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
Lesson 5-2Study Guide and Intervention . . . . . . . . 251–252Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 253Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 254Reading to Learn Mathematics . . . . . . . . . . 255Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 256
Lesson 5-3Study Guide and Intervention . . . . . . . . 257–258Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 259Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 260Reading to Learn Mathematics . . . . . . . . . . 261Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 262
Lesson 5-4Study Guide and Intervention . . . . . . . . 263–264Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 265Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 266Reading to Learn Mathematics . . . . . . . . . . 267Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 268
Lesson 5-5Study Guide and Intervention . . . . . . . . 269–270Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 271Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 272Reading to Learn Mathematics . . . . . . . . . . 273Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 274
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
(continued on the next page)
© Glencoe/McGraw-Hill x Glencoe Geometry
Theorem or Postulate Found on Page Description/Illustration/Abbreviation
Theorem 5.8Exterior Angle Inequality Theorem
Theorem 5.9
Theorem 5.10
Theorem 5.11Triangle Inequality Theorem
Theorem 5.12
Theorem 5.13SAS Inequality/Hinge Theorem
Theorem 5.14SSS Inequality
Learning to Read MathematicsProof Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
55
Study Guide and InterventionBisectors, Medians, and Altitudes
NAME ______________________________________________ DATE ____________ PERIOD _____
5-15-1
© Glencoe/McGraw-Hill 245 Glencoe Geometry
Less
on
5-1
Perpendicular Bisectors and Angle Bisectors A perpendicular bisector of aside of a triangle is a line, segment, or ray that is perpendicular to the side and passesthrough its midpoint. Another special segment, ray, or line is an angle bisector, whichdivides an angle into two congruent angles.
Two properties of perpendicular bisectors are:(1) a point is on the perpendicular bisector of a segment if and only if it is equidistant from
the endpoints of the segment, and(2) the three perpendicular bisectors of the sides of a triangle meet at a point, called the
circumcenter of the triangle, that is equidistant from the three vertices of the triangle.
Two properties of angle bisectors are:(1) a point is on the angle bisector of an angle if and only if it is equidistant from the sides
of the angle, and(2) the three angle bisectors of a triangle meet at a point, called the incenter of the
triangle, that is equidistant from the three sides of the triangle.
BD��� is the perpendicularbisector of A�C�. Find x.
BD��� is the perpendicular bisector of A�C�, soAD � DC.3x � 8 � 5x � 6
14 � 2x7 � x
3x � 8
5x � 6B
C
D
A
MR��� is the angle bisectorof �NMP. Find x if m�1 � 5x � 8 andm�2 � 8x � 16.
MR��� is the angle bisector of �NMP, so m�1 � m�2.5x � 8 � 8x � 16
24 � 3x8 � x
12
N R
PM
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find the value of each variable.
1. 2. 3.
DE��� is the perpendicular �CDF is equilateral. DF��� bisects �CDE.bisector of A�C�.
4. For what kinds of triangle(s) can the perpendicular bisector of a side also be an anglebisector of the angle opposite the side?
5. For what kind of triangle do the perpendicular bisectors intersect in a point outside thetriangle?
FE
DC(4x � 30)�
8x �D
F
C
E10y � 46x �
3x �
8y
CE
DA
B
7x � 96x � 2
© Glencoe/McGraw-Hill 246 Glencoe Geometry
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
© Glencoe/McGraw-Hill 247 Glencoe Geometry
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
© Glencoe/McGraw-Hill 248 Glencoe Geometry
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
© Glencoe/McGraw-Hill 249 Glencoe Geometry
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
© Glencoe/McGraw-Hill 250 Glencoe Geometry
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
© Glencoe/McGraw-Hill 251 Glencoe Geometry
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
3. all angles whose measures are less than m�1
4. all angles whose measures are greater than m�1
5. all angles whose measures are less than m�7
6. all angles whose measures are greater than m�2
7. all angles whose measures are greater than m�5
8. all angles whose measures are less than m�4
9. all angles whose measures are less than m�1
10. all angles whose measures are greater than m�4
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
© Glencoe/McGraw-Hill 252 Glencoe Geometry
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
Study Guide and Intervention (continued)
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
© Glencoe/McGraw-Hill 253 Glencoe Geometry
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.
5. all angles whose measures are less than m�1
6. all angles whose measures are less than m�9
7. all angles whose measures are greater than m�5
8. all angles whose measures are greater than m�8
Determine the relationship between the measures of the given angles.
9. m�ABD, m�BAD 10. m�ADB, m�BAD
11. m�BCD, m�CDB 12. m�CBD, m�CDB
Determine the relationship between the lengths of the given sides.
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
© Glencoe/McGraw-Hill 254 Glencoe Geometry
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.
5. all angles whose measures are less than m�1
6. all angles whose measures are less than m�3
7. all angles whose measures are greater than m�7
8. all angles whose measures are greater than m�2
Determine the relationship between the measures of the given angles.
9. m�QRW, m�RWQ 10. m�RTW, m�TWR
11. m�RST, m�TRS 12. m�WQR, m�QRW
Determine the relationship between the lengths of the given sides.
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
© Glencoe/McGraw-Hill 255 Glencoe Geometry
Less
on
5-2
Pre-Activity How can you tell which corner is bigger?
Read the introduction to Lesson 5-2 at the top of page 247 in your textbook.
• 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
© Glencoe/McGraw-Hill 256 Glencoe Geometry
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
NAME ______________________________________________ DATE ____________ PERIOD _____
5-25-2
Study Guide and InterventionIndirect Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
5-35-3
© Glencoe/McGraw-Hill 257 Glencoe Geometry
Less
on
5-3
Indirect Proof with Algebra One way to prove that a statement is true is to assumethat its conclusion is false and then show that this assumption leads to a contradiction ofthe hypothesis, a definition, postulate, theorem, or other statement that is accepted as true.That contradiction means that the conclusion cannot be false, so the conclusion must betrue. This is known as indirect proof.
Steps for Writing an Indirect Proof
1. Assume that the conclusion is false.2. Show that this assumption leads to a contradiction.3. Point out that the assumption must be false, and therefore, the conclusion must be true.
Given: 3x � 5 � 8Prove: x � 1
Step 1 Assume that x is not greater than 1. That is, x � 1 or x � 1.Step 2 Make a table for several possibilities for x � 1 or x � 1. The
contradiction is that when x � 1 or x � 1, then 3x � 5 is notgreater than 8.
Step 3 This contradicts the given information that 3x � 5 � 8. Theassumption that x is not greater than 1 must be false, which means that the statement “x � 1” must be true.
Write the assumption you would make to start an indirect proof of each statement.
1. If 2x � 14, then x � 7.
2. For all real numbers, if a � b � c, then a � c � b.
Complete the proof.Given: n is an integer and n2 is even.Prove: n is even.
3. Assume that
4. Then n can be expressed as 2a � 1 by
5. n2 � Substitution
6. � Multiply.
7. � Simplify.
8. � 2(2a2 � 2a) � 1
9. 2(2a2 � 2a)� 1 is an odd number. This contradicts the given that n2 is even,
so the assumption must be
10. Therefore,
x 3x � 5
1 8
0 5
�1 2
�2 �1
�3 �4
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 258 Glencoe Geometry
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
Study Guide and Intervention (continued)
Indirect Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
5-35-3
ExercisesExercises
ExampleExample
Skills PracticeIndirect Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
5-35-3
© Glencoe/McGraw-Hill 259 Glencoe Geometry
Less
on
5-3
Write the assumption you would make to start an indirect proof of each statement.
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
© Glencoe/McGraw-Hill 260 Glencoe Geometry
Write the assumption you would make to start an indirect proof of each statement.
1. B�D� bisects �ABC.
2. RT � TS
PROOF Write an indirect proof.
3. Given: �4x � 2 � �10Prove: x � 3
4. Given: m�2 � m�3 � 180Prove: a ⁄|| b
5. PHYSICS Sound travels through air at about 344 meters per second when thetemperature is 20°C. If Enrique lives 2 kilometers from the fire station and it takes 5 seconds for the sound of the fire station siren to reach him, how can you proveindirectly that it is not 20°C when Enrique hears the siren?
12
3
a
b
Practice Indirect Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
5-35-3
Reading to Learn MathematicsIndirect Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
5-35-3
© Glencoe/McGraw-Hill 261 Glencoe Geometry
Less
on
5-3
Pre-Activity How is indirect proof used in literature?
Read the introduction to Lesson 5-3 at the top of page 255 in your textbook.
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
© Glencoe/McGraw-Hill 262 Glencoe Geometry
More CounterexamplesSome statements in mathematics can be proven false by counterexamples.Consider the following statement.
For any numbers a and b, a � b � b � a.
You can prove that this statement is false in general if you can find oneexample for which the statement is false.
Let a � 7 and b � 3. Substitute these values in the equation above.
7 � 3 � 3 � 74 � �4
In general, for any numbers a and b, the statement a � b � b � a is false.You can make the equivalent verbal statement: subtraction is not acommutative operation.
In each of the following exercises a, b, and c are any numbers. Prove that the statement is false by counterexample.
1. a � (b � c) � (a � b) � c 2. a (b c) � (a b) c
3. a b � b a 4. a (b � c) � (a b) � (a c)
5. a � (bc) � (a � b)(a � c) 6. a2 � a2 � a4
7. Write the verbal equivalents for Exercises 1, 2, and 3.
8. For the Distributive Property a(b � c) � ab � ac it is said that multiplicationdistributes over addition. Exercises 4 and 5 prove that some operations do notdistribute. Write a statement for each exercise that indicates this.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
5-35-3
Study Guide and InterventionThe Triangle Inequality
NAME ______________________________________________ DATE ____________ PERIOD _____
5-45-4
© Glencoe/McGraw-Hill 263 Glencoe Geometry
Less
on
5-4
The Triangle Inequality If you take three straws of lengths 8 inches, 5 inches, and 1 inch and try to make a triangle with them, you will find that it is not possible. Thisillustrates the Triangle Inequality Theorem.
Triangle Inequality The sum of the lengths of any two sides of aTheorem triangle is greater than the length of the third side.
The measures of two sides of a triangle are 5 and 8. Find a rangefor the length of the third side.By the Triangle Inequality, all three of the following inequalities must be true.
5 � x � 8 8 � x � 5 5 � 8 � xx � 3 x � �3 13 � x
Therefore x must be between 3 and 13.
Determine whether the given measures can be the lengths of the sides of atriangle. Write yes or no.
1. 3, 4, 6 2. 6, 9, 15
3. 8, 8, 8 4. 2, 4, 5
5. 4, 8, 16 6. 1.5, 2.5, 3
Find the range for the measure of the third side given the measures of two sides.
7. 1 and 6 8. 12 and 18
9. 1.5 and 5.5 10. 82 and 8
11. Suppose you have three different positive numbers arranged in order from least togreatest. What single comparison will let you see if the numbers can be the lengths ofthe sides of a triangle?
BC
A
a
cb
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 264 Glencoe Geometry
Distance Between a Point and a Line
Study Guide and Intervention (continued)
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
NAME ______________________________________________ DATE ____________ PERIOD _____
5-45-4
© Glencoe/McGraw-Hill 265 Glencoe Geometry
Less
on
5-4
Determine whether the given measures can be the lengths of the sides of atriangle. Write yes or no.
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)
© Glencoe/McGraw-Hill 266 Glencoe Geometry
Determine whether the given measures can be the lengths of the sides of atriangle. Write yes or no.
1. 9, 12, 18 2. 8, 9, 17
3. 14, 14, 19 4. 23, 26, 50
5. 32, 41, 63 6. 2.7, 3.1, 4.3
7. 0.7, 1.4, 2.1 8. 12.3, 13.9, 25.2
Find the range for the measure of the third side of a triangle given the measuresof two sides.
9. 6 and 19 10. 7 and 29
11. 13 and 27 12. 18 and 23
13. 25 and 38 14. 31 and 39
15. 42 and 6 16. 54 and 7
ALGEBRA Determine whether the given coordinates are the vertices of a triangle.Explain.
17. R(1, 3), S(4, 0), T(10, �6) 18. W(2, 6), X(1, 6), Y(4, 2)
19. P(�3, 2), L(1, 1), M(9, �1) 20. B(1, 1), C(6, 5), D(4, �1)
21. GARDENING Ha Poong has 4 lengths of wood from which he plans to make a border for atriangular-shaped herb garden. The lengths of the wood borders are 8 inches, 10 inches,12 inches, and 18 inches. How many different triangular borders can Ha Poong make?
Practice The Triangle Inequality
NAME ______________________________________________ DATE ____________ PERIOD _____
5-45-4
Reading to Learn MathematicsThe Triangle Inequality
NAME ______________________________________________ DATE ____________ PERIOD _____
5-45-4
© Glencoe/McGraw-Hill 267 Glencoe Geometry
Less
on
5-4
Pre-Activity How can you use the Triangle Inequality Theorem when traveling?
Read the introduction to Lesson 5-4 at the top of page 261 in your textbook.
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
© Glencoe/McGraw-Hill 268 Glencoe Geometry
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
NAME ______________________________________________ DATE ____________ PERIOD _____
5-45-4
Study Guide and InterventionInequalities Involving Two Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
5-55-5
© Glencoe/McGraw-Hill 269 Glencoe Geometry
Less
on
5-5
SAS Inequality The following theorem involves the relationship between the sides oftwo triangles and an angle in each triangle.
If two sides of a triangle are congruent to two sides of another triangle and the included angle in one triangle has a
SAS Inequality/Hinge Theorem greater measure than the included angle in the other, then the third side of the If R�S� � A�B�, S�T� � B�C�, andfirst triangle is longer than the third side m�S � m�B, then RT � AC.
of the second triangle.
Write an inequality relating the lengths of C�D� and A�D�.Two sides of �BCD are congruent to two sides of �BAD and m�CBD � m�ABD. By the SAS Inequality/Hinge Theorem,CD � AD.
Write an inequality relating the given pair of segment measures.
1. 2.
MR, RP AD, CD
3. 4.
EG, HK MR, PR
Write an inequality to describe the possible values of x.
5. 6.
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
© Glencoe/McGraw-Hill 270 Glencoe Geometry
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
Study Guide and Intervention (continued)
Inequalities Involving Two Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
5-55-5
ExampleExample
ExercisesExercises
Skills PracticeInequalities Involving Two Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
5-55-5
© Glencoe/McGraw-Hill 271 Glencoe Geometry
Less
on
5-5
Write an inequality relating the given pair of angles or segment measures.
1. m�BXA, m�DXA
2. BC, DC
Write an inequality relating the given pair of angles or segment measures.
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
© Glencoe/McGraw-Hill 272 Glencoe Geometry
Write an inequality relating the given pair of angles or segment measures.
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 _____
5-55-5
© Glencoe/McGraw-Hill 273 Glencoe Geometry
Less
on
5-5
Pre-Activity How does a backhoe work?
Read the introduction to Lesson 5-5 at the top of page 267 in your textbook.
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
© Glencoe/McGraw-Hill 274 Glencoe Geometry
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
© Glencoe/McGraw-Hill 275 Glencoe Geometry
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.
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NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 276 Glencoe Geometry
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
© Glencoe/McGraw-Hill 277 Glencoe Geometry
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.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�
For Questions 5–7, use the figure to determine which is a true statement for the given information.
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.
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NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 278 Glencoe Geometry
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
15. Choose the assumption you would make to start an indirect proof of x � 2.A. x � 2 B. x � 2 C. x � 2 D. x � 2
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
© Glencoe/McGraw-Hill 279 Glencoe Geometry
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.
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���
For Questions 5–7, use the figure to determine which is a true statement for the given information.
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.
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8.
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NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 280 Glencoe Geometry
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
15. Choose the assumption you would make to start an indirect proof of x � 1.A. x � 1 B. x � 1 C. x � 1 D. x � 1
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
© Glencoe/McGraw-Hill 281 Glencoe Geometry
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
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NAME DATE PERIOD
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© Glencoe/McGraw-Hill 282 Glencoe Geometry
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
© Glencoe/McGraw-Hill 283 Glencoe Geometry
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
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NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 284 Glencoe Geometry
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
© Glencoe/McGraw-Hill 285 Glencoe Geometry
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.
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NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 286 Glencoe Geometry
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
© Glencoe/McGraw-Hill 287 Glencoe Geometry
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
© Glencoe/McGraw-Hill 288 Glencoe Geometry
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
© Glencoe/McGraw-Hill 289 Glencoe Geometry
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
© Glencoe/McGraw-Hill 290 Glencoe Geometry
Chapter 5 Quiz (Lesson 5–4)
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
Chapter 5 Quiz (Lesson 5–5)
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
© Glencoe/McGraw-Hill 291 Glencoe Geometry
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.
© Glencoe/McGraw-Hill 292 Glencoe Geometry
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)
© Glencoe/McGraw-Hill 293 Glencoe Geometry
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
© Glencoe/McGraw-Hill 294 Glencoe Geometry
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.
© Glencoe/McGraw-Hill A2 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Bis
ecto
rs,M
edia
ns,
and
Alt
itu
des
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
5-1
5-1
©G
lenc
oe/M
cGra
w-H
ill24
5G
lenc
oe G
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
gle
mee
t at
a p
oin
t,ca
lled
th
eci
rcu
mce
nte
rof
th
e tr
ian
gle,
that
is
equ
idis
tan
t fr
om t
he
thre
e ve
rtic
es o
f th
e tr
ian
gle.
Tw
o pr
oper
ties
of
angl
e bi
sect
ors
are:
(1)
a po
int
is o
n t
he
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
ula
rb
isec
tor
of A �
C�.F
ind
x.
BD
���
is t
he
perp
endi
cula
r bi
sect
or o
f A �
C�,s
oA
D�
DC
.3x
�8
�5x
�6
14 �
2x7
�x
3x �
8
5x �
6B
C
D
A
MR
���
is t
he
angl
e b
isec
tor
of �
NM
P.F
ind
xif
m�
1 �
5x�
8 an
dm
�2
�8x
�16
.
MR
���
is t
he
angl
e bi
sect
or o
f �
NM
P,s
o m
�1
�m
�2.
5x�
8 �
8x�
1624
�3x
8 �
x
12
NR
PM
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
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
ctor
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�
8 y
CED
A
B 7x �
96x
� 2
©G
lenc
oe/M
cGra
w-H
ill24
6G
lenc
oe G
eom
etry
Med
ian
s an
d A
ltit
ud
esA
med
ian
is a
lin
e se
gmen
t th
at c
onn
ects
th
e ve
rtex
of
atr
ian
gle
to t
he
mid
poin
t of
th
e op
posi
te s
ide.
Th
e th
ree
med
ian
s of
a t
rian
gle
inte
rsec
t at
th
ece
ntr
oid
of t
he
tria
ngl
e.
Cen
tro
idT
he c
entr
oid
of a
tria
ngle
is lo
cate
d tw
o th
irds
of t
he d
ista
nce
from
aT
heo
rem
vert
ex t
o th
e m
idpo
int
of t
he s
ide
oppo
site
the
ver
tex
on a
med
ian.
AL
��2 3� A
E,
BL
��2 3� B
F,
CL
��2 3� C
D
Poi
nts
R,S
,an
d T
are
the
mid
poi
nts
of
A �B�
,B�C�
and
A�C�
,res
pec
tive
ly.F
ind
x,y
,an
d z
.
CU
��2 3�
CR
BU
��2 3�
BT
AU
��2 3�
AS
6x�
�2 3�(6
x�
15)
24�
�2 3�(2
4 �
3y�
3)6z
�4
��2 3� (
6z�
4 �
11)
9x�
6x�
1536
�24
�3y
�3
�3 2� (6z
�4)
�6z
�4
�11
3x�
1536
�21
�3y
9z�
6�
6z�
15x
�5
15�
3y3z
�9
5�
yz
�3
Fin
d t
he
valu
e of
eac
h v
aria
ble
.
1.x
�4
2.x
�6;
y�
5
B�D�
is a
med
ian
.A
B�
CB
;D,E
,an
d F
are
mid
poin
ts.
3.x
�3;
y�
54.
x�
12;
y�
5;z
�2
EH
�F
H�
HG
5.x
�2;
y�
2;z
�2
6.x
�6;
y�
5;z
�8
Vis
th
e ce
ntr
oid
of �
RS
T;
Dis
th
e ce
ntr
oid
of �
AB
C.
TP
�18
;MS
�15
;RN
�24
7.F
or w
hat
kin
d of
tri
angl
e ar
e th
e m
edia
ns
and
angl
e bi
sect
ors
the
sam
e se
gmen
ts?
equ
ilate
ral t
rian
gle
8.F
or w
hat
kin
d of
tri
angl
e is
th
e ce
ntr
oid
outs
ide
the
tria
ngl
e?n
ot
po
ssib
le
P
M
V
T
N
RS
y x
z
G
FE
B
AC
24
329z
� 6
6z6x
8y
MJ
PN
O
LK3y
� 5
2 x6z12
24 10H
GF
E
7x �
4
9x �
2 5 y
DB
EF
AC
9x �
6
10x
3y15D
BA
C
6x �
3
7x �
1
AC
T
SR
U
B
3y �
3
6x
1524
11
6z �
4
AC
F
ED
L
Bce
ntro
id
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Bis
ecto
rs,M
edia
ns,
and
Alt
itu
des
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
5-1
5-1
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 5-1)
© Glencoe/McGraw-Hill A3 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Bis
ecto
rs,M
edia
ns,
and
Alt
itu
des
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
5-1
5-1
©G
lenc
oe/M
cGra
w-H
ill24
7G
lenc
oe G
eom
etry
Lesson 5-1
ALG
EBR
AF
or E
xerc
ises
1–4
,use
th
e gi
ven
in
form
atio
n t
o fi
nd
eac
h v
alu
e.
1.F
ind
xif
E �G�
is a
med
ian
of
�D
EF
.2.
Fin
d x
and
RT
if S�
U�is
a m
edia
n o
f �
RS
T.
x�
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
ises
5–7
,use
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
),an
d I
(3,6
).F
ind
the
coor
din
ates
of
the
poi
nts
of
con
curr
ency
of
�H
IJ.
11.o
rth
ocen
ter
12.c
entr
oid
13.c
ircu
mce
nte
r
(3,1
)��1 30 �
,2�
��7 2� ,�5 2� �5y
� 6
8x �
16
7 z �
4
24
18
RQ
A
ab
c
P
y �
1
2z2.
8
23.
6
x
L
NQ
RB
P
M
( 3x
� 3
) �x �
8 x �
9
I
JH
K
AD4x
� 1
2x �
6B
G EF
C
RU 5x �
30
2x �
24
S
T
DG 3x �
1
5x �
17
E
F
©G
lenc
oe/M
cGra
w-H
ill24
8G
lenc
oe G
eom
etry
ALG
EBR
AIn
�A
BC
,B�F�
is t
he
angl
e b
isec
tor
of �
AB
C,A�
E�,B�
F�,
and
C �D�
are
med
ian
s,an
d P
is t
he
cen
troi
d.
1.F
ind
xif
DP
�4x
�3
and
CP
�30
.4.
5
2.F
ind
yif
AP
�y
and
EP
�18
.36
3.F
ind
zif
FP
�5z
�10
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
Answers (Lesson 5-1)
© Glencoe/McGraw-Hill A4 Glencoe Geometry
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
Answers (Lesson 5-1)
© Glencoe/McGraw-Hill A5 Glencoe Geometry
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.
�T
,�R
,�S
R�S�
,S�T�,
R�T�
�C
,�B
,�A
Det
erm
ine
the
rela
tion
ship
bet
wee
n t
he
mea
sure
s of
th
e gi
ven
an
gles
.
4.�
R,�
RU
Sm
�R
�m
�R
US
5.�
T,�
US
Tm
�T
�m
�U
ST
6.�
UV
S,�
Rm
�U
VS
�m
�R
Det
erm
ine
the
rela
tion
ship
bet
wee
n t
he
len
gth
s of
th
e gi
ven
sid
es.
7.A �
C�,B�
C�A
C�
BC
8.B�
C�,D�
B�B
C�
DB
9.A�
C�,D�
B�A
C�
DB
ABC
D30
�
30� 30
�
90�
RV
S
UT
2513
2424
22
21.6
35
AC
B
3.8
4.3
4.0
RT
S
60�
80�
40�
TS
R48
cm
23.7
cm
35 c
m
Answers (Lesson 5-2)
© Glencoe/McGraw-Hill A6 Glencoe Geometry
Skil
ls P
ract
ice
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
3G
lenc
oe G
eom
etry
Lesson 5-2
Det
erm
ine
wh
ich
an
gle
has
th
e gr
eate
st m
easu
re.
1.�
1,�
3,�
42.
�4,
�5,
�7
�1
�4
3.�
2,�
3,�
64.
�5,
�6,
�8
�6
�8
Use
th
e E
xter
ior
An
gle
Ineq
ual
ity
Th
eore
m t
o li
st a
ll
angl
es t
hat
sat
isfy
th
e st
ated
con
dit
ion
.
5.al
l an
gles
wh
ose
mea
sure
s ar
e le
ss t
han
m�
1
�2,
�3,
�4,
�5,
�7,
�8
6.al
l an
gles
wh
ose
mea
sure
s ar
e le
ss t
han
m�
9
�2,
�4,
�6,
�7
7.al
l an
gles
wh
ose
mea
sure
s ar
e gr
eate
r th
an m
�5
�1,
�3
8.al
l an
gles
wh
ose
mea
sure
s ar
e gr
eate
r th
an m
�8
�1,
�3,
�5
Det
erm
ine
the
rela
tion
ship
bet
wee
n t
he
mea
sure
s of
th
e gi
ven
an
gles
.
9.m
�A
BD
,m�
BA
D10
.m�
AD
B,m
�B
AD
m�
AB
D�
m�
BA
Dm
�A
DB
�m
�B
AD
11.m
�B
CD
,m�
CD
B12
.m�
CB
D,m
�C
DB
m�
BC
D�
m�
CD
Bm
�C
BD
�m
�C
DB
Det
erm
ine
the
rela
tion
ship
bet
wee
n t
he
len
gth
s of
th
e gi
ven
sid
es.
13.L �
M�,L�
P�14
.M�P�
,M�N�
LM
�L
PM
P�
MN
15.M�
N�,N�
P�16
.M�P�
,L�P�
MN
�N
PM
P�
LP
83�
57�
79�
44�
59�
38�
LN
P
M
2334
4139
35A
BC
D
1
24
6
7
89
35
12
46
78
35
©G
lenc
oe/M
cGra
w-H
ill25
4G
lenc
oe G
eom
etry
Det
erm
ine
wh
ich
an
gle
has
th
e gr
eate
st m
easu
re.
1.�
1,�
3,�
42.
�4,
�8,
�9
�1
�4
3.�
2,�
3,�
74.
�7,
�8,
�10
�7
�10
Use
th
e E
xter
ior
An
gle
Ineq
ual
ity
Th
eore
m t
o li
st
all
angl
es t
hat
sat
isfy
th
e st
ated
con
dit
ion
.
5.al
l an
gles
wh
ose
mea
sure
s ar
e le
ss t
han
m�
1
�3,
�4,
�5,
�7,
�8
6.al
l an
gles
wh
ose
mea
sure
s ar
e le
ss t
han
m�
3
�5,
�7,
�8
7.al
l an
gles
wh
ose
mea
sure
s ar
e gr
eate
r th
an m
�7
�1,
�3,
�5,
�9
8.al
l an
gles
wh
ose
mea
sure
s ar
e gr
eate
r th
an m
�2
�6,
�9
Det
erm
ine
the
rela
tion
ship
bet
wee
n t
he
mea
sure
s of
th
e gi
ven
an
gles
.
9.m
�Q
RW
,m�
RW
Q10
.m�
RT
W,m
�T
WR
m�
QR
W�
�R
WQ
m�
RT
W�
�T
WR
11.m
�R
ST
,m�
TR
S12
.m�
WQ
R,m
�Q
RW
m�
RS
T�
�T
RS
m�
WQ
R�
�Q
RW
Det
erm
ine
the
rela
tion
ship
bet
wee
n t
he
len
gth
s of
th
e gi
ven
sid
es.
13.D �
H�,G�
H�14
.D�E�
,D�G�
DH
�G
HD
E�
DG
15.E�
G�,F�
G�16
.D�E�
,E�G�
EG
�F
GD
E�
EG
17.S
POR
TST
he
figu
re s
how
s th
e po
siti
on o
f th
ree
tree
s on
on
e pa
rt o
f a
Fri
sbee
™ c
ours
e.A
t w
hic
h t
ree
posi
tion
is
the
angl
e be
twee
n t
he
tree
s th
e gr
eate
st?
2
53 ft
40 ft
3
2
1
37.5
ft
120�
32�
48�
113�
17�
H
DE
F
G
3447
45
44
22
1435
Q
R
S
TW
12
46
78
9
35
12
46
78
910
3
5
Pra
ctic
e (
Ave
rag
e)
Ineq
ual
itie
s an
d T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
5-2
5-2
Answers (Lesson 5-2)
© Glencoe/McGraw-Hill A7 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csIn
equ
alit
ies
and
Tri
ang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
5-2
5-2
©G
lenc
oe/M
cGra
w-H
ill25
5G
lenc
oe G
eom
etry
Lesson 5-2
Pre-
Act
ivit
yH
ow c
an y
ou t
ell
wh
ich
cor
ner
is
big
ger?
Rea
d th
e in
trod
uct
ion
to
Les
son
5-2
at
the
top
of p
age
247
in y
our
text
book
.
•W
hic
h s
ide
of t
he
pati
o is
opp
osit
e th
e la
rges
t co
rner
?th
e 51
-fo
ot
sid
e•
Whi
ch s
ide
of t
he p
atio
is o
ppos
ite
the
smal
lest
cor
ner?
the
45-f
oo
t si
de
Rea
din
g t
he
Less
on
1.N
ame
the
prop
erty
of
ineq
ual
ity
that
is
illu
stra
ted
by e
ach
of
the
foll
owin
g.
a.If
x�
8 an
d 8
�y,
then
x�
y.Tr
ansi
tive
Pro
per
tyb
.If
x�
y,th
en x
�7.
5 �
y�
7.5.
Su
btr
acti
on
Pro
per
tyc.
If x
�y,
then
�3x
��
3y.
Mu
ltip
licat
ion
Pro
per
tyd
.If
xis
an
y re
al n
um
ber,
x�
0,x
�0,
or x
�0.
Co
mp
aris
on
Pro
per
ty
2.U
se t
he d
efin
itio
n of
ineq
ualit
y to
wri
te a
n eq
uati
onth
at s
how
s th
at e
ach
ineq
ualit
y is
tru
e.
a.20
�12
20 �
12 �
8b
.101
�99
101
�99
�2
c.8
��
28
��
2 �
10d
.7 �
�7
7 �
�7
�14
e.�
11 �
�12
�11
��
12 �
1f.
�30
��
45�
30 �
�45
�15
3.In
th
e fi
gure
,m�
IJK
�45
an
d m
�H
�m
�I.
a.A
rran
ge t
he
foll
owin
g an
gles
in
ord
er f
rom
lar
gest
to
smal
lest
:�I,
�IJ
K,�
H,�
IJH
�IJ
H,�
IJK
,�H
,�I
b.
Arr
ange
th
e si
des
of �
HIJ
in o
rder
fro
m s
hor
test
to
lon
gest
.
H�J�,
I�J�,H�
I�c.
Is �
HIJ
an a
cute
,rig
ht,
or o
btu
se t
rian
gle?
Exp
lain
you
r re
ason
ing.
Ob
tuse
;sa
mp
le a
nsw
er:
�IJ
His
ob
tuse
bec
ause
m
�IJ
H�
180
�m
�IJ
K�
135.
Th
eref
ore
,�H
IJis
ob
tuse
bec
ause
ith
as a
n o
btu
se a
ng
le.
d.
Is �
HIJ
scal
ene,
isos
cele
s,or
equ
ilat
eral
? E
xpla
in y
our
reas
onin
g.
Sca
len
e;sa
mp
le a
nsw
er:
the
thre
e an
gle
s o
f �
HIJ
all h
ave
dif
fere
nt
mea
sure
s,so
th
e si
des
op
po
site
th
em m
ust
hav
e d
iffe
ren
t le
ng
ths.
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r a
new
geo
met
ric
theo
rem
is
to r
elat
e it
to
a th
eore
m y
oule
arn
ed e
arli
er.E
xpla
in h
ow t
he
Ext
erio
r A
ngl
e In
equ
alit
y T
heo
rem
is
rela
ted
to t
he
Ext
erio
r A
ngl
e T
heo
rem
,an
d w
hy
the
Ext
erio
r A
ngl
e In
equ
alit
y T
heo
rem
mu
st b
e tr
ue
ifth
e E
xter
ior
An
gle
Th
eore
m i
s tr
ue.
Sam
ple
an
swer
:Th
e E
xter
ior
An
gle
Th
eore
m s
ays
that
th
e m
easu
re o
f an
exte
rio
r an
gle
of
a tr
ian
gle
is e
qu
al t
o t
he
sum
of
the
mea
sure
s o
f th
etw
o r
emo
te in
teri
or
ang
les,
wh
ile t
he
Ext
erio
r A
ng
le In
equ
alit
y T
heo
rem
says
th
at t
he
mea
sure
of
an e
xter
ior
ang
le is
gre
ater
th
an t
he
mea
sure
of
eith
er r
emo
te in
teri
or
ang
le.I
f a
nu
mb
er is
eq
ual
to
th
e su
m o
f tw
op
osi
tive
nu
mb
ers,
it m
ust
be
gre
ater
th
an e
ach
of
tho
se t
wo
nu
mb
ers.K
JH
I
©G
lenc
oe/M
cGra
w-H
ill25
6G
lenc
oe G
eom
etry
Co
nst
ruct
ion
Pro
blem
Th
e di
agra
m b
elow
sh
ows
segm
ent
AB
adja
cen
t to
a c
lose
d re
gion
.Th
epr
oble
m r
equ
ires
th
at y
ou c
onst
ruct
an
oth
er s
egm
ent
XY
to t
he
righ
t of
th
ecl
osed
reg
ion
su
ch t
hat
poi
nts
A,B
,X,a
nd
Yar
e co
llin
ear.
You
are
not
all
owed
to t
ouch
or
cros
s th
e cl
osed
reg
ion
wit
h y
our
com
pass
or
stra
igh
tedg
e.
Fol
low
th
ese
inst
ruct
ion
s to
con
stru
ct a
seg
men
t X
Yso
th
at i
t is
coll
inea
r w
ith
seg
men
t A
B.
1.C
onst
ruct
th
e pe
rpen
dicu
lar
bise
ctor
of
A �B�
.Lab
el t
he
mid
poin
t as
poi
nt
C,
and
the
lin
e as
m.
2.M
ark
two
poin
ts P
and
Qon
lin
e m
that
lie
wel
l ab
ove
the
clos
ed r
egio
n.
Con
stru
ct t
he
perp
endi
cula
r bi
sect
or n
of P�
Q�.L
abel
th
e in
ters
ecti
on o
fli
nes
man
d n
as p
oin
t D
.
3.M
ark
poin
ts R
and
Son
lin
e n
that
lie
wel
l to
th
e ri
ght
of t
he
clos
edre
gion
.Con
stru
ct t
he
perp
endi
cula
r bi
sect
or k
of R�
S�.L
abel
th
ein
ters
ecti
on o
f li
nes
nan
d k
as p
oin
t E
.
4.M
ark
poin
t X
on l
ine
kso
th
at X
is b
elow
lin
e n
and
so t
hat
E�X�
isco
ngr
uen
t to
D �C�
.
5.M
ark
poin
ts T
and
Von
lin
e k
and
on o
ppos
ite
side
s of
X,s
o th
at X�
T�an
dX �
V�ar
e co
ngr
uen
t.C
onst
ruct
th
e pe
rpen
dicu
lar
bise
ctor
�of
T�V�
.Cal
l th
epo
int
wh
ere
the
lin
e �
hit
s th
e bo
un
dary
of
the
clos
ed r
egio
n p
oin
t Y
.X �Y�
corr
espo
nds
to
the
new
roa
d.
Q Pm
k
�
nD
RE T X V
YB
AC
S
Exis
ting
Road
Clos
ed R
egio
n(L
ake)
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
5-2
5-2
Answers (Lesson 5-2)
© Glencoe/McGraw-Hill A8 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Ind
irec
t P
roo
f
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
5-3
5-3
©G
lenc
oe/M
cGra
w-H
ill25
7G
lenc
oe G
eom
etry
Lesson 5-3
Ind
irec
t Pr
oo
f w
ith
Alg
ebra
On
e w
ay t
o pr
ove
that
a s
tate
men
t is
tru
e is
to
assu
me
that
its
con
clu
sion
is
fals
e an
d th
en s
how
th
at t
his
ass
um
ptio
n l
eads
to
a co
ntr
adic
tion
of
the
hyp
oth
esis
,a d
efin
itio
n,p
ostu
late
,th
eore
m,o
r ot
her
sta
tem
ent
that
is
acce
pted
as
tru
e.T
hat
con
trad
icti
on m
ean
s th
at t
he
con
clu
sion
can
not
be
fals
e,so
th
e co
ncl
usi
on m
ust
be
tru
e.T
his
is
know
n a
s in
dir
ect
pro
of.
Ste
ps
for
Wri
tin
g a
n In
dir
ect
Pro
of
1.A
ssum
e th
at t
he c
oncl
usio
n is
fal
se.
2.S
how
tha
t th
is a
ssum
ptio
n le
ads
to a
con
trad
ictio
n.3.
Poi
nt o
ut t
hat
the
assu
mpt
ion
mus
t be
fal
se,
and
ther
efor
e, t
he c
oncl
usio
n m
ust
be t
rue.
Giv
en:3
x�
5 �
8P
rove
:x�
1S
tep
1A
ssu
me
that
xis
not
gre
ater
th
an 1
.Th
at i
s,x
�1
or x
�1.
Ste
p 2
Mak
e a
tabl
e fo
r se
vera
l po
ssib
ilit
ies
for
x�
1 or
x�
1.T
he
con
trad
icti
on i
s th
at w
hen
x�
1 or
x�
1,th
en 3
x�
5 is
not
grea
ter
than
8.
Ste
p 3
Th
is c
ontr
adic
ts t
he
give
n i
nfo
rmat
ion
th
at 3
x�
5 �
8.T
he
assu
mpt
ion
th
at x
is n
ot g
reat
er t
han
1 m
ust
be
fals
e,w
hic
h
mea
ns
that
th
e st
atem
ent
“x�
1”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
ch s
tate
men
t.
1.If
2x
�14
,th
en x
�7.
x�
7
2.F
or a
ll r
eal
nu
mbe
rs,i
f a
�b
�c,
then
a�
c�
b.a
�c
�b
Com
ple
te t
he
pro
of.
Giv
en:n
is a
n i
nte
ger
and
n2
is e
ven
.P
rove
:nis
eve
n.
3.A
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
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill25
8G
lenc
oe G
eom
etry
Ind
irec
t Pr
oo
f w
ith
Geo
met
ryT
o w
rite
an
in
dire
ct p
roof
in
geo
met
ry,y
ou a
ssu
me
that
th
e co
ncl
usi
on i
s fa
lse.
Th
en y
ou s
how
th
at t
he
assu
mpt
ion
lea
ds t
o a
con
trad
icti
on.
Th
e co
ntr
adic
tion
sh
ows
that
th
e co
ncl
usi
on c
ann
ot b
e fa
lse,
so i
t m
ust
be
tru
e.
Giv
en:m
�C
�10
0P
rove
:�A
is n
ot a
rig
ht
angl
e.S
tep
1A
ssu
me
that
�A
is a
rig
ht
angl
e.
Ste
p 2
Sh
ow t
hat
th
is l
eads
to
a co
ntr
adic
tion
.If
�A
is a
rig
ht
angl
e,th
en m
�A
�90
an
d m
�C
�m
�A
�10
0 �
90 �
190.
Th
us
the
sum
of
the
mea
sure
s of
th
e an
gles
of
�A
BC
is g
reat
er t
han
180
.
Ste
p 3
Th
e co
ncl
usi
on t
hat
th
e su
m o
f th
e m
easu
res
of t
he
angl
es o
f �
AB
Cis
gre
ater
th
an 1
80 i
s a
con
trad
icti
on o
f a
know
n p
rope
rty.
Th
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
.
Com
ple
te t
he
pro
of.
Giv
en:�
1 �
�2
and
D�G�
is n
ot c
ongr
uen
t to
F�G�
.P
rove
:D �E�
is n
ot c
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.
4.E�
G��
E�G�
Ref
lexi
ve P
rop
erty
5.�
ED
G�
�E
FG
SA
S
6.D�
G��
F�G�C
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
Answers (Lesson 5-3)
© Glencoe/McGraw-Hill A9 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Ind
irec
t P
roo
f
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
5-3
5-3
©G
lenc
oe/M
cGra
w-H
ill25
9G
lenc
oe G
eom
etry
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
©G
lenc
oe/M
cGra
w-H
ill26
0G
lenc
oe G
eom
etry
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.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
Answers (Lesson 5-3)
© Glencoe/McGraw-Hill A10 Glencoe Geometry
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
©G
lenc
oe/M
cGra
w-H
ill26
1G
lenc
oe G
eom
etry
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
©G
lenc
oe/M
cGra
w-H
ill26
2G
lenc
oe G
eom
etry
Mo
re C
ou
nte
rexa
mp
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
Answers (Lesson 5-3)
© Glencoe/McGraw-Hill A11 Glencoe Geometry
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
Answers (Lesson 5-4)
© Glencoe/McGraw-Hill A12 Glencoe Geometry
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
Answers (Lesson 5-4)
© Glencoe/McGraw-Hill A13 Glencoe Geometry
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
es t
hro
ug
h t
he
thir
d v
erte
x.
a�
ba
�b
170
3812
193
ZX
Y
GD
EF
©G
lenc
oe/M
cGra
w-H
ill26
8G
lenc
oe G
eom
etry
Co
nst
ruct
ing
Tri
ang
les
Th
e m
easu
rem
ents
of
the
sid
es o
f a
tria
ngl
e ar
e gi
ven
.If
a tr
ian
gle
hav
ing
sid
esw
ith
th
ese
mea
sure
men
ts i
s n
ot p
ossi
ble
,th
en w
rite
im
pos
sibl
e.If
a t
rian
gle
isp
ossi
ble
,dra
w i
t an
d m
easu
re e
ach
an
gle
wit
h a
pro
trac
tor.
1.A
R�
5 cm
m�
A�
302.
PI
�8
cmm
�P
�
RT
�3
cmm
�R
�90
IN�
3 cm
m�
I�
AT
�6
cmm
�T
�60
PN
�2
cmm
�N
�
imp
oss
ible
3.O
N�
10 c
mm
�O
�4.
TW
�6
cmm
�T
�11
5
NE
�5.
3 cm
m�
N�
WO
�7
cmm
�W
�15
GE
�4.
6 cm
m�
E�
TO
�2
cmm
�O
�50
imp
oss
ible
5.B
A�
3.l
cmm
�B
�16
36.
AR
�4
cmm
�A
�90
AT
�8
cmm
�A
�11
RM
�5
cmm
�R
�37
BT
�5
cmm
�T
�6
AM
�3
cmm
�M
�53
M
RA
T
BA
W
T
O
AR T
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
5-4
5-4
Answers (Lesson 5-4)
© Glencoe/McGraw-Hill A14 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Ineq
ual
itie
s In
volv
ing
Tw
o T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
5-5
5-5
©G
lenc
oe/M
cGra
w-H
ill26
9G
lenc
oe G
eom
etry
Lesson 5-5
SAS
Ineq
ual
ity
Th
e fo
llow
ing
theo
rem
in
volv
es t
he
rela
tion
ship
bet
wee
n t
he
side
s of
two
tria
ngl
es a
nd
an a
ngl
e in
eac
h t
rian
gle.
If tw
o si
des
of a
tria
ngle
are
con
grue
nt
to t
wo
side
s of
ano
ther
tria
ngle
and
the
in
clud
ed a
ngle
in o
ne t
riang
le h
as a
S
AS
Ineq
ual
ity/
Hin
ge
Th
eore
mgr
eate
r m
easu
re t
han
the
incl
uded
ang
le
in t
he o
ther
, th
en t
he t
hird
sid
e of
the
If
R�S�
�A�
B�,
S�T�
�B�
C�,
and
first
tria
ngle
is lo
nger
tha
n th
e th
ird s
ide
m�
S�
m�
B,
then
RT
�A
C.
of t
he s
econ
d tr
iang
le.
Wri
te a
n i
neq
ual
ity
rela
tin
g th
e le
ngt
hs
of C �
D�an
d A�
D�.
Tw
o si
des
of �
BC
Dar
e co
ngr
uen
t to
tw
o si
des
of �
BA
Dan
d m
�C
BD
�m
�A
BD
.By
the
SA
S I
neq
ual
ity/
Hin
ge T
heo
rem
,C
D�
AD
.
Wri
te a
n i
neq
ual
ity
rela
tin
g th
e gi
ven
pai
r of
seg
men
t m
easu
res.
1.2.
MR
,RP
AD
,CD
MR
�R
PA
D�
CD
3.4.
EG
,HK
MR
,PR
EG
�H
KM
R�
PR
Wri
te a
n i
neq
ual
ity
to d
escr
ibe
the
pos
sib
le v
alu
es o
f x.
5.6.
x�
12.5
x�
1.6
62�
65�
2.7
cm1.
8 cm
1.8
cm( 3
x �
2.1
) cm
115�12
0�24
cm
24 c
m40
cm
( 4x
� 1
0) c
m
MR
NP
48�46
�
2025
20
EG
HKJ
F60
�
62�
10
10
42
42
C ADB
22�
38�
N
R
P
M
21�
19�
BD A
28� 22
�
C
ST
80�
R
BC
60�A
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill27
0G
lenc
oe G
eom
etry
SSS
Ineq
ual
ity
Th
e co
nve
rse
of t
he
Hin
ge T
heo
rem
is
also
use
ful
wh
en t
wo
tria
ngl
esh
ave
two
pair
s of
con
gru
ent
side
s.
If tw
o si
des
of a
tria
ngle
are
con
grue
nt t
o tw
o si
des
of a
noth
er t
riang
le a
nd t
he t
hird
sid
e in
one
tria
ngle
SS
S In
equ
alit
yis
long
er t
han
the
third
sid
e in
the
oth
er,
then
the
an
gle
betw
een
the
pair
of c
ongr
uent
sid
es in
the
fir
st t
riang
le is
gre
ater
tha
n th
e co
rres
pond
ing
angl
e in
the
sec
ond
tria
ngle
.If
NM
�S
R, M
P�
RT
, and
NP
�S
T, t
hen
m�
M�
m�
R.
Wri
te a
n i
neq
ual
ity
rela
tin
g th
e m
easu
res
of
�A
BD
and
�C
BD
.T
wo
side
s of
�A
BD
are
con
gru
ent
to t
wo
side
s of
�C
BD
,an
d A
D�
CD
.B
y th
e S
SS
In
equ
alit
y,m
�A
BD
�m
�C
BD
.
Wri
te a
n i
neq
ual
ity
rela
tin
g th
e gi
ven
pai
r of
an
gle
mea
sure
s.
1.2.
m�
MP
R,m
�N
PR
m�
AB
D,m
�C
BD
m�
MP
R�
m�
NP
Rm
�A
BD
�m
�C
BD
3.4.
m�
C,m
�Z
m�
XY
W,m
�W
YZ
m�
C�
m�
Zm
�X
YW
�m
�W
YZ
Wri
te a
n i
neq
ual
ity
to d
escr
ibe
the
pos
sib
le v
alu
es o
f x.
5.6.
12 �
x�
116
1 �
x�
1233�
60 c
m
60 c
m
36 c
m
30 c
m( 3
x �
3) �
(1 – 2x �
6)�
52�
30
30
28
12
42
28
ZW
XY
30C
AX
B30
5048
2424
ZY
1116
2626
B
CD
A
13
10
M
R
NP
13 16C D A
B
3838
2323
3336
TR
SN
MP
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Ineq
ual
itie
s In
volv
ing
Tw
o T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
5-5
5-5
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 5-5)
© Glencoe/McGraw-Hill A15 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Ineq
ual
itie
s In
volv
ing
Tw
o T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
5-5
5-5
©G
lenc
oe/M
cGra
w-H
ill27
1G
lenc
oe G
eom
etry
Lesson 5-5
Wri
te a
n i
neq
ual
ity
rela
tin
g th
e gi
ven
pai
r of
an
gles
or
seg
men
t m
easu
res.
1.m
�B
XA
,m�
DX
A
m�
BX
A�
m�
DX
A
2.B
C,D
C
BC
�D
C
Wri
te a
n i
neq
ual
ity
rela
tin
g th
e gi
ven
pai
r of
an
gles
or
segm
ent
mea
sure
s.
3.m
�S
TR
,m�
TR
U4.
PQ
,RQ
m�
ST
R�
m�
TR
UP
Q�
RQ
5.In
th
e fi
gure
,B�A�
,B�D�
,B�C�
,an
d B�
E�ar
e co
ngr
uen
t an
d A
C�
DE
.H
ow d
oes
m�
1 co
mpa
re w
ith
m�
3? E
xpla
in y
our
thin
kin
g.
m�
1 �
m�
3;F
rom
th
e g
iven
info
rmat
ion
an
d t
he
SS
S In
equ
alit
y T
heo
rem
,it
follo
ws
that
in �
AB
Can
d �
DB
Ew
e h
ave
m�
AB
C�
m�
DB
E.S
ince
m
�A
BC
�m
�1
�m
�2
and
m�
DB
E�
m�
3 �
m�
2,it
fo
llow
s th
at m
�1
�m
�2
�m
�3
�m
�2.
Su
btr
act
m�
2 fr
om
eac
h s
ide
of
the
last
ineq
ual
ity
to g
et
m�
1 �
m�
3.
6.W
rite
a t
wo-
colu
mn
pro
of.
Giv
en:B �
A��
D�A�
BC
�D
CP
rove
:m�
1 �
m�
2
Pro
of:
Sta
tem
ents
Rea
son
s
1.B�
A��
D�A�
1.G
iven
2.B
C�
DC
2.G
iven
3.A�
C��
A�C�
3.R
efle
xive
Pro
per
ty4.
m�
1 �
m�
24.
SS
S In
equ
alit
y
1 2
B
A
D
C
12
3
B
AD
C
E
95�
77
85�
PR
SQ31
30
2222
RS
UT
6
98
3
3
B
AC
D
X
©G
lenc
oe/M
cGra
w-H
ill27
2G
lenc
oe G
eom
etry
Wri
te a
n i
neq
ual
ity
rela
tin
g th
e gi
ven
pai
r of
an
gles
or
segm
ent
mea
sure
s.
1.A
B,B
K2.
ST
,SR
AB
�B
KS
T�
SR
3.m
�C
DF
,m�
ED
F4.
m�
R,m
�T
m�
CD
F�
m�
ED
Fm
�R
�m
�T
5.W
rite
a t
wo-
colu
mn
pro
of.
Giv
en:G
is t
he
mid
poin
t of
D �F�
.m
�1
�m
�2
Pro
ve:E
D�
EF
Pro
of:
Sta
tem
ents
Rea
son
s
1.G
is t
he
mid
po
int
of
D�F�.
1.G
iven
2.D�
G��
F�G�2.
Def
init
ion
of
mid
po
int
3.E�
G��
E�G�
3.R
efle
xive
Pro
per
ty4.
m�
1 �
m�
24.
Giv
en5.
ED
�E
F5.
SA
S In
equ
alit
y
6.TO
OLS
Reb
ecca
use
d a
spri
ng
clam
p to
hol
d to
geth
er a
ch
air
leg
she
repa
ired
wit
h w
ood
glu
e.W
hen
sh
e op
ened
th
e cl
amp,
she
not
iced
th
at t
he
angl
e be
twee
n t
he
han
dles
of
the
clam
pde
crea
sed
as t
he
dist
ance
bet
wee
n t
he
han
dles
of
the
clam
pde
crea
sed.
At
the
sam
e ti
me,
the
dist
ance
bet
wee
n t
he
grip
pin
g en
ds o
f th
e cl
amp
incr
ease
d.W
hen
sh
e re
leas
ed t
he
han
dles
,th
e di
stan
ce b
etw
een
th
e gr
ippi
ng
end
of t
he
clam
p de
crea
sed
and
the
dist
ance
bet
wee
n t
he
han
dles
in
crea
sed.
Is t
he
clam
p an
exa
mpl
e of
th
e S
AS
or
SS
S I
neq
ual
ity?
SA
S In
equ
alit
y
12
DF
E
G
2021
RT
S
JK
1414
14
13
12C
F
E
D
( x �
3) �
( x �
3) �
1010
RT
S
Q
40�
30�
60�
AK
M
B
Pra
ctic
e (
Ave
rag
e)
Ineq
ual
itie
s In
volv
ing
Tw
o T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
5-5
5-5
Answers (Lesson 5-5)
© Glencoe/McGraw-Hill A16 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csIn
equ
alit
ies
Invo
lvin
g T
wo
Tri
ang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
5-5
5-5
©G
lenc
oe/M
cGra
w-H
ill27
3G
lenc
oe G
eom
etry
Lesson 5-5
Pre-
Act
ivit
yH
ow d
oes
a b
ack
hoe
wor
k?
Rea
d th
e in
trod
uct
ion
to
Les
son
5-5
at
the
top
of p
age
267
in y
our
text
book
.
Wh
at i
s th
e m
ain
kin
d of
tas
k th
at b
ackh
oes
are
use
d to
per
form
?B
ackh
oes
are
use
d m
ain
ly f
or
dig
gin
g.
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.W
rite
a c
oncl
usi
on t
hat
you
can
dra
w f
rom
th
e gi
ven
in
form
atio
n.
Th
en n
ame
the
theo
rem
th
at ju
stif
ies
you
r co
ncl
usi
on.
a.L �
M��
O�P�
,M�N�
�P�
Q�,a
nd
LN
�O
Qm
�M
�m
�P
;S
SS
Ineq
ual
ity
Th
eore
mb
.L�
M��
O�P�
,M�N�
�P�
Q�,a
nd
m�
P�
m�
MO
Q�
LN
(or
LN
�O
Q);
SA
S In
equ
alit
y T
heo
rem
(o
r H
ing
e T
heo
rem
)c.
LM
�8,
LN
�15
,OP
�8,
OQ
�15
,m�
L�
22,a
nd
m�
O�
21M
N�
PQ
;S
AS
Ineq
ual
ity
Th
eore
m (
or
Hin
ge
Th
eore
m)
2.In
th
e fi
gure
,�E
FG
is i
sosc
eles
wit
h b
ase
F�G�
and
Fis
th
e m
idpo
int
of D �
G�.D
eter
min
e w
het
her
eac
h o
f th
e fo
llow
ing
is
a va
lid
con
clu
sion
th
at y
ou c
an d
raw
bas
ed o
n t
he
give
n
info
rmat
ion
.(W
rite
val
idor
in
vali
d.)
If
the
con
clu
sion
is
vali
d,id
enti
fy t
he
defi
nit
ion
,pro
pert
y,po
stu
late
,or
theo
rem
th
at
supp
orts
it.
a.�
3 �
�4
valid
;Is
osc
eles
Tri
ang
le T
heo
rem
b.
DF
�G
Fva
lid;
def
init
ion
of
mid
po
int
c.�
DE
Fis
iso
scel
es.
inva
lidd
.m
�3
�m
�1
valid
;E
xter
ior
An
gle
Ineq
ual
ity
Th
eore
me.
m�
2 �
m�
4va
lid;
Ext
erio
r A
ng
le In
equ
alit
y T
heo
rem
f.m
�2
�m
�3
valid
;S
ub
stit
uti
on
Pro
per
ty (
usi
ng
co
ncl
usi
on
s fr
om
par
ts
g.D
E�
EG
valid
;S
AS
Ineq
ual
ity
Th
eore
m (
or
Hin
ge
Th
eore
m)
a a
nd
e)
h.
DE
�F
Gin
valid
Hel
pin
g Y
ou
Rem
emb
er3.
A g
ood
way
to
rem
embe
r so
met
hin
g is
to
thin
k of
it
in c
oncr
ete
term
s.H
ow c
an y
ouil
lust
rate
th
e H
inge
Th
eore
m w
ith
eve
ryda
y ob
ject
s?S
amp
le a
nsw
er:
Pu
t tw
op
enci
ls o
n a
des
kto
p s
o t
hat
th
e er
aser
s to
uch
.As
you
incr
ease
or
dec
reas
e th
e m
easu
re o
f th
e an
gle
fo
rmed
by
the
pen
cils
,th
e d
ista
nce
bet
wee
n t
he
po
ints
of
the
pen
cils
incr
ease
s o
r d
ecre
ases
acc
ord
ing
ly.
FG
D
E
12
34
NQ
PM
LO
©G
lenc
oe/M
cGra
w-H
ill27
4G
lenc
oe G
eom
etry
Dra
win
g a
Dia
gra
mIt
is
use
ful
and
ofte
n n
eces
sary
to
draw
a d
iagr
am o
f th
e si
tuat
ion
bein
g de
scri
bed
in a
pro
blem
.Th
e vi
sual
izat
ion
of
the
prob
lem
is
hel
pfu
l in
th
e pr
oces
s of
pro
blem
sol
vin
g.
Th
e ro
ads
con
nec
tin
g th
e to
wn
s of
Kin
gs,
Ch
ana,
and
Hol
com
b f
orm
a t
rian
gle.
Dav
is J
un
ctio
n i
slo
cate
d i
n t
he
inte
rior
of
this
tri
angl
e.T
he
dis
tan
ces
from
Dav
is J
un
ctio
n t
o K
ings
,Ch
ana,
and
Hol
com
b a
re 3
km
,4
km
,an
d 5
km
,res
pec
tive
ly.J
ane
beg
ins
at H
olco
mb
an
dd
rive
s d
irec
tly
to C
han
a,th
en t
o K
ings
,an
d t
hen
bac
k t
oH
olco
mb
.At
the
end
of
her
tri
p,s
he
figu
res
she
has
tra
vele
d25
km
alt
oget
her
.Has
sh
e fi
gure
d t
he
dis
tan
ce c
orre
ctly
?
To
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
Tri
angl
e In
equ
alit
y T
heo
rem
,th
e di
stan
ce f
rom
Hol
com
b to
Ch
ana
is l
ess
than
9 k
m.S
imil
arly
,th
e di
stan
ce f
rom
Ch
ana
to K
ings
is
less
th
an 7
km
,an
d th
edi
stan
ce f
rom
Kin
gs t
o H
olco
mb
is l
ess
than
8 k
m.
Th
eref
ore,
Jan
e m
ust
hav
e tr
avel
ed l
ess
than
(9
�7
�8)
km
or
24
km v
ersu
s h
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
emen
ts i
s tr
ue.
Dra
w a
nd
lab
el a
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
alf
the
sum
of
the
len
gth
s of
th
e ot
her
tw
o si
des.
If B�
D�is
th
e al
titu
de,
then
it is
tru
e th
at B�
D��
A�C�
.T
hen
�B
DC
and
�B
DA
are
rig
ht
tria
ng
les.
By
Th
eore
m 6
-8,B
D�
BC
an
d B
D�
BA
.Usi
ng
Th
eore
m 6
-2,2
BD
�B
A�
BC
.Th
us,
BD
��1 2�
(BA
�B
C).
2.If
poi
nt
Qis
in
th
e in
teri
or o
f
AB
Can
d on
th
e an
gle
bise
ctor
of �
B,t
hen
Qis
equ
idis
tan
t fr
om A�
B�an
d C�
B�.(
Hin
t:D
raw
Q�D�
and
Q�E�
such
th
at Q�
D��
A�B�
and
Q�E�
�C�
B�.)
If Q
is o
n t
he
bis
ecto
r o
f �
B,Q�
D��
A�B�
,an
d
Q�E�
�C�
B�,t
hen
�Q
EB
��
QD
Bby
HA
.Th
us,
Q�E�
�Q�
D�by
CP
CT
C,w
hic
h m
ean
s th
at Q
iseq
uid
ista
nt
fro
m A�
B�an
d C�
B�.
CE
B
A
Q
D
AD
C
B
King
s
Davi
sJu
nctio
n
Chan
aHo
lcom
b
3 km
5 km
4 km
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
5-5
5-5
Exam
ple
Exam
ple
Answers (Lesson 5-5)
© Glencoe/McGraw-Hill A17 Glencoe Geometry
Chapter 5 Assessment Answer Key Form 1 Form 2APage 275 Page 276 Page 277
(continued on the next page)
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
© Glencoe/McGraw-Hill A18 Glencoe Geometry
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
© Glencoe/McGraw-Hill A19 Glencoe Geometry
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
© Glencoe/McGraw-Hill A20 Glencoe Geometry
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
© Glencoe/McGraw-Hill A21 Glencoe Geometry
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�
© Glencoe/McGraw-Hill A22 Glencoe Geometry
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
© Glencoe/McGraw-Hill A23 Glencoe Geometry
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.
© Glencoe/McGraw-Hill A24 Glencoe Geometry
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
© Glencoe/McGraw-Hill A25 Glencoe Geometry
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
© Glencoe/McGraw-Hill A26 Glencoe Geometry
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