he

side

an

d pa

sses

thro

ugh

its

mid

poin

t.A

not

her

spe

cial

seg

men

t,ra

y,or

lin

e is

an

an

gle

bis

ecto

r,w

hic

hdi

vide

s an

an

gle

into

tw

o co

ngr

uen

t an

gles

.

Tw

o pr

oper

ties

of

perp

endi

cula

r bi

sect

ors

are:

(1)

a po

int

is o

n t

he

perp

endi

cula

r bi

sect

or o

f a

segm

ent

if a

nd

only

if

it i

s eq

uid

ista

nt

from

the

endp

oin

ts o

f th

e se

gmen

t,an

d(2

) th

e th

ree

perp

endi

cula

r bi

sect

ors

of t

he

side

s of

a t

rian

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mee

t at

a p

oin

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lled

th

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that

is

equ

idis

tan

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om t

he

thre

e ve

rtic

es o

f th

e tr

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gle.

Tw

o pr

oper

ties

of

angl

e bi

sect

ors

are:

(1)

a po

int

is o

n t

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angl

e bi

sect

or o

f an

an

gle

if a

nd

only

if

it i

s eq

uid

ista

nt

from

th

e si

des

of t

he

angl

e,an

d(2

) th

e th

ree

angl

e bi

sect

ors

of a

tri

angl

e m

eet

at a

poi

nt,

call

ed t

he

ince

nte

rof

th

etr

ian

gle,

that

is

equ

idis

tan

t fr

om t

he

thre

e si

des

of t

he

tria

ngl

e.

BD

� ��

is t

he

per

pen

dic

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rb

isec

tor

of A �

C�.F

ind

x.

BD

��

is t

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perp

endi

cula

r bi

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or o

f A �

C�,s

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8

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C

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MR

��

is t

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e b

isec

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of �

NM

P.F

ind

xif

m�

1 �

5x�

8 an

dm

�2

�8x

�16

.

MR

��

is t

he

angl

e bi

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or o

f �

NM

P,s

o m

�1

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�2.

5x�

8 �

8x�

1624

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8 �

x

12

NR

PM

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ple1

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ple2

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ple2

Exer

cises

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cises

Fin

d t

he

valu

e of

eac

h v

aria

ble

.

1.2.

3.

DE

��

is t

he

perp

endi

cula

r �

CD

Fis

equ

ilat

eral

.D

F��

�bi

sect

s �

CD

E.

bise

ctor

of

A �C�

.x

�10

;y

�2

x�

7.5

x�

7

4.F

or w

hat

kin

ds o

f tr

ian

gle(

s) c

an t

he

perp

endi

cula

r bi

sect

or o

f a

side

als

o be

an

an

gle

bise

ctor

of

the

angl

e op

posi

te t

he

side

?is

osc

eles

tri

ang

le,e

qu

ilate

ral t

rian

gle

5.F

or w

hat

kin

d of

tri

angl

e do

th

e pe

rpen

dicu

lar

bise

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s in

ters

ect

in a

poi

nt

outs

ide

the

tria

ngl

e?o

btu

se t

rian

gle

FE

DC

( 4x

� 3

0)�8 x

�D

F CE 10y

� 4

6 x�

3x�

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med

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is a

lin

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t th

at c

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ects

th

e ve

rtex

of

atr

ian

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to t

he

mid

poin

t of

th

e op

posi

te s

ide.

Th

e th

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med

ian

s of

a t

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inte

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th

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ntr

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of t

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he c

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of a

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is lo

cate

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o th

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vert

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on a

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BL

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D

Poi

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are

the

mid

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of

A �B�

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and

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,res

pec

tive

ly.F

ind

x,y

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d z

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CU

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BU

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6x�

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d t

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.

1.x

�4

2.x

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y�

5

B�D�

is a

med

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B�

CB

;D,E

,an

d F

are

mid

poin

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3.x

�3;

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x�

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8.F

or w

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kin

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ide

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z

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L

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)

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ME

____

____

____

____

____

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IOD

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5-1

5-1

Exam

ple

Exam

ple

Exer

cises

Exer

cises

Answers (Lesson 5-1)


An

swer

s

Skil

ls P

ract

ice

Bis

ecto

rs,M

edia

ns,

and

Alt

itu

des

NA

ME

____

____

____

____

____

____

____

____

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AT

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ER

IOD

____

_

5-1

5-1

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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

�D

EF

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Fin

d x

and

RT

if S�

U�is

a m

edia

n o

f �

RS

T.

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9x

�18

;R

T�

120

3.F

ind

xan

d E

Fif

B�D�

is a

n a

ngl

e bi

sect

or.

4.F

ind

xan

d IJ

if H�

K�is

an

alt

itu

de o

f �

HIJ

.

x�

3.5;

EF

�13

x�

29;

IJ�

57

ALG

EBR

AF

or E

xerc

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5–7

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th

e fo

llow

ing

info

rmat

ion

.In

�L

MN

,P,Q

,an

d R

are

the

mid

poin

ts o

f L �

M�,M�

N�,a

nd

L�N�

,re

spec

tive

ly.

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 b

isec

tors

of

�P

QR

and

mee

t 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

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d I

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).F

ind

the

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din

ates

of

the

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nts

of

con

curr

ency

of

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11.o

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ter

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8

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6

x

L

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( 3x

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) �x �

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9

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G EF

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T

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is t

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of �

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C,A�

E�,B�

F�,

and

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are

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ian

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1.F

ind

xif

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and

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.4.

5

2.F

ind

yif

AP

�y

and

EP

�18

.36

3.F

ind

zif

FP

�5z

�10

an

d B

P�

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

? E

xpla

in.

Yes;

sin

ce x

�40

an

d B�

F�is

an

an

gle

bis

ecto

r,it

fo

llow

s th

at m

�B

AF

�70

an

d m

�A

BF

�20

.So

m�

AF

B�

90,a

nd

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 b

isec

tor.

7.F

ind

xif

EH

�16

an

d F

H�

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

rth

ocen

ter

11.c

entr

oid

12.c

ircu

mce

nte

r

��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

nst

ruct

a m

obil

e ou

t of

fla

t tr

ian

gles

so

that

th

e su

rfac

esof

th

e tr

ian

gles

han

g pa

rall

el t

o th

e fl

oor

wh

en t

he

mob

ile

is s

usp

ende

d.H

ow c

anN

abu

ko b

e ce

rtai

n t

hat

sh

e h

angs

th

e tr

ian

gles

to

ach

ieve

th

is e

ffec

t?S

he

nee

ds

to h

ang

eac

h t

rian

gle

fro

m it

s ce

nte

r o

f g

ravi

ty o

r ce

ntr

oid

,w

hic

h is

th

e p

oin

t at

wh

ich

th

e th

ree

med

ian

s o

f th

e tr

ian

gle

inte

rsec

t.

DI

HF

G

E

SR

P TX

AC F

E DP

B

Pra

ctic

e (

Ave

rag

e)

Bis

ecto

rs,M

edia

ns,

and

Alt

itu

des

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-1

5-1



Readin

g t

o L

earn

Math

em

ati

csB

isec

tors

,Med

ian

s,an

d A

ltit

ud

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

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

uct

ion

to

Les

son

5-1

at

the

top

of p

age

238

in y

our

text

book

.

Dra

w a

ny

tria

ngl

e an

d co

nn

ect

each

ver

tex

to t

he

mid

poin

t of

th

e op

posi

tesi

de t

o fo

rm t

he

thre

e m

edia

ns

of t

he

tria

ngl

e.Is

th

e po

int

wh

ere

the

thre

em

edia

ns

inte

rsec

t th

e m

idpo

int

of e

ach

of

the

med

ian

s?S

amp

le a

nsw

er:

No

;th

e in

ters

ecti

on

po

int

app

ears

to

be

mo

re t

han

hal

fway

fro

m e

ach

ver

tex

to t

he

mid

po

int

of

the

op

po

site

sid

e.

Rea

din

g t

he

Less

on

1.U

nde

rlin

e th

e co

rrec

t w

ord

or p

hra

se t

o co

mpl

ete

each

sen

ten

ce.

a.T

hre

e or

mor

e li

nes

th

at i

nte

rsec

t at

a c

omm

on p

oin

t ar

e ca

lled

(par

alle

l/per

pen

dicu

lar/

con

curr

ent)

lin

es.

b.

An

y po

int

on t

he

perp

endi

cula

r bi

sect

or o

f a

segm

ent

is

(par

alle

l to

/con

gru

ent

to/e

quid

ista

nt

from

) th

e en

dpoi

nts

of

the

segm

ent.

c.A

(n)

(alt

itu

de/a

ngl

e bi

sect

or/m

edia

n/p

erpe

ndi

cula

r bi

sect

or)

of a

tri

angl

e is

a

segm

ent

draw

n f

rom

a v

erte

x of

th

e tr

ian

gle

perp

endi

cula

r to

th

e li

ne

con

tain

ing

the

oppo

site

sid

e.

d.

The

poi

nt o

f co

ncur

renc

y of

the

thr

ee p

erpe

ndic

ular

bis

ecto

rs o

f a

tria

ngle

is

call

ed t

he(o

rth

ocen

ter/

circ

um

cen

ter/

cen

troi

d/in

cen

ter)

.

e.A

ny

poin

t in

th

e in

teri

or o

f an

an

gle

that

is

equ

idis

tan

t fr

om t

he

side

s of

th

at a

ngl

e li

es o

n t

he

(med

ian

/an

gle

bise

ctor

/alt

itu

de).

f.T

he

poin

t of

con

curr

ency

of

the

thre

e an

gle

bise

ctor

s of

a t

rian

gle

is c

alle

d th

e(o

rth

ocen

ter/

circ

um

cen

ter/

cen

troi

d/in

cen

ter)

.

2.In

th

e fi

gure

,Eis

th

e m

idpo

int

of A �

B�,F

is t

he

mid

poin

t of

B�C�

,an

d G

is t

he

mid

poin

t of

A �C�

.

a.N

ame

the

alti

tude

s of

�A

BC

.A�

C�,B�

C�,C�

D�b

.N

ame

the

med

ian

s of

�A

BC

.A�

F�,B�

G�,C�

E�c.

Nam

e th

e ce

ntr

oid

of �

AB

C.

Hd

.N

ame

the

orth

ocen

ter

of �

AB

C.

Ce.

If A

F�

12 a

nd

CE

�9,

fin

d 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

eth

ing

is t

o ex

plai

n i

t to

som

eon

e el

se.S

upp

ose

that

acl

assm

ate

is h

avin

g tr

oubl

e re

mem

beri

ng

wh

eth

er t

he

cen

ter

of g

ravi

ty o

f a

tria

ngl

e is

the

orth

ocen

ter,

the

cen

troi

d,th

e in

cen

ter,

or t

he

circ

um

cen

ter

of t

he

tria

ngl

e.S

ugg

est

aw

ay t

o re

mem

ber

wh

ich

poi

nt

it i

s.S

amp

le a

nsw

er:T

he

term

s ce

ntr

oid

and

cen

ter

of

gra

vity

mea

n t

he

sam

e th

ing

an

d in

bo

th t

erm

s,th

e le

tter

s“c

ent”

com

e at

th

e b

egin

nin

g o

f th

e te

rms.

AB

C

FG

ED

H

©G

lenc

oe/M

cGra

w-H

ill25

0G

lenc

oe G

eom

etry

Insc

rib

ed a

nd

Cir

cum

scri

bed

Cir

cles

Th

e th

ree

angl

e bi

sect

ors

of a

tri

angl

e in

ters

ect

in a

sin

gle

poin

t ca

lled

th

e in

cen

ter.

Th

ispo

int

is t

he

cen

ter

of a

cir

cle

that

just

tou

ches

th

e th

ree

side

s of

th

e tr

ian

gle.

Exc

ept

for

the

thre

e po

ints

wh

ere

the

circ

le t

ouch

es t

he

side

s,th

e ci

rcle

is

insi

de t

he

tria

ngl

e.T

he

circ

le i

ssa

id t

o be

in

scri

bed

in t

he

tria

ngl

e.

1.W

ith

a c

ompa

ss a

nd

a st

raig

hte

dge,

con

stru

ct t

he

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

bise

ctor

s of

�P

and

�Q

.Lab

el t

he

poin

t

wh

ere

the

bise

ctor

s m

eet

A.

Ste

p 2

Con

stru

ct a

per

pen

dicu

lar

segm

ent

from

Ato

R �Q�

.Use

th

e le

tter

Bto

lab

el t

he

poin

t w

her

e th

e pe

rpen

dicu

lar

segm

ent

inte

rsec

ts R �

Q�.

Ste

p 3

Use

a c

ompa

ss t

o dr

aw t

he

circ

le w

ith

cen

ter

at A

and

radi

us

A �B�

.

Con

stru

ct t

he

insc

rib

ed c

ircl

e in

eac

h t

rian

gle.

2.3.

Th

e th

ree

perp

endi

cula

r bi

sect

ors

of t

he

side

s of

a t

rian

gle

also

mee

t in

a s

ingl

e po

int.

Th

ispo

int

is t

he

cen

ter

of t

he

circ

um

scri

bed

circ

le,w

hic

h p

asse

s th

rou

gh e

ach

ver

tex

of t

he

tria

ngl

e.E

xcep

t fo

r th

e th

ree

poin

ts w

her

e th

e ci

rcle

tou

ches

th

e tr

ian

gle,

the

circ

le i

sou

tsid

e th

e tr

ian

gle.

4.F

ollo

w t

he

step

s be

low

to

con

stru

ct t

he

circ

um

scri

bed

circ

le

for

�F

GH

.S

tep

1C

onst

ruct

th

e pe

rpen

dicu

lar

bise

ctor

s of

F �G�

and

F�H�

.U

se t

he

lett

er A

to l

abel

th

e po

int

wh

ere

the

perp

endi

cula

r bi

sect

ors

mee

t.S

tep

2D

raw

th

e ci

rcle

th

at h

as c

ente

r A

and

radi

us

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

AT

E__

____

____

__P

ER

IOD

____

_

5-1

5-1



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Ineq

ual

itie

s an

d T

rian

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

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

in

equ

alit

ies,

incl

udi

ng

the

Tra

nsi

tive

,Add

itio

n,

Su

btra

ctio

n,M

ult

ipli

cati

on,a

nd

Div

isio

n P

rope

rtie

s of

In

equ

alit

y,ca

n b

e u

sed

wit

hm

easu

res

of a

ngl

es a

nd

segm

ents

.Th

ere

is a

lso

a C

ompa

riso

n P

rope

rty

of I

neq

ual

ity.

For

an

y re

al n

um

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 i

nequ

alit

y in

volv

ing

an e

xter

ior

angl

e.

If an

ang

le is

an

exte

rior

angl

e of

aE

xter

ior

An

gle

tria

ngle

, th

en it

s m

easu

re is

gre

ater

tha

n In

equ

alit

y T

heo

rem

the

mea

sure

of

eith

er o

f its

cor

resp

ondi

ng

rem

ote

inte

rior

angl

es.

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

mea

sure

of

an e

xter

ior

angl

e is

gre

ater

th

an t

he

mea

sure

of

eith

er r

emot

e in

teri

or a

ngl

e.S

o m

�3

�m

�1

and

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 an

gles

wh

ose

mea

sure

s ar

e le

ss t

han

m�

1�

3,�

4

2.al

l an

gles

wh

ose

mea

sure

s ar

e gr

eate

r th

an m

�3

�1,

�5

3.al

l an

gles

wh

ose

mea

sure

s ar

e le

ss t

han

m�

1�

5,�

6

4.al

l an

gles

wh

ose

mea

sure

s ar

e gr

eate

r th

an m

�1

�7

5.al

l an

gles

wh

ose

mea

sure

s ar

e le

ss t

han

m�

7�

1,�

3,�

5,�

6,�

TU

V

6.al

l an

gles

wh

ose

mea

sure

s ar

e gr

eate

r th

an m

�2

�4

7.al

l an

gles

wh

ose

mea

sure

s ar

e gr

eate

r th

an m

�5

�1,

�7,

�T

UV

8.al

l an

gles

wh

ose

mea

sure

s ar

e le

ss t

han

m�

4�

2,�

3

9.al

l an

gles

who

se m

easu

res

are

less

tha

n m

�1

�4,

�5,

�7,

�N

PR

10.a

ll a

ngl

es w

hos

e m

easu

res

are

grea

ter

than

m�

4�

1,�

8,�

OP

N,�

RO

Q

RO

QN

P3

456

Exer

cise

s 9–

10

78

21

S

XT

WV

3 4

5

67

21

U

Exer

cise

s 3–

8

MJ

K

3

45

21L Ex

erci

ses

1–2

HE

F3

4

21

G

AC

D1

B

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-H

ill25

2G

lenc

oe G

eom

etry

An

gle

-Sid

e R

elat

ion

ship

sW

hen

th

e si

des

of t

rian

gles

are

n

ot c

ongr

uen

t,th

ere

is a

rel

atio

nsh

ip b

etw

een

th

e si

des

and

angl

es o

f th

e tr

ian

gles

.

•If

on

e si

de o

f a

tria

ngl

e is

lon

ger

than

an

oth

er s

ide,

then

th

e an

gle

oppo

site

th

e lo

nge

r si

de h

as a

gre

ater

mea

sure

th

an t

he

If A

C�

AB

, th

en m

�B

�m

�C

.

angl

e op

posi

te t

he

shor

ter

side

.If

m�

A�

m�

C,

then

BC

�A

B.

•If

on

e an

gle

of a

tri

angl

e h

as a

gre

ater

mea

sure

th

an a

not

her

an

gle,

then

th

e si

de o

ppos

ite

the

grea

ter

angl

e is

lon

ger

than

th

e si

de o

ppos

ite

the

less

er a

ngl

e.

BC

A

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Ineq

ual

itie

s an

d T

rian

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-2

5-2

Lis

t th

e an

gles

in

ord

erfr

om l

east

to

grea

test

mea

sure

.

�T

,�R

,�S

RT

9 cm

6 cm

7 cm

S

Lis

t th

e si

des

in

ord

erfr

om s

hor

test

to

lon

gest

.

C �B�

,A�B�

,A�C�

AB

C

20�

35�

125�

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Lis

t th

e an

gles

or

sid

es i

n o

rder

fro

m l

east

to

grea

test

mea

sure

.

1.2.

3.

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5-2

5-2



Stu

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5-3

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ssu

me

that

nis

no

t ev

en.T

hat

is,a

ssu

me

nis

od

d.

4.T

hen

nca

n b

e ex

pres

sed

as 2

a�

1 by

the

mea

nin

g o

f o

dd

nu

mb

er.

5.n

2�

(2a

�1)

2S

ubs

titu

tion

6.�

(2a

�1)

(2a

�1)

Mu

ltip

ly.

7.�

4a2

�4a

�1

Sim

plif

y.

8.�

2(2a

2�

2a)

�1

Dis

trib

uti

ve P

rop

erty

9.2(

2a2

�2a

)�1

is a

n o

dd n

um

ber.

Th

is c

ontr

adic

ts t

he

give

n t

hat

n2

is e

ven

,

so t

he

assu

mpt

ion

mu

st b

e fa

lse.

10.T

her

efor

e,n

is e

ven

.

x3x

�5

18

05

�1

2

�2

�1

�3

�4

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ple

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th

e co

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on i

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en y

ou s

how

th

at t

he

assu

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ds t

o a

con

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e co

ntr

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th

e co

ncl

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e fa

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t m

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is l

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ntr

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is a

rig

ht

angl

e,th

en m

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an

d m

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190.

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us

the

sum

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mea

sure

s of

th

e an

gles

of

�A

BC

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reat

er t

han

180

.

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p 3

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e co

ncl

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hat

th

e su

m o

f th

e m

easu

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of t

he

angl

es o

f �

AB

Cis

gre

ater

th

an 1

80 i

s a

con

trad

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f a

know

n p

rope

rty.

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e as

sum

ptio

n t

hat

�A

is a

rig

ht

angl

e m

ust

be

fals

e,w

hic

h

mea

ns

that

th

e st

atem

ent

“�A

is n

ot a

rig

ht

angl

e”m

ust

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

uen

t to

V�E�

,th

en �

AV

Eis

not

iso

scel

es.

�A

VE

is is

osc

eles

.

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ple

te t

he

pro

of.

Giv

en:�

1 �

�2

and

D�G�

is n

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ongr

uen

t to

F�G�

.P

rove

:D �E�

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ongr

uen

t to

F�E�

.

3.A

ssu

me

that

D�E�

�F�E�

.A

ssu

me

the

con

clu

sion

is

fals

e.

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G��

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Ref

lexi

ve P

rop

erty

5.�

ED

G�

�E

FG

SA

S

6.D�

G��

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PC

TC

7.T

his

con

trad

icts

th

e gi

ven

in

form

atio

n,s

o th

e as

sum

ptio

n m

ust

be fa

lse.

8.T

her

efor

e,D�

E�is

no

t co

ng

ruen

t to

F�E�

.

12

DG

FE

AB

C

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Ind

irec

t P

roo

f

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-3

5-3

Exer

cises

Exer

cises

Exam

ple

Exam

ple



An

swer

s

Skil

ls P

ract

ice

Ind

irec

t P

roo

f

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

E__

____

____

__P

ER

IOD

____

_

5-3

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

pen

dicu

lar

to l

ine

b.L

ine

ais

no

t p

erp

end

icu

lar

to li

ne

b.

4.�

5 is

su

pple

men

tary

to

�6.

�5

is n

ot

sup

ple

men

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

:A

ssu

me

x�

2.S

tep

2:

If x

�2,

then

x2

�4.

Bu

t if

x2

�4,

it f

ollo

ws

that

x2

�8

�12

.T

his

co

ntr

adic

ts t

he

giv

en f

act

that

x2

�8

�12

.S

tep

3:

Sin

ce t

he

assu

mp

tio

n o

f x

�2

lead

s to

a c

on

trad

icti

on

,it

mu

st

be

fals

e.T

her

efo

re, x

�2

mu

st b

e tr

ue.

6.G

iven

:�D

��

F.

Pro

ve:D

E�

EF

Pro

of:

Ste

p 1

:A

ssu

me

DE

�E

F.

Ste

p 2

:If

DE

�E

F,th

en D�

E��

E�F�by

th

e d

efin

itio

n o

f co

ng

ruen

t se

gm

ents

.B

ut

if D�

E��

E�F�,

then

�D

��

Fby

th

e Is

osc

eles

Tri

ang

le T

heo

rem

.T

his

co

ntr

adic

ts t

he

giv

en in

form

atio

n t

hat

�D

��

F.

Ste

p 3

:S

ince

th

e as

sum

pti

on

th

at D

E�

EF

lead

s to

a c

on

trad

icti

on

,it

mu

st b

e fa

lse.

Th

eref

ore

,it

mu

st b

e tr

ue

that

DE

E

F.

DF

E

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Wri

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he

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mp

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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�

do

es n

ot

bis

ect

�A

BC

.

2.R

T�

TS

RT

T

S

PRO

OF

Wri

te a

n i

nd

irec

t p

roof

.

3.G

iven

:�4x

�2

��

10P

rove

:x�

3P

roo

f:S

tep

1:

Ass

um

e x

�3.

Ste

p 2

:If

x�

3,th

en �

4x

�12

.Bu

t �

4x

�12

imp

lies

that

�

4x�

2

�10

,wh

ich

co

ntr

adic

ts t

he

giv

en in

equ

alit

y.S

tep

3:

Sin

ce t

he

assu

mp

tio

n t

hat

x�

3 le

ads

to a

co

ntr

adic

tio

n,

it m

ust

be

tru

e th

at x

�3.

4.G

iven

:m�

2 �

m�

3 �

180

Pro

ve: a

⁄|| bP

roo

f:S

tep

1:

Ass

um

e a

|| b.

Ste

p 2

:If

a|| b

,th

en t

he

con

secu

tive

inte

rio

r an

gle

s �

2 an

d �

3 ar

esu

pp

lem

enta

ry.T

hu

s m

�2

�m

�3

�18

0.T

his

co

ntr

adic

ts t

he

giv

en s

tate

men

t th

at m

�2

�m

�3

18

0.S

tep

3:

Sin

ce t

he

assu

mp

tio

n le

ads

to a

co

ntr

adic

tio

n,t

he

stat

emen

t a

|| bm

ust

be

fals

e.T

her

efo

re,a

⁄|| bm

ust

be

tru

e.

5.PH

YSI

CS

Sou

nd

trav

els

thro

ugh

air

at

abou

t 34

4 m

eter

s pe

r se

con

d w

hen

th

ete

mpe

ratu

re i

s 20

°C.I

f E

nri

que

live

s 2

kilo

met

ers

from

th

e fi

re s

tati

on a

nd

it t

akes

5

seco

nds

for

th

e so

un

d of

th

e fi

re s

tati

on s

iren

to

reac

h h

im,h

ow c

an y

ou p

rove

indi

rect

ly t

hat

it

is n

ot 2

0°C

wh

en E

nri

que

hea

rs t

he

sire

n?

Ass

um

e th

at it

is 2

0°C

wh

en E

nri

qu

e h

ears

th

e si

ren

,th

en s

ho

w t

hat

at

this

tem

per

atu

re it

will

tak

e m

ore

th

an 5

sec

on

ds

for

the

sou

nd

of

the

sire

n t

o r

each

him

.Sin

ce t

he

assu

mp

tio

n is

fal

se,y

ou

will

hav

e p

rove

dth

at it

is n

ot

20°C

wh

en E

nri

qu

e h

ears

th

e si

ren

.

1 23

a b

Pra

ctic

e (

Ave

rag

e)

Ind

irec

t P

roo

f

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-3

5-3



Readin

g t

o L

earn

Math

em

ati

csIn

dir

ect

Pro

of

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-3

5-3

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Lesson 5-3

Pre-

Act

ivit

yH

ow i

s in

dir

ect

pro

of u

sed

in

lit

erat

ure

?

Rea

d th

e in

trod

uct

ion

to

Les

son

5-3

at

the

top

of p

age

255

in y

our

text

book

.

How

cou

ld t

he

auth

or o

f a

mu

rder

mys

tery

use

in

dire

ct r

easo

nin

g to

sh

owth

at a

par

ticu

lar

susp

ect

is n

ot g

uil

ty?

Sam

ple

an

swer

:A

ssu

me

that

the

per

son

is g

uilt

y.T

hen

sh

ow t

hat

th

is a

ssu

mp

tion

co

ntr

adic

tsev

iden

ce t

hat

has

bee

n g

ath

ered

ab

ou

t th

e cr

ime.

Rea

din

g t

he

Less

on

1.S

uppl

y th

e m

issi

ng w

ords

to

com

plet

e th

e lis

t of

ste

ps in

volv

ed in

wri

ting

an

indi

rect

pro

of.

Ste

p 1

Ass

um

e th

at t

he

con

clu

sion

is

.

Ste

p 2

Sh

ow t

hat

th

is a

ssu

mpt

ion

lea

ds t

o a

of t

he

or s

ome

oth

er f

act,

such

as

a de

fin

itio

n,p

ostu

late

,

,or

coro

llar

y.

Ste

p 3

Poi

nt

out

that

th

e as

sum

ptio

n m

ust

be

and,

ther

efor

e,th

e

con

clu

sion

mu

st b

e .

2.S

tate

th

e as

sum

ptio

n t

hat

you

wou

ld m

ake

to s

tart

an

in

dire

ct p

roof

of

each

sta

tem

ent.

a.If

�6x

�30

,th

en x

��

5.x

�

5b

.If

nis

a m

ult

iple

of

6,th

en n

is a

mu

ltip

le o

f 3.

nis

no

t a

mu

ltip

le o

f 3.

c.If

aan

d b

are

both

odd

,th

en a

bis

odd

.ab

is e

ven

.ab

is g

reat

erd

.If

ais

pos

itiv

e an

d b

is n

egat

ive,

then

ab

is n

egat

ive.

than

or

equ

al t

o 0

.e.

If F

is b

etw

een

Ean

d D

,th

en E

F�

FD

�E

D.

EF

�F

D

ED

f.In

a p

lan

e,if

tw

o li

nes

are

per

pen

dicu

lar

to t

he

sam

e li

ne,

then

th

ey a

re p

aral

lel.

Two

lin

es a

re n

ot

par

alle

l.g.

Ref

er t

o th

e fi

gure

.h

.R

efer

to

the

figu

re.

If A

B�

AC

,th

en m

�B

�m

�C

.In

�P

QR

,PR

�Q

R�

QP

.m

�B

m

�C

PR

�Q

R�

QP

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r a

new

con

cept

in m

athe

mat

ics

is t

o re

late

it t

o so

met

hing

you

hav

eal

read

y le

arne

d.H

ow i

s th

e pr

oces

s of

ind

irec

t pr

oof

rela

ted

to t

he r

elat

ions

hip

betw

een

aco

ndi

tion

al s

tate

men

t an

d it

s co

ntr

apos

itiv

e?S

amp

le a

nsw

er:T

he

con

trap

osi

tive

of

the

con

dit

ion

al s

tate

men

t p

→q

is t

he

stat

emen

t �

q→

�p

.In

an

ind

irec

t p

roo

f o

f a

con

dit

ion

al s

tate

men

t p

→q

,yo

u a

ssu

me

that

qis

fals

e an

d s

ho

w t

hat

th

is im

plie

s th

at p

is f

alse

,th

at is

,yo

u s

ho

w t

hat

�

q→

�p

is t

rue.

Bec

ause

a s

tate

men

t is

log

ical

ly e

qu

ival

ent

to it

sco

ntr

apo

siti

ve,p

rovi

ng

th

e co

ntr

apo

siti

ve is

tru

e is

a w

ay o

f p

rovi

ng

th

eo

rig

inal

co

nd

itio

nal

is t

rue.

PRQ

AC

B

tru

efa

lse

theo

rem

hyp

oth

esis

con

trad

icti

on

fals

e

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Mo

re C

ou

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rexa

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les

Som

e st

atem

ents

in

mat

hem

atic

s ca

n b

e pr

oven

fal

se b

y co

un

tere

xam

ple

s.C

onsi

der

the

foll

owin

g st

atem

ent.

For

an

y n

um

bers

aan

d b,

a�

b�

b�

a.

You

can

pro

ve t

hat

th

is s

tate

men

t is

fal

se i

n g

ener

al i

f yo

u c

an f

ind

one

exam

ple

for

wh

ich

th

e st

atem

ent

is f

alse

.

Let

a�

7 an

d b

�3.

Su

bsti

tute

th

ese

valu

es i

n t

he

equ

atio

n a

bove

.

7 �

3 �

3 �

74

� �

4

In g

ener

al,f

or a

ny

nu

mbe

rs a

and

b,th

e st

atem

ent

a�

b�

b�

ais

fal

se.

You

can

mak

e th

e eq

uiv

alen

t ve

rbal

sta

tem

ent:

subt

ract

ion

is

not

aco

mm

uta

tive

ope

rati

on.

In e

ach

of

the

foll

owin

g ex

erci

ses

a,b

,an

d c

are

any

nu

mb

ers.

Pro

ve t

hat

th

e st

atem

ent

is f

alse

by

cou

nte

rexa

mp

le.

Sam

ple

an

swer

s ar

e g

iven

.

1.a

�(b

�c)

� (

a�

b) �

c2.

a

(b

c) �

(a

b)

c

6 �

(4 �

2) �

(6 �

4) �

26

�(4

�2)

� (

6 �

4) �

26

�2

� 2

�2

�6 2��

�1 2.5 �4

0

3

0.7

5

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�

a4

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

th

e ve

rbal

equ

ival

ents

for

Exe

rcis

es 1

,2,a

nd

3.

1.S

ub

trac

tio

n is

no

t an

ass

oci

ativ

e o

per

atio

n.

2.D

ivis

ion

is n

ot

an a

sso

ciat

ive

op

erat

ion

.3.

Div

isio

n is

no

t a

com

mu

tati

ve o

per

atio

n.

8.F

or t

he

Dis

trib

uti

ve P

rope

rty

a(b

�c)

�ab

�ac

it i

s sa

id t

hat

mu

ltip

lica

tion

dist

ribu

tes

over

add

itio

n.E

xerc

ises

4 a

nd

5 pr

ove

that

som

e op

erat

ion

s do

not

dist

ribu

te.W

rite

a s

tate

men

t fo

r ea

ch e

xerc

ise

that

in

dica

tes

this

.

4.D

ivis

ion

do

es n

ot

dis

trib

ute

ove

r ad

dit

ion

.5.

Ad

dit

ion

do

es n

ot

dis

trib

ute

ove

r m

ult

iplic

atio

n.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-3

5-3



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Th

e Tr

ian

gle

Ineq

ual

ity

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

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 l

engt

hs

8 in

ches

,5 i

nch

es,a

nd

1 in

ch a

nd

try

to m

ake

a tr

ian

gle

wit

h t

hem

,you

wil

l fi

nd

that

it

is n

ot p

ossi

ble.

Th

isil

lust

rate

s th

e T

rian

gle

Ineq

ual

ity

Th

eore

m.

Tria

ng

le In

equ

alit

yT

he s

um o

f th

e le

ngth

s of

any

tw

o si

des

of a

Th

eore

mtr

iang

le is

gre

ater

tha

n th

e le

ngth

of

the

third

sid

e.

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

equ

alit

y,al

l th

ree

of t

he

foll

owin

g in

equ

alit

ies

mu

st b

e tr

ue.

5 �

x�

88

�x

�5

5 �

8 �

xx

�3

x�

�3

13 �

x

Th

eref

ore

xm

ust

be

betw

een

3 a

nd

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

no

6.1.

5,2.

5,3

yes

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

18

5 �

n�

76

�n

�30

9.1.

5 an

d 5.

5 10

.82

and

8

4 �

n�

774

�n

�90

11.S

upp

ose

you

hav

e th

ree

diff

eren

t po

siti

ve n

um

bers

arr

ange

d in

ord

er f

rom

lea

st t

ogr

eate

st.W

hat

sin

gle

com

pari

son

wil

l le

t yo

u s

ee i

f th

e n

um

bers

can

be

the

len

gth

s of

the

side

s of

a t

rian

gle?

Fin

d t

he

sum

of

the

two

sm

alle

r n

um

ber

s.If

th

at s

um

is g

reat

er t

han

th

ela

rges

t n

um

ber

,th

en t

he

thre

e n

um

ber

s ca

n b

e th

e le

ng

ths

of

the

sid

eso

f a

tria

ng

le.

BC

A

a

cb

Exer

cises

Exer

cises

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 G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Th

e Tr

ian

gle

Ineq

ual

ity

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-4

5-4

Th

e pe

rpen

dicu

lar

segm

ent

from

a p

oin

t to

a li

ne

is t

he

shor

test

seg

men

t fr

om t

he

poin

t to

th

e li

ne.

P�C�

is t

he s

hort

est

segm

ent

from

Pto

AB

�� .

Th

e pe

rpen

dicu

lar

segm

ent

from

a p

oin

t to

a pl

ane

is t

he

shor

test

seg

men

t fr

om t

he

poin

t to

th

e pl

ane.

Q�T�

is t

he s

hort

est

segm

ent

from

Qto

pla

ne N

.

Q TN

B

P CA G

iven

:Poi

nt

Pis

eq

uid

ista

nt

from

th

e si

des

of

an

an

gle.

Pro

ve:B �

A��

C�A�

Pro

of:

1.D

raw

B �P�

and

C�P�

⊥to

1.

Dis

t.is

mea

sure

d th

e si

des

of �

RA

S.

alon

g a

⊥.

2.�

PB

Aan

d �

PC

Aar

e ri

ght

angl

es.

2.D

ef.o

f ⊥

lin

es3.

�A

BP

and

�A

CP

are

righ

t tr

ian

gles

.3.

Def

.of

rt.�

4.�

PB

A�

�P

CA

4.R

t.an

gles

are

�.

5.P

is e

quid

ista

nt

from

th

e si

des

of �

RA

S.

5.G

iven

6.B �

P��

C�P�

6.D

ef.o

f eq

uid

ista

nt

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

CT

C3.

�A

CB

��

RT

S3.

CP

CT

C4.

�A

CB

and

�A

CD

are

a li

nea

r pa

ir;

4.D

ef.o

f lin

ear

pai

r�

RT

San

d �

RT

Uar

e a

lin

ear

pair

.

5.�

AC

Ban

d �

AC

Dar

e su

pple

men

tary

;5.

Lin

ear

pai

rs a

re s

up

pl.

�R

TS

and

�R

TU

are

supp

lem

enta

ry.

6.�

AC

D�

�R

TU

6.A

ngl

es s

upp

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

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Th

e Tr

ian

gle

Ineq

ual

ity

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

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,

26n

o

5.9,

10,2

0n

o6.

15,1

7,19

yes

7.14

,17,

31n

o8.

6,7,

12ye

s

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

14

4 �

n�

147

�n

�21

11.8

an

d 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�

,an

d

No

;S

T�

2�2�,

TU

�4�

2�,an

d

AC

��

17�,s

o A

B�

BC

�A

C,

SU

�6�

2�,so

ST

�T

U�

SU

.A

B�

AC

�B

C,a

nd

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

;H

I�2�

5�,IJ

�4�

5�,an

d

Yes;

DE

��

41�,E

F�

2,an

d

HJ

�6�

5�,so

HI�

IJ�

HJ.

DF

��

29�,s

o D

E�

EF

�D

F,

DE

�D

F�

EF

,an

d D

F�

EF

�D

E.

©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,

17n

o

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.

1n

o8.

12.3

,13.

9,25

.2ye

s

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

2913

�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

;R

S�

3�2�,

ST

�6�

2�,an

d

Yes;

WX

�1,

XY

�5,

and

R

T�

9�2�,

so R

S�

ST

�R

T.

WY

�2�

5�,so

WX

�X

Y�

WY

,W

X�

WY

�X

Y,a

nd

W

Y�

XY

�W

X.

19.P

(�3,

2),L

(1,1

),M

(9,�

1)20

.B(1

,1),

C(6

,5),

D(4

,�1)

No

;P

L�

�17�

,LM

�2

�17�

,an

d

Yes;

BC

��

41�,C

D�

2�10�

,an

dP

M�

3 �

17�,s

o P

L�

LM

�P

M.

BD

��

13�,s

o B

C�

CD

�B

D,

BC

�B

D�

CD

,an

d B

D�

CD

�B

C.

21.G

AR

DEN

ING

Ha

Poo

ng h

as 4

leng

ths

of w

ood

from

whi

ch h

e pl

ans

to m

ake

a bo

rder

for

atr

ian

gula

r-sh

aped

her

b ga

rden

.Th

e le

ngt

hs

of t

he

woo

d bo

rder

s ar

e 8

inch

es,1

0 in

ches

,12

in

ches

,an

d 18

in

ches

.How

man

y di

ffer

ent

tria

ngu

lar

bord

ers

can

Ha

Poo

ng

mak

e?3

Pra

ctic

e (

Ave

rag

e)

Th

e Tr

ian

gle

Ineq

ual

ity

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-4

5-4



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csT

he

Tria

ng

le In

equ

alit

y

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

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

uct

ion

to

Les

son

5-4

at

the

top

of p

age

261

in y

our

text

book

.

In a

ddit

ion

to

the

grea

ter

dist

ance

in

volv

ed i

n f

lyin

g fr

om C

hic

ago

toC

olu

mbu

s th

rou

gh I

ndi

anap

olis

rat

her

th

an f

lyin

g n

onst

op,w

hat

are

tw

oot

her

rea

son

s th

at i

t w

ould

tak

e lo

nge

r to

get

to

Col

um

bus

if y

ou t

ake

two

flig

hts

rat

her

th

an o

ne?

Sam

ple

an

swer

:ti

me

nee

ded

fo

r an

ext

rata

keo

ff a

nd

lan

din

g;

layo

ver

tim

e in

Ind

ian

apo

lis b

etw

een

th

etw

o f

ligh

ts

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.

Wh

ich

sta

tem

ents

are

tru

e?C

,D,F

A.

DE

�E

F�

FD

B.D

E�

EF

�F

D

C.

EG

�E

F�

FG

D.E

D�

DG

�E

G

E.

Th

e sh

orte

st d

ista

nce

fro

m D

to E

G��

� is

DF

.

F.T

he

shor

test

dis

tan

ce f

rom

Dto

EG

� �� i

s D

G.

2.C

ompl

ete

each

sen

ten

ce a

bou

t �

XY

Z.

a.If

XY

�8

and

YZ

�11

,th

en t

he

ran

ge o

f va

lues

for

XZ

is

�X

Z�

.

b.

If X

Y�

13 a

nd

XZ

�25

,th

en Y

Zm

ust

be

betw

een

an

d .

c.If

�X

YZ

is i

sosc

eles

wit

h �

Zas

th

e ve

rtex

an

gle,

and

XZ

�8.

5,th

en t

he

ran

ge o

f

valu

es f

or X

Yis

�

XY

�.

d.

If X

Z�

aan

d Y

Z�

b,w

ith

b�

a,th

en t

he r

ange

for

XY

is

�X

Y�

.

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

a n

ew t

heo

rem

is

to s

tate

it

info

rmal

ly i

n d

iffe

ren

t w

ords

.How

cou

ld y

ou r

esta

te t

he

Tri

angl

e In

equ

alit

y T

heo

rem

?

Sam

ple

an

swer

:Th

e si

de

that

co

nn

ects

on

e ve

rtex

of

a tr

ian

gle

to

ano

ther

is a

sh

ort

er p

ath

bet

wee

n t

he

two

ver

tice

s th

an t

he

pat

h t

hat

go

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Readin

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bein

g de

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in a

pro

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.Th

e vi

sual

izat

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of

the

prob

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pfu

l in

th

e pr

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s of

pro

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sol

vin

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e ro

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wn

s of

Kin

gs,

Ch

ana,

and

Hol

com

b f

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rian

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Dav

is J

un

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n i

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cate

d i

n t

he

inte

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of

this

tri

angl

e.T

he

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from

Dav

is J

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o K

ings

,Ch

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and

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com

b a

re 3

km

,4

km

,an

d 5

km

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pec

tive

ly.J

ane

beg

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at H

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mb

an

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tly

to C

han

a,th

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o K

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d t

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bac

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.At

the

end

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tri

p,s

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figu

res

she

has

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d25

km

alt

oget

her

.Has

sh

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gure

d t

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dis

tan

ce c

orre

ctly

?

To

solv

e th

is p

robl

em,a

dia

gram

can

be

draw

n.B

ased

on

th

is d

iagr

am a

nd

the

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angl

e In

equ

alit

y T

heo

rem

,th

e di

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ce f

rom

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com

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ess

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9 k

m.S

imil

arly

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is

less

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an 7

km

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than

8 k

m.

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eref

ore,

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e m

ust

hav

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ed l

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than

(9

�7

�8)

km

or

24

km v

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er c

alcu

late

d di

stan

ce o

f 25

km

.

Exp

lain

wh

y ea

ch o

f th

e fo

llow

ing

stat

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ts i

s tr

ue.

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w a

nd

lab

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dia

gram

to

be

use

d i

n t

he

exp

lan

atio

n.

1.If

an

alt

itu

de i

s dr

awn

to

one

side

of

a tr

ian

gle,

then

th

ele

ngt

h o

f th

e al

titu

de i

s le

ss t

han

on

e-h

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sum

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the

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des.

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at B�

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and

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m 6

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BC

an

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ng

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m 6

-2,2

BD

�B

A�

BC

.Th

us,

BD

��1 2�

(BA

�B

C).

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poi

nt

Qis

in

th

e in

teri

or o

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AB

Can

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e an

gle

bise

ctor

of �

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om A�

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t:D

raw

Q�D�

and

Q�E�

such

th

at Q�

D��

A�B�

and

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�C�

B�.)

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is o

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bis

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f �

B,Q�

D��

A�B�

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hen

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CT

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AD

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n

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3 km

5 km

4 km

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t

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ME

____

____

____

____

____

____

____

____

____

____

____

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AT

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____

____

__P

ER

IOD

____

_

5-5

5-5

Exam

ple

Exam

ple



Chapter 5 Assessment Answer Key Form 1 Form 2APage 275 Page 276 Page 277


An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

A

C

D

B

C

B

A

C

B

C

D

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

A

B

B

C

B

A

D

D

A

6, �1

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

B

D

A

C

B

A

C

A

A

D

A


Chapter 5 Assessment Answer KeyForm 2A (continued) Form 2BPage 278 Page 279 Page 280

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

B

A

D

B

C

D

C

B

C

9, �2

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

C

D

B

A

A

C

D

B

C

A

C

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

D

A

B

A

D

C

B

A

C

160


Chapter 5 Assessment Answer KeyForm 2CPage 281 Page 282

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

AD��

x � 8; AC�� is the ⊥bisector of B�D�.

4

��136�, �

232��

25

135 � m�A � 0

�I, �H, �G

P�Q�, P�R�, Q�R�

X�Y�

4 is not a factor of n.

A�B� is not a median.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

�X � �Z

13 m and 33 m

PT

34

EF � GH

m�1 � m�2

Definition of �segments

Reflexive Prop.

SSS Inequality

y � �c �

ba

�x


Chapter 5 Assessment Answer KeyForm 2DPage 283 Page 284

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

LM��

x � 5; RS�� is the ⊥bisector of P�Q�.

8

��92

�, �32

��

20

127 � m�X � 0

�T, �V, �U

F�H�, G�H�, G�F�

L�M�

n2 is not an evennumber.

A�D� is not analtitude.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

S�V� ⊥ P�Q�

15 ft and 43 ft

BE

35

m�1 � m�2

BC � ED

Def. of midpoint

Reflexive Prop.

SAS Inequality

x � �12

�a


Chapter 5 Assessment Answer KeyForm 3Page 285 Page 286

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

15

��3183�, ��

3123��

32

146 � m�L � 0

�H, �I, �G

Q�R�, P�Q�, P�R�

shortest: V�Y�;longest: V�W�

x � 3

x � y � 7 and4y � 3x � 14 so x � 2, y � 5, and

TC � TA � 22. So,T lies on BD��.

The � bisectors arenot concurrent.

x � 3

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

140

�B � �E

5 in. and 53 in.

YW

x � 5

Def. of � segments

Addition Prop. ofInequality

Reflexive Prop.

SSS Inequality

y � �c �

d2a

�x � �c

2�ad

2a�


Chapter 5 Assessment Answer KeyPage 287, Open-Ended Assessment

Scoring Rubric

Score General Description Specific Criteria

• Shows thorough understanding of the concepts ofbisectors, medians, altitudes, inequalities in triangles,indirect proof, the Triangle Inequality, SAS Inequality, andSSS Inequality.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Figures are accurate and appropriate.• Goes beyond requirements of some or all problems.

• Shows an understanding of the concepts of bisectors,medians, altitudes, inequalities in triangles, indirect proof,the Triangle Inequality, SAS Inequality, and SSSInequality.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Figures are mostly accurate and appropriate.• Satisfies all requirements of problems.

• Shows an understanding of most of the concepts ofbisectors, medians, altitudes, inequalities in triangles,indirect proof, the Triangle Inequality, SAS Inequality, andSSS Inequality.

• 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 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 bisectors, medians, altitudes, inequalities in triangles,indirect proof, the Triangle Inequality, SAS Inequality, andSSS Inequality.

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Figures are inaccurate or inappropriate.• Does not satisfy requirements of problems.• No answer 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


An

swer

s

Chapter 5 Assessment Answer KeyPage 287, Open-Ended Assessment

Sample Answers

1. As the sticks are pulled apart the anglegets greater and the rubber band will bestretched and become longer. This situationillustrates the SAS Inequality Theorem.

2. Ashley is correct, FG�� and JK�� are 4 inchesapart. The shortest distance from a pointto a line is the perpendicular distance.Since E�H� is perpendicular to both lines, itsmeasure is the shortest distance fromFG�� to JK��.

3. The segment from B to AC�� could intersectAC�� in two different points because thelength of the segment, 6, is more than the perpendicular distance from B to AC��, 5,and less than the length of A�B�, 10. B�D� caneither slant in towards A or out towardsC as shown in this figure.

4. a. The student should draw a righttriangle.

b. The student should draw an obtusetriangle.

c. The student should draw an acutetriangle.

d. The student should draw anequilateral triangle.

5. An altitude of �ABC extends from avertex and is perpendicular to the oppositeside of the triangle as shown in figure I.A perpendicular bisector of a side isperpendicular to the side but it alsointersects the midpoint of the side andbut does not necessarily intersect theopposite vertex of the triangle as shownin figure II.

6. The SAS Inequality Theorem requires twotriangles that have two pairs of congruentsides and the included angles are related.Then the third side of each triangle willalso be related in the same way as theincluded angles. See �ABC and �EDF.Since �C � �F, then AB � ED.

The theorem that states that the largerangle is opposite the longer side refers tosides and angles within one triangle. Forexample, in �XYZ, since 6 is the longestside, �X will have the greatest measure.

7. If x � 3, then x � 5; x � 5.

Z4

63

X

Y

C 40� B

A

F 50� D

E

A C

altitude

Figure I

B

A C

perpendicularbisector

Figure II

B

altitudes

1066 5

A

B

CD

In addition to the scoring rubric found on page A22, the following sample answers may be used as guidance in evaluating open-ended assessment items.


Chapter 5 Assessment Answer KeyVocabulary Test/Review Quiz 1 Quiz 3Page 288 Page 289 Page 290

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

false, median

false, orthocenter

true

false, centroid

true

false, incenter

true

true

false, 3

true

three or morelines intersecting

at a common point

1.

2.

3.

4.

5.

circumcenter

true

centroid

x � �1

P�Q�

1.

2.

3.

4.

5.

The conclusionis false.

x � 6

A�B� � B�C�

Assume that x 10.That is, assume that

x 10.

C�B� � C�A�

Quiz 2Page 289

Quiz 4Page 290

1.

2.

3.

4.

5.

AC � AB � AD

Yes; AB � AC � BC,BC � AC � AB,

and AB � BC � AC.

AE

2 � x � 16

C

1.

2.

3.

4.

5.

m�1 � m�2

AB � DE

GH � 7

A�E� � A�E�

SAS Inequality


Chapter 5 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 291 Page 292

An

swer

s

Part I

Part II

6.

7.

8.

9.

180 � x � 50

x2 4

m�C m�A

B�D� bisects �ABC.

1.

2.

3.

4.

5.

C

A

D

B

D

1.

2.

3.

4.

5. and 6.

7.

8.

9.

10.

11.

12.

13.

14.

3; 4

24 units

m�ABC �m�1 � m�2

Angle AdditionPostulate

�5�

scalene103

�DBE and �FEC(�1, 3)

�PMQ � �PQM

yes

6 � n � 12

(2, �1)

(�2, �5)

x

y

O


Chapter 5 Assessment Answer KeyStandardized Test Practice

Page 293 Page 294

1.

2.

3.

4.

5.

6.

7.

8. E F G H

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 D9. 10.

11. 12.

13.

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

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

14.

15.

6

10

4 1 7 2 0 0 5

1 7

3 4

8

chapter 5 resource masters - math problem solving©glencoe/mcgraw-hill iv glencoe geometry...

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