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Notes
Triangles
Introduction
In this unit students identify and
classify triangles by various
methods. They learn how to test
for and prove triangle congruence
and how to write coordinate
proofs. Bisectors, medians, and
altitudes of triangles are
introduced. Students explore
inequalities and triangles and
learn to use indirect proof.
You can use triangles
and their properties to
model and analyze
many real-world
situations. In this
unit, you will learn
about relationships in
and among triangles,
including congruence
and similarity.
Students apply their knowledge
of ratios and proportions to
similar figures and scale factors.
They explore proportional parts
of triangles and proportional
relationships between similar
triangles. Finally, students focus
on right triangles. They use the
Pythagorean Theorem and are
introduced to trigonometric ratios.
About the Photograph The large
photograph is the Air Force
Academy Chapel in Colorado
Springs, Colorado. Its 17-spire
structure is designed to resemble
aircraft in synchronized flight.
Have students use the Internet to
find out more about this building.
Chapter 4
Congruent Triangles
Chapter 5
Assessment Options
Relationships in Triangles
Unit 2 Test Pages 413–414
of the Chapter 7 Resource Masters
may be used as a test or review
for Unit 2. This assessment contains both multiple-choice and
short answer items.
Chapter 6
Proportions and Similarity
Chapter 7
Right Triangles and Trigonometry
174 Unit 2 Triangles
ExamView® Pro
This CD-ROM can be used to
create additional unit tests and
review worksheets.
174
Unit 2 Triangles
An online, research-based, instructional,
assessment, and intervention tool that provides
specific feedback on student mastery of state and
national standards, instant remediation, and a
management system to track performance. For
more information, contact mhdigitallearning.com.
Real-Life Geometry Videos
What’s Math Got to Do With It?
Real-Life Geometry Videos engage
students, showing them how math
is used in everyday situations. Use
Video 2 with this unit.
Teaching
Suggestions
Have students study the
USA TODAY Snapshot.
• Ask them which region of
the United States has the
lowest percentage of children
using the Internet. South
• Have students speculate
about why the Northeast has
the greatest percentage of
children using the Internet.
• What percentage of children
in the Midwest use the
Internet? 32.1%
Who Is Behind This
Geometry Concept Anyway?
USA TODAY Snapshots®
Have you ever wondered who first developed
some of the ideas you are learning in your geometry
class? Today, many students use the Internet for
learning and research. In this project, you will be
using the Internet to research a topic in geometry.
You will then prepare a portfolio or poster to display
your findings.
Where children use the Internet
Percentage of children 3-17
using the Internet at home,
by region:
Northeast
35.5%
Midwest
West
Lesson
Page
4-6
218
5-1
241
6-6
325
7-1
347
32.1%
29.0%
Log on to www.geometryonline.com/webquest.
Begin your WebQuest by reading the Task.
Continue working on
your WebQuest as you
study Unit 2.
Additional USA TODAY
Snapshots appearing in Unit 2:
Chapter 4 Gross Domestic
Product slides in
2001 (p. 206)
Chapter 5 Sources of college
information (p. 259)
Chapter 6 Workplace manners
declining (p. 296)
Chapter 7 Bruins bring skills
to Major League
Soccer (p. 347)
South
27.6%
Source: Census Bureau
By Sam Ward, USA TODAY
Unit 2 Triangles
175
Internet Project
Problem-Based Learning A WebQuest is an online project in which
students do research on the Internet, gather data, and make presentations
using word processing, graphing, page-making, or presentation software. In
each chapter, students advance to the next step in their WebQuest. At the
end of Chapter 7, the project culminates with a presentation of their findings.
Teaching notes and sample answers are available in the WebQuest and
Project Resources.
Unit 2 Triangles 175
Congruent Triangles
Chapter Overview and Pacing
Year-long pacing: pages T20–T21.
PACING (days)
Regular
Block
LESSON OBJECTIVES
Basic/
Average
Advanced
Basic/
Average
Advanced
1
1
0.5
0.5
2 (with 4-2
Preview)
1
1 (with 4-2
Preview)
0.5
Congruent Triangles (pp. 192–198)
• Name and label corresponding parts of congruent triangles.
• Identify congruence transformations.
2
2
1
1
Proving Congruence—SSS, SAS (pp. 200–206)
• Use the SSS Postulate to test for triangle congruence.
• Use the SAS Postulate to test for triangle congruence.
2
2
1
1
Classifying Triangles (pp. 178–183)
• Identify and classify triangles by angles.
• Identify and classify triangles by sides.
Angles of Triangles (pp. 184–191)
Preview: Use a model to find the relationships among the measures of the interior
angles of a triangle.
• Apply the Angle Sum Theorem.
• Apply the Exterior Angle Theorem.
2 (with 4-5 2 (with 4-5 1 (with 4-5 1 (with 4-5
Follow-Up) Follow-Up) Follow-Up) Follow-Up)
Proving Congruence—ASA, AAS (pp. 207–215)
• Use the ASA Postulate to test for triangle congruence.
• Use the AAS Theorem to test for triangle congruence.
Follow-Up: Use models to explore congruence in right triangles.
Isosceles Triangles (pp. 216–221)
• Use properties of isosceles triangles.
• Use properties of equilateral triangles.
2
2
1
1
Triangles and Coordinate Proof (pp. 222–226)
• Position and label triangles for use in coordinate proofs.
• Write coordinate proofs.
2
2
1
1
Study Guide and Practice Test (pp. 227–231)
Standardized Test Practice (pp. 232–233)
1
1
1
0.5
Chapter Assessment
1
1
0.5
0.5
15
14
8
7
TOTAL
An electronic version of this chapter is available on StudentWorksTM. This backpack solution CD-ROM
allows students instant access to the Student Edition, lesson worksheet pages, and web resources.
176A Chapter 4 Congruent Triangles
Timesaving Tools
™
All-In-One Planner
and Resource Center
Chapter Resource Manager
See pages T5 and T21.
183–184
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CHAPTER 4 RESOURCE MASTERS
1–2
239
239, 241
81–84
Materials
4-1
4-1
patty paper, protractor, grid paper
GCC 23
4-2
4-2
(Preview: protractor, scissors)
straightedge
SC 7
4-3
4-3
straightedge
4-4
4-4
straightedge, compass, scissors
4-5
4-5
4-6
4-6
ruler, scissors, protractor,
tracing paper
4-7
4-7
grid paper, straightedge
1–2
SC 8
10
straightedge, compass, scissors,
patty paper, protractor, ruler
(Follow-Up: ruler, protractor)
225–238,
242–244
*Key to Abbreviations: GCC Graphing Calculator and Computer Masters
SC School-to-Career Masters
Chapter 4 Congruent Triangles
176B
Mathematical Connections
and Background
Continuity of Instruction
Prior Knowledge
In Chapter 1, students calculated distance
between points on the coordinate plane
using the Distance Formula. In Chapter 3,
they learned that an intersection between
parallel lines and a transversal creates a
variety of congruent and supplementary
angles.
Classifying Triangles
Triangles can be classified based on their angle
measures. In an acute triangle, all of the angles are acute.
In an obtuse triangle, one of the angles is obtuse. In a
right triangle, one angle measures 90. When all of the
angles of an acute triangle are congruent, it is called an
equiangular triangle.
Triangles can also be classified according to
their number of congruent sides. No two sides of a
scalene triangle are congruent. At least two sides of an
isosceles triangle are congruent. All of the sides of an
equilateral triangle are congruent. Equilateral triangles
are a special kind of isosceles triangle.
Angles of Triangles
This Chapter
In this chapter, students prove triangles
congruent using SSS, SAS, ASA, and AAS.
They also learn how to write two new types
of proofs, the flow proof and the coordinate
proof. Students classify triangles according to
their angles or sides and apply the Angle Sum
Theorem and the Exterior Angle Theorem.
The special properties of isosceles and equilateral triangles are introduced, and students
are expected to use those properties in proofs.
Students also position and label triangles for
use in coordinate proofs.
The Angle Sum Theorem states that the sum of
the measures of the interior angles of a triangle is always
180. This theorem can be applied to any triangle. It also
leads to the Third Angle Theorem: If two angles of one
triangle are congruent to two angles of a second triangle,
then the third angles of the triangles are congruent.
Each angle of a triangle has an exterior angle,
which is formed by one side of the triangle and the
extension of another side. The interior angles of the
triangle not adjacent to a given exterior angle are called
remote interior angles. The measure of an exterior angle
of a triangle is equal to the sum of the measures of the
two remote interior angles. This is the Exterior Angle
Theorem.
This lesson also introduces flow proofs. A flow
proof organizes a series of statements in logical order.
Arrows are used to indicate the order of the statements.
Flow proofs can be written horizontally or vertically.
Congruent Triangles
Future Connections
Students will need the knowledge gained
from this chapter to master the skills taught
later in this course. They must understand
triangle congruence to be successful scholars
of trigonometry, a precursor to calculus and
all courses in higher math.
176C Chapter 4 Congruent Triangles
Two triangles are congruent if and only if their
corresponding parts are congruent. Certain transformations, including a slide, flip, and turn, do not affect
congruence. These transformations are called congruence
transformations.
Congruence of triangles, like that of angles and
segments, is reflexive, symmetric, and transitive.
Proving Congruence—SSS, SAS
Isosceles Triangles
In this lesson you will construct a triangle in
which three sides are congruent to the three sides of a
given triangle. This activity demonstrates the SideSide-Side Postulate. Also written as SSS, it states that
if the sides of one triangle are congruent to the sides
of a second triangle, then the triangles are congruent.
You will also construct a triangle in which two
sides are congruent to two sides of a given triangle and
the included angle is congruent to the included angle
in the given triangle. This activity demonstrates the
Side-Angle-Side Postulate, also written SAS. It states
that if two sides and the included angle of one triangle
are congruent to two sides and the included angle of
another triangle, then the triangles are congruent.
Isosceles triangles have special terminology for
their parts. The angle formed by the congruent sides
is called the vertex angle. The two angles formed by
the base and one of the congruent sides are called base
angles. The congruent sides are called legs. Isosceles
triangles also have special properties recognized in
Isosceles Triangle Theorem and its converse: If two
sides of a triangle are congruent, then the angles
opposite those sides are congruent.
This theorem leads to corollaries about the
angles of an equilateral triangle. The first states that a
triangle is equilateral if and only if it is equiangular.
The second states that each angle of an equilateral
triangle measures 60°.
Proving Congruence—ASA, AAS
Triangles and Coordinate Proof
The Angle-Side-Angle Postulate, written as
ASA, works because the measures of two angles of a
triangle and the side between them form a unique
triangle. The postulate states that if two angles and
the included side of one triangle are congruent to two
angles and the included side of another triangle, then
the triangles are congruent.
The Angle-Angle-Side, or AAS, Theorem
follows from the ASA Postulate: If two angles and a
nonincluded side of one triangle are congruent to the
corresponding two angles and side of a second
triangle, then the two triangles are congruent.
Right triangles have their own theorems to
prove congruence. One of those is the LL Congruence
Theorem, which is the SAS Postulate applied to right
triangles. It states that if the legs of one right triangle
are congruent to the corresponding legs of another
right triangle, then the triangles are congruent. The
HA Theorem is based on the AAS Theorem: If the
hypotenuse and acute angle of one right triangle are
congruent to the hypotenuse and corresponding acute
angle of another right triangle, then the two triangles
are congruent. The LA Theorem states that if one leg
and an acute angle of one right triangle are congruent
to the corresponding leg and acute angle of another
right triangle, then the triangles are congruent. The HL
Postulate is based on SSA, a test that only works for
right triangles. It states that if the hypotenuse and leg
of one right triangle are congruent to the hypotenuse
and corresponding leg of another right triangle, then
the triangles are congruent.
The coordinate plane can be used in
combination with algebra in a new method of proof
called coordinate proof. Before beginning a coordinate proof, you will need to place the figure in the
coordinate plane. It is important that you use coordinates that make computation as simple as possible.
Using the origin as a vertex or center will help, and
you should place at least one side of a polygon on an
axis. If possible, keep the figure within the first
quadrant.
Once the triangle is placed, you can proceed
with the proof. The Distance Formula, Slope Formula,
and Midpoint Formula are often used in coordinate
proof.
Chapter 4 Congruent Triangles
176D
and Assessment
Key to Abbreviations:
TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters
ASSESSMENT
INTERVENTION
Type
Student Edition
Teacher Resources
Ongoing
Prerequisite Skills, pp. 177, 183,
191, 198, 206, 213, 221
Practice Quiz 1, p. 198
Practice Quiz 2, p. 221
5-Minute Check Transparencies
Prerequisite Skills Workbook, pp. 1–2, 81–84
Quizzes, CRM pp. 239–240
Mid-Chapter Test, CRM p. 241
Study Guide and Intervention, CRM pp. 183–184,
189–190, 195–196, 201–202, 207–208,
213–214, 219–220
Mixed
Review
pp. 183, 191, 198, 206, 213,
221, 226
Cumulative Review, CRM p. 242
Error
Analysis
Find the Error, pp. 188, 203
Common Misconceptions, p. 178
Find the Error, TWE pp. 189, 203
Unlocking Misconceptions, TWE p. 202
Tips for New Teachers, TWE p. 210
Standardized
Test Practice
pp. 183, 191, 198, 206, 213,
217, 219, 221, 226, 231, 232,
233
TWE pp. 232–233
Standardized Test Practice, CRM pp. 243–244
Open-Ended
Assessment
Writing in Math, pp. 183, 191,
198, 205, 213, 221, 226
Open Ended, pp. 180, 188, 195,
203, 210, 219, 224
Standardized Test, p. 233
Modeling: TWE p. 198
Speaking: TWE pp. 183, 206, 221, 226
Writing: TWE pp. 191, 213
Open-Ended Assessment, CRM p. 237
Chapter
Assessment
Study Guide, pp. 227–230
Practice Test, p. 231
Multiple-Choice Tests (Forms 1, 2A, 2B),
CRM pp. 225–230
Free-Response Tests (Forms 2C, 2D, 3),
CRM pp. 231–236
Vocabulary Test/Review, CRM p. 238
For more information on
Yearly ProgressPro, see p. 174.
Geometry Lesson
4-1
4-2
4-3
4-4
4-5
4-6
4-7
Yearly ProgressPro Skill Lesson
Classifying Triangles
Angles of Triangles
Congruent Triangles
Proving Congruence—SSS, SAS
Proving Congruence—ASA, AAS
Isosceles Triangles
Triangles and Coordinate Proof
GeomPASS: Tutorial Plus,
Lesson 10
www.geometryonline.com/
self_check_quiz
www.geometryonline.com/
extra_examples
Standardized Test Practice
CD-ROM
www.geometryonline.com/
standardized_test
ExamView® Pro (see below)
MindJogger Videoquizzes
www.geometryonline.com/
vocabulary_review
www.geometryonline.com/
chapter_test
ExamView® Pro
Use the networkable ExamView® Pro to:
• Create multiple versions of tests.
• Create modified tests for Inclusion students.
• Edit existing questions and add your own questions.
• Use built-in state curriculum correlations to create
tests aligned with state standards.
• Apply art to your test from a program bank of artwork.
For more information on Intervention and Assessment, see pp. T8–T11.
176E Chapter 4 Congruent Triangles
Technology/Internet
Reading and Writing in Mathematics
Glencoe Geometry provides numerous opportunities to incorporate reading and writing
into the mathematics classroom.
Student Edition
Additional Resources
• Foldables Study Organizer, p. 177
• Concept Check questions require students to verbalize
and write about what they have learned in the lesson.
(pp. 180, 188, 195, 203, 210, 219, 224)
• Reading Mathematics, p. 199
• Writing in Math questions in every lesson, pp. 183, 191,
198, 205, 213, 221, 226
• Reading Study Tip, pp. 186, 207
• WebQuest, p. 216
Teacher Wraparound Edition
• Foldables Study Organizer, pp. 177, 227
• Study Notebook suggestions, pp. 181, 184, 188, 194,
199, 203, 210, 214, 219, 224
• Modeling activities, p. 198
• Speaking activities, pp. 183, 206, 221, 226
• Writing activities, pp. 191, 213
• ELL Resources, pp. 176, 182, 190, 197, 199, 205,
212, 220, 225, 227
• Vocabulary Builder worksheets require students to
define and give examples for key vocabulary terms as
they progress through the chapter. (Chapter 4 Resource
Masters, pp. vii-viii)
• Proof Builder helps students learn and understand
theorems and postulates from the chapter. (Chapter 4
Resource Masters, pp. ix–x)
• Reading to Learn Mathematics master for each lesson
(Chapter 4 Resource Masters, pp. 187, 193, 199, 205,
211, 217, 223)
• Vocabulary PuzzleMaker software creates crossword,
jumble, and word search puzzles using vocabulary lists
that you can customize.
• Teaching Mathematics with Foldables provides
suggestions for promoting cognition and language.
• Reading Strategies for the Mathematics Classroom
• WebQuest and Project Resources
For more information on Reading and Writing in Mathematics, see pp. T6–T7.
Lesson 4-1
Lesson 4-2
Lesson 4-7
Building on Prior
Knowledge
Using Manipulatives
Flexible Groups
If possible, pair each English Language
Learner with a bilingual student. Give
each pair of students posterboard to cut
into several right triangles in different
sizes, each with the right angle labeled.
Have students measure the acute angles
with a protractor. Ask them to explain in
their own words why Corollaries 4.1 and
4.2 are true.
Understanding how to best place a triangle
on the coordinate plane can be a difficult
concept for students. Divide the class into
small groups. Give each group a coordinate
plane made from posterboard, and several
different triangles (isosceles, equilateral,
right triangles, scalene, etc.). Have each
group find as many ways as possible to
place each triangle on the coordinate plane.
Then have each group name the coordinates
of the vertices of the triangles.
Students often get confused as to what
scale to use when measuring with a
protractor. Ask students to label the angles
as acute or obtuse before they begin to
measure. Then they can determine which
measure is reasonable for the angle they
are measuring.
Chapter 4 Congruent Triangles
176F
Congruent
Triangles
Notes
Have students read over the list
of objectives and make a list of
any words with which they are
not familiar.
Point out to students that this is
only one of many reasons why
each objective is important.
Others are provided in the
introduction to each lesson.
• Lesson 4-1 Classify triangles.
• Lesson 4-2 Apply the Angle Sum Theorem
and the Exterior Angle Theorem.
• Lesson 4-3 Identify corresponding parts of
congruent triangles.
• Lessons 4-4 and 4-5 Test for triangle
congruence using SSS, SAS, ASA, and AAS.
• Lesson 4-6 Use properties of isosceles and
equilateral triangles.
• Lesson 4-7 Write coordinate proofs.
Key Vocabulary
•
•
•
•
•
exterior angle (p. 186)
flow proof (p. 187)
corollary (p. 188)
congruent triangles (p. 192)
coordinate proof (p. 222)
Triangles are found everywhere you look. Triangles with the
same size and shape can even be found on the tail of a whale.
You will learn more about orca whales in Lesson 4-4.
Lesson
4-1
4-2
Preview
4-2
4-3
4-4
4-5
4-5
Follow-Up
4-6
4-7
NCTM
Standards
Local
Objectives
3, 6, 8, 9, 10
3, 6, 10
3, 6, 8, 9, 10
3, 6, 8, 9, 10
3, 6, 7, 8, 9, 10
3, 6, 7, 8, 9, 10
3, 7, 10
3, 6, 7, 8, 9, 10
3, 6, 7, 8, 9, 10
Key to NCTM Standards:
1=Number & Operations, 2=Algebra,
3=Geometry, 4=Measurement,
5=Data Analysis & Probability, 6=Problem
Solving, 7=Reasoning & Proof,
8=Communication, 9=Connections,
10=Representation
176
Chapter 4 Congruent Triangles
176 Chapter 4 Congruent Triangles
Daniel J. Cox/Getty Images
Vocabulary Builder
ELL
The Key Vocabulary list introduces students to some of the main vocabulary terms
included in this chapter. For a more thorough vocabulary list with pronunciations of
new words, give students the Vocabulary Builder worksheets found on pages vii and
viii of the Chapter 4 Resource Masters. Encourage them to complete the definition
of each term as they progress through the chapter. You may suggest that they add
these sheets to their study notebooks for future reference when studying for the
Chapter 4 test.
Prerequisite Skills To be successful in this chapter, you’ll need to master
these skills and be able to apply them in problem-solving situations. Review
these skills before beginning Chapter 4.
For Lesson 4-1
Solve Equations
Solve each equation. (For review, see pages 737 and 738.)
1
1
1. 2x 18 5 ⫺6
2. 3m 16 12 9
2
23
3. 4y 12 16 1
4. 10 8 3z ⫺
3
1 3
2
5. 6 2a 2
6. b 9 15 ⫺36
2 4
3
For Lessons 4-2, 4-4, and 4-5
Congruent Angles
Name the indicated angles or pairs of angles if p 㛳 q and m 㛳 ᐉ.
(For review, see Lesson 3-1.)
5 6
9 1
7. angles congruent to ⬔8 ⬔2, ⬔12, ⬔15, ⬔6, ⬔9, ⬔3, ⬔13
7 8
2 10
11 3
13
8. angles congruent to ⬔13 ⬔2, ⬔12, ⬔15, ⬔6, ⬔9, ⬔3, ⬔8
⬔12, ⬔15
9. angles supplementary to ⬔1 ⬔6, ⬔9, ⬔3, ⬔13, ⬔2, ⬔8,
⬔7, ⬔10
10. angles supplementary to ⬔12 ⬔4, ⬔16, ⬔11, ⬔14, ⬔5, ⬔1,
For Lessons 4-3 and 4-7
m
4 12
15 16
14
p
ᐉ
q
This section provides a review of
the basic concepts needed before
beginning Chapter 4. Page
references are included for
additional student help.
Additional review is provided in
the Prerequisite Skills Workbook,
pages 1–2, 81–84.
Prerequisite Skills in the Getting
Ready for the Next Lesson section
at the end of each exercise set
review a skill needed in the next
lesson.
For
Lesson
Prerequisite
Skill
4-2
Angles Formed by Parallel
Lines and a Transversal, p. 183
Properties of Congruence,
p. 191
Distance Formula, p. 198
Bisectors of Segments and
Angles, p. 206
Classification of triangles by
sides, p. 213
Finding Midpoints, p. 221
Distance Formula
4-3
Find the distance between each pair of points. Round to the nearest tenth.
(For review, see Lesson 1-3.)
11. (6, 8), (4, 3) 11.2
12. (15, 12), (6, 18) 21.8
13. (11, 8), (3, 4) 14.6
14. (10, 4), (8, 7) 21.1
4-4
4-5
4-6
Triangles Make this Foldable to help you organize your notes. Begin with two
4-7
sheets of grid paper and one sheet of construction paper.
Fold and Cut
Staple and Label
Staple the edge
to form a booklet.
Then label each
page with a
lesson number
and title.
Stack the grid
paper on the
construction
paper. Fold
diagonally as
shown and cut
off the excess.
Congruent
Triangles
Reading and Writing As you read and study the chapter, use your journal for sketches and examples of
terms associated with triangles and sample proofs.
Chapter 4 Congruent Triangles 177
TM
For more information
about Foldables, see
Teaching Mathematics
with Foldables.
Writer’s Journal Use this Foldable for student writing about triangles.
After students make their Foldable journal, have them label the
pages to correspond to the seven lessons in this chapter. Students
use their Foldable to take notes, define terms, record concepts, and
write examples. Writer’s journals can also be used by students to
record the direction and progress of learning, describe positive and
negative experiences during learning, write about personal associations
and experiences called to mind during learning, and to list examples
of ways in which new knowledge has or will be used in their
daily lives.
Chapter 4 Congruent Triangles 177
Lesson
Notes
Classifying Triangles
• Identify and classify triangles by angles.
1 Focus
5-Minute Check
Transparency 4-1 Use as a
quiz or review of Chapter 3.
Mathematical Background notes
are available for this lesson on
p. 176C.
are triangles important
in construction?
Ask students:
• Why does the shape of a triangle
offer good support for a roof?
As gravity and the weight of roof
material act on the two top legs of
the triangle, the third leg prevents
them from collapsing or spreading
apart. Triangular supports within
the truss offer even more support.
• How would you classify the
angles of a triangular truss
with angle measures 96°, 42°,
and 42°? obtuse, acute, acute
• Identify and classify triangles by sides.
Vocabulary
•
•
•
•
•
•
•
acute triangle
obtuse triangle
right triangle
equiangular triangle
scalene triangle
isosceles triangle
equilateral triangle
are triangles important
in construction?
Many structures use triangular shapes as
braces for construction. The roof sections of
houses are made of triangular trusses that
support the roof and the house.
CLASSIFY TRIANGLES BY ANGLES Recall that a triangle is a three-sided
polygon. Triangle ABC, written ABC, has parts that are named using the letters
A, B, and C.
B
B, B
C, and C
A.
• The sides of ABC are A
• The vertices are A, B, and C.
• The angles are ABC or B, BCA or C,
A
C
and BAC or A.
There are two ways to classify triangles. One way is by their angles. All triangles
have at least two acute angles, but the third angle is used to classify the triangle.
Study Tip
Common
Misconceptions
These classifications are
distinct groups. For
example, a triangle cannot
be right and acute.
Classifying Triangles by Angles
In an acute triangle , all
of the angles are acute.
In an obtuse triangle ,
one angle is obtuse.
67˚
42˚
13˚
37˚
76˚
all angle measures 90
In a right triangle , one
angle is right.
142˚
25˚
one angle measure 90
90˚
48˚
one angle measure 90
An acute triangle with all angles congruent is an equiangular triangle .
Example 1 Classify Triangles by Angles
ARCHITECTURE The roof of this house is
made up of three different triangles. Use a
protractor to classify DFH, DFG, and
HFG as acute, equiangular, obtuse, or right.
DFH has all angles with measures
less than 90, so it is an acute triangle.
DFG and HFG both have one angle with
measure equal to 90. Both of these are right
triangles.
178
Chapter 4 Congruent Triangles
(t)Martin Jones/CORBIS, (b)David Scott/Index Stock
Resource Manager
Workbook and Reproducible Masters
Chapter 4 Resource Masters
• Study Guide and Intervention, pp. 183–184
• Skills Practice, p. 185
• Practice, p. 186
• Reading to Learn Mathematics, p. 187
• Enrichment, p. 188
Prerequisite Skills Workbook, pp. 1–2
Teaching Geometry With Manipulatives
Masters, pp. 1, 16, 69
Transparencies
5-Minute Check Transparency 4-1
Answer Key Transparencies
Technology
Interactive Chalkboard
F
D
G
H
CLASSIFY TRIANGLES BY SIDES Triangles can also be classified according
to the number of congruent sides they have. To indicate that sides of a triangle are
congruent, an equal number of hash marks are drawn on the corresponding sides.
Classifying Triangles by Sides
No two sides of a
scalene triangle are
congruent.
At least two sides of an
isosceles triangle are
congruent.
All of the sides of an
equilateral triangle are
congruent.
2 Teach
CLASSIFY TRIANGLES
BY ANGLES
In-Class Example
Power
Point®
1 ARCHITECTURE The
triangular truss below is
modeled for steel construction.
Use a protractor to classify
JMN, JKO, and OLN as
acute, equiangular, obtuse or
right.
An equilateral triangle is a special kind of isosceles triangle.
M
L
1. Yes, all edges of the paper are an equal length.
Equilateral Triangles
Model
Analyze 2–3. See students’ work.
• Align three pieces of patty
paper as indicated. Draw
a dot at X.
• Fold the patty paper
through X and Y and
through X and Z.
1. Is XYZ equilateral? Explain.
J
2. Use three pieces of patty paper to make a
triangle that is isosceles, but not equilateral.
X
Y
K
Z
Example 2 Classify Triangles by Sides
CLASSIFY TRIANGLES
BY SIDES
A
In-Class Examples
B
C
Power
Point®
Teaching Tip
Explain to
students that all equilateral
triangles are isosceles, but not all
isosceles triangles are equilateral.
D
E
Example 3 Find Missing Values
2 Identify the indicated
ALGEBRA Find x and the measure of each side of equilateral
triangle RST if RS x 9, ST 2x, and RT 3x 9.
Since RST is equilateral, RS ST.
R
triangles in the figure if
V
VX
UX
.
U
S
x+9
x 9 2x Substitution
9 x Subtract x from each side.
V
2x
3x – 9
Y T
T
Next, substitute to find the length of each side.
RS x 9
ST 2x
RT 3x 9
9 9 or 18
2(9) or 18
3(9) 9 or 18
For RST, x 9, and the measure of each side is 18.
www.geometryonline.com/extra_examples
N
JMN is obtuse. JKO is right.
OLN is equiangular.
3. Use three pieces of patty paper to make a
scalene triangle.
Identify the indicated type of triangle in the figure.
a. isosceles triangles
b. scalene triangles
Isosceles triangles have
Scalene triangles have
at least two sides
no congruent sides.
congruent. So, ABD
AEB, AED, ACB,
and EBD are isosceles.
ACD, BCE, and
DCE are scalene.
O
Lesson 4-1 Classifying Triangles 179
Geometry Activity
Materials: patty paper, pencil
• Have students use a protractor to measure the angles of XYZ, and ask them to
consider whether an equilateral triangle could be obtuse or right.
• Students can fold a corner of the patty paper to form a right triangle. They can also
use a ruler and protractor to draw and label different combinations of triangles,
such as an acute scalene triangle, an obtuse isosceles triangle, and so on.
U
W
Z
X
a. isosceles triangles UTX, UVX
b. scalene triangles VYX, ZTX,
VZU, YTU, VWX, ZUX,
YXU
3 ALGEBRA Find d and the
measure of each side of
equilateral triangle KLM if
KL d 2, LM 12 d,
and KM 4d 13. d 5 and
the measure of each side is 7.
Lesson 4-1 Classifying Triangles 179
In-Class Example
Example 4 Use the Distance Formula
Power
Point®
4 COORDINATE GEOMETRY
Find the measures of the
sides of RST. Classify the
triangle by sides.
Look Back
COORDINATE GEOMETRY Find the measures of the
sides of DEC. Classify the triangle by sides.
Use the Distance Formula to find the lengths of each side.
To review the Distance
Formula, see Lesson 1-3.
2)2 (3 2
)2
EC (5
Study Tip
y
S (4, 4)
y
D
ED (5 3)2 (3 9
)2
49 1
64 36
50
100
E
C
O
DC (3 2
)2 (9
2)2
x
1 49
O
50
x
Since E
C and D
C have the same length, DEC is isosceles.
T
(8,–1)
R
(–1, –3)
RS 74
; ST 41
;
RT 85
; RST is scalene.
Concept Check
1. Explain how a triangle can be classified in two ways. 1–2. See margin.
2. OPEN ENDED
Determine whether each of the following statements is always, sometimes, or
never true. Explain. 3. Always; equiangular triangles have three acute angles.
3. Equiangular triangles are also acute.
4. Right triangles are acute. Never; right triangles have one right angle and acute
Answers
1. Triangles are classified by sides
and angles. For example, a
triangle can have a right angle
and have no two sides congruent.
2. Sample answer:
Draw a triangle that is isosceles and right.
triangles have all acute angles.
Guided Practice
GUIDED PRACTICE KEY
Exercises
Examples
5–8
9, 10
11
12
1
3
4
2
Use a protractor to classify each triangle as acute, equiangular, obtuse, or right.
obtuse 6.
equiangular
5.
7. Identify the obtuse triangles if
MJK KLM, mMJK 126,
and mJNM 52.
J
K
N
M
8. Identify the right triangles if
, G
H
DF
, and G
I EF
.
IJ 㛳 GH
E
MJK,
KLM,
JKN,
LMN
G
D H
L
9. ALGEBRA Find x, JM, MN, and
JN if JMN is an isosceles triangle
with JM
M
N
.
J
J
R
4x
3x – 9
x–2
F
10. ALGEBRA Find x, QR, RS, and QS
if QRS is an equilateral triangle.
M
2x – 5
GHD,
GHJ,
IJF,
EIG
I
N
x 4, JM 3, MN 3, JN 2
180 Chapter 4 Congruent Triangles
1
2
Q
2x + 1
6x – 1
Differentiated Instruction
Naturalist Certain forms of algae are triangular in structure. Three-sided
leaves are said to have a triangular shape. Some wings of birds and
insects are triangular. Blue spruce trees grow in a triangular shape. Cats
have triangular ears. Students can use these examples, find more
throughout the chapter, or come up with their own ideas, and classify
triangles found in nature.
180
Chapter 4 Congruent Triangles
S
x , QR 2, RS 2, QS 2
11. Find the measures of the sides of TWZ with vertices at T(2, 6), W(4, 5), and
Z(3, 0). Classify the triangle. TW 125 , WZ 74 , TZ 61; scalene
Application
12. QUILTING The star-shaped composite quilting square is
made up of four different triangles. Use a ruler to classify
the four triangles by sides. 8 scalene triangles (green),
8 isosceles triangles in the middle (blue), 4 isosceles
triangles around the middle (yellow) and 4 isosceles at
the corners of the square (purple)
Study Notebook
★ indicates increased difficulty
Practice and Apply
For
Exercises
See
Examples
13–18
19, 21–25
26–29
30, 31
32–37,
40, 41
1
1, 2
3
2
4
Use a protractor to classify each triangle as acute, equiangular, obtuse, or right.
right 14.
acute 15.
acute
13.
16.
Extra Practice
17.
obtuse 18.
right
About the Exercises…
19. ASTRONOMY On May 5, 2002, Venus,
Saturn, and Mars were aligned in a
triangular formation. Use a protractor
or ruler to classify the triangle formed
by sides and angles. equilateral, equiangular
Organization by Objective
• Classify Triangles by
Angles: 13–18, 19, 21
• Classify Triangles by Sides:
19, 21, 22–39
Mars
Saturn
Venus
20. RESEARCH Use the Internet or other resource to find out how astronomers can
predict planetary alignment. See students’ work.
21. ARCHITECTURE The restored and decorated Victorian houses in San Francisco
are called the “Painted Ladies.” Use a protractor to classify the triangles
indicated in the photo by sides and angles. isosceles, acute
22. AGB, AGC, DGB, DGC
The Painted Ladies are
located in Alamo Square.
The area is one of 11
designated historic
districts in San Francisco.
Source: www.sfvisitor.org
27. x 5, MN 9,
MP 9, NP 9
28. x 8, QR 14,
RS 14, QS 14
Have students—
• add the definitions/examples of
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 4.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
obtuse
See page 760.
Architecture
3 Practice/Apply
Identify the indicated type of triangles in the figure
if A
BD
DC
CA
BC
AD
B
and .
22. right
23. obtuse BAC, CDB
24. scalene AGB, AGC, 25. isosceles ABD, ACD,
DGB, DGC
B
A
G
D
BAC, CDB
ALGEBRA Find x and the measure of each side of the triangle.
C
26. GHJ is isosceles, with HG
JG
, GH x 7, GJ 3x 5,
and HJ x 1. x 6, GH 13, GJ 13, HJ 5
27. MPN is equilateral with MN 3x 6, MP x 4, and NP 2x 1.
28. QRS is equilateral. QR is two less than two times a number, RS is six more
than the number, and QS is ten less than three times the number.
LJ. JL is five less than two times a number. JK is
29. JKL is isosceles with KJ three more than the number. KL is one less than the number. Find the measure
of each side. x 8, JL 11, JK 11, KL 7
Odd/Even Assignments
Exercises 13–18, 22–29, and
32–41 are structured so that
students practice the same
concepts whether they are
assigned odd or even
problems.
Alert! Exercise 20 requires the
Internet or other research
materials.
Assignment Guide
Basic: 13–37 odd, 42–57
Average: 13–41 odd, 42–57
Advanced: 14–40 even, 42–52
(optional: 53–57)
Lesson 4-1 Classifying Triangles 181
Joseph Sohm/Stock Boston
Interactive
Chalkboard
PowerPoint®
Presentations
This CD-ROM is a customizable Microsoft® PowerPoint®
presentation that includes:
• Step-by-step, dynamic solutions of each In-Class Example
from the Teacher Wraparound Edition
• Additional, Try These exercises for each example
• The 5-Minute Check Transparencies
• Hot links to Glencoe Online Study Tools
Lesson 4-1 Classifying Triangles 181
NAME ______________________________________________ DATE
★ 30. CRYSTAL The top of the crystal bowl shown is
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
4-1
Study
Guide and
circular. The diameter at the top of the bowl is
MN
MN. P is the midpoint of M
N
, and O
P
.
If MN 24 and OP 12, determine whether MPO
and NPO are equilateral.
p. 183
(shown)
Classifying
Triangles and p. 184
Classify Triangles by Angles
One way to classify a triangle is by the measures
of its angles.
• If one of the angles of a triangle is an obtuse angle, then the triangle is an obtuse triangle.
• If one of the angles of a triangle is a right angle, then the triangle is a right triangle.
a.
Lesson 4-1
• If all three angles of an acute triangle are congruent, then the triangle is an equiangular triangle.
Classify each triangle.
A
B
60
E
120
35
D
25
F
The triangle has one angle that is obtuse. It is an obtuse triangle.
c.
G
90
H
60
30
J
The triangle has one right angle. It is a right triangle.
Exercises
Classify each triangle as acute, equiangular, obtuse, or right.
1. K
2. N
30
67
L
23
65 65
Lexington to Nashville, 265 miles from Cairo to Lexington, and 144 miles from
Cairo to Nashville.
Q
60
60
R
obtuse
60
S
equiangular
5. W
T
6.
B
60
45
50
X
V
acute
Gl
3.
O
P
M
right
4.
U
30
120
90
45
90
28
F
Y
right
92
D
obtuse
NAME
______________________________________________
DATE
/M
G
Hill
183
____________
Gl PERIOD
G _____
Skills
Practice,
4-1
Practice
(Average)
p. 185 and
Practice,
186 (shown)
Classifyingp.
Triangles
Use a protractor to classify each triangle as acute, equiangular, obtuse, or right.
1.
2.
3.
obtuse
acute
B
E
5. obtuse
ABC, CDE
BED, BDC
6. scalene
A
D
C
7. isosceles
ABC, CDE
32. AB 29
, BC 4, AC 29
;
isosceles
Find the measures of the sides of ABC and classify each triangle by its sides.
32. A(5, 4), B(3, 1), C(7, 1)
33. A(4, 1), B(5, 6), C(3, 7)
34. A(7, 9), B(7, 1), C(4, 1)
35. A(3, 1), B(2, 1), C(2, 3)
33. AB 106
,
BC 233
, AC 65; scalene
36. A(0, 5), B(53, 2), C(0,1)
37. A(9, 0), B5, 63
, C(1, 0)
ABD, BED, BDC
ALGEBRA Find x and the measure of each side of the triangle.
8. FGH is equilateral with FG x 5, GH 3x 9, and FH 2x 2.
x 7, FG 12, GH 12, FH 12
9. LMN is isosceles, L is the vertex angle, LM 3x 2, LN 2x 1, and MN 5x 2.
x 3, LM 7, LN 7, MN 13
Find the measures of the sides of KPL and classify each triangle by its sides.
10. K(3, 2) P(2, 1), L(2, 3)
KP 26
, PL 42
, LK 26
; isosceles
11. K(5, 3), P(3, 4), L(1, 1)
KP 53
, PL 5, LK 213
; scalene
★
★
34. AB 10, BC 11,
AC 221
; scalene
right
Identify the indicated type of triangles if
A
B
A
D
B
D
D
C
, BE
E
D
, AB
⊥B
C
, and ED
⊥D
C
.
4. right
38. PROOF Write a two-column
proof to prove that EQL is
equiangular. See p. 233A.
35. AB 29
,
BC 4, AC 29
;
isosceles
E
M
Q
N
P
L
R
U
I
37. AB 124
,
40. COORDINATE GEOMETRY
BC 124
, AC 8; ★
Show that S is the midpoint of
isosceles
R
T
and U is the midpoint of T
V
.
★ 41. COORDINATE GEOMETRY
Show that ADC is isosceles.
See p. 233A.
See p. 233A.
KP 210
, PL 52
, LK 52
; isosceles
T (4, 14) y
13. DESIGN Diana entered the design at the right in a logo contest
sponsored by a wildlife environmental group. Use a protractor.
How many right angles are there? 5
NAME
______________________________________________
DATE
/M
G
Hill
186
y
D
12
S (7, 8)
____________
Gl PERIOD
G _____
Reading
4-1
Readingto
to Learn
Learn Mathematics
ELL
Mathematics,
p. 187
Classifying Triangles
Pre-Activity
39. PROOF Write a paragraph proof
to prove that RPM is an obtuse
triangle if mNPM 33. See p. 233A.
33˚
36. AB 84
, BC 84, AC 6;
isosceles
12. K(2, 6), P(4, 0), L(3, 1)
Gl
O
31. MAPS The total distance from Nashville,
Lexington
Tennessee, to Cairo, Illinois, to Lexington,
Kentucky, and back to Nashville, Tennessee,
is 593 miles. The distance from Cairo to
Cairo
Lexington is 81 more miles than the
distance from Lexington to Nashville. The
Nashville
distance from Cairo to Nashville is 40 miles
less than the distance from Nashville to
Lexington. Classify the triangle formed by its sides. Scalene; it is 184 miles from
C
All three angles are congruent, so all three angles have measure 60°.
The triangle is an equiangular triangle.
b.
N
P
No, MO NO 288
• If all three of the angles of a triangle are acute angles, then the triangle is an acute triangle.
Example
M
U (0, 8)
R (10, 2)
Why are triangles important in construction?
8 4
Read the introduction to Lesson 4-1 at the top of page 178 in your textbook.
• Why are triangles used for braces in construction rather than other shapes?
( a2 , b)
4
V (4, 2)
O
4
8
(0, 0)
A
x
O
C
(a, 0) x
Sample answer: Triangles lie in a plane and are rigid shapes.
• Why do you think that isosceles triangles are used more often than
scalene triangles in construction? Sample answer: Isosceles
42. CRITICAL THINKING K
L
is a segment representing one side of isosceles right
LM
triangle KLM, with K(2, 6), and L(4, 2). KLM is a right angle, and K
L
.
Describe how to find the coordinates of vertex M and name these coordinates.
triangles are symmetrical.
Reading the Lesson
1. Supply the correct numbers to complete each sentence.
a. In an obtuse triangle, there are
2 acute angle(s),
0 right angle(s), and
Use the Distance Formula and Slope Formula; (0, 0) or (8, 4).
1 obtuse angle(s).
b. In an acute triangle, there are
3 acute angle(s),
0 right angle(s), and
0 obtuse angle(s).
c. In a right triangle, there are
2 acute angle(s),
182 Chapter 4 Congruent Triangles
1 right angle(s), and
0 obtuse angle(s).
2. Determine whether each statement is always, sometimes, or never true.
a. A right triangle is scalene. sometimes
NAME ______________________________________________ DATE
b. An obtuse triangle is isosceles. sometimes
4-1
Enrichment
Enrichment,
c. An equilateral triangle is a right triangle. never
d. An equilateral triangle is isosceles. always
____________ PERIOD _____
p. 188
e. An acute triangle is isosceles. sometimes
f. A scalene triangle is obtuse. sometimes
Reading Mathematics
3. Describe each triangle by as many of the following words as apply: acute, obtuse, right,
scalene, isosceles, or equilateral.
a.
b.
70
80
30
acute, scalene
c.
135
obtuse, isosceles
4
3
5
right, scalene
Helping You Remember
4. A good way to remember a new mathematical term is to relate it to a nonmathematical
definition of the same word. How is the use of the word acute, when used to describe
acute pain, related to the use of the word acute when used to describe an acute angle or
an acute triangle? Sample answer: Both are related to the meaning of acute
as sharp. An acute pain is a sharp pain, and an acute angle can be
thought of as an angle with a sharp point. In an acute triangle all of the
angles are acute.
182
Chapter 4 Congruent Triangles
When you read geometry, you may need to draw a diagram to make the text
easier to understand.
Example
Consider three points, A, B, and C on a coordinate grid.
The y-coordinates of A and B are the same. The x-coordinate of B is
greater than the x-coordinate of A. Both coordinates of C are greater
than the corresponding coordinates of B. Is triangle ABC acute, right,
or obtuse?
To answer this question, first draw a sample triangle
that fits the description.
Side AB must be a horizontal segment because the
y-coordinates are the same. Point C must be located
to the right and up from point B.
From the diagram you can see that triangle ABC
must be obtuse.
y
Q
A
B
O
x
43. WRITING IN MATH
Answer the question that was posed at the beginning
of the lesson. See margin.
Why are triangles important in construction?
4 Assess
Include the following in your answer:
• describe how to classify triangles, and
• if one type of triangle is used more often in architecture than other types.
Standardized
Test Practice
Open-Ended Assessment
44. Classify ABC with vertices A(1, 1), B(1, 3), and C(3, 1). C
A scalene acute
B equilateral
C isosceles acute D isosceles right
45. ALGEBRA Find the value of y if the mean of x, y, 15, and 35 is 25 and the mean
of x, 15, and 35 is 27. B
A 18
B 19
C 31
D 36
Maintain Your Skills
Mixed Review
Graph each line. Construct a perpendicular segment through the given point.
Then find the distance from the point to the line. (Lesson 3-6)
46. y x 2, (2, 2) 18 47. x y 2, (3, 3)
46–48. See margin for graphs.
Find x so that p 㛳 q . (Lesson 3-5)
49. 15
50. 45
p
q
p
110˚
q
8
48. y 7, (6, 2) 9
Getting Ready for
Lesson 4-2
51. 44
Prerequisite Skill Students will
learn about angles of triangles in
Lesson 4-2. They will use angle
relationships with the Angle Sum
Theorem and the Exterior Angle
Theorem to find angle measures.
Use Exercises 53–57 to determine
your students’ familiarity with
angles formed by parallel lines
and a transversal.
57˚
p
(3x – 50)˚
(4x + 10)˚
(2x – 5)˚
q
(3x – 9)˚
For this proof, the reasons in the right column are not in the proper order. Reorder
the reasons to properly match the statements in the left column. (Lesson 2-6)
52. Given: 3x 4 x 10 Order should be: 1. Given
Prove: x 3
Proof:
Statements
a. 3x 4 x 10
b. 2x 4 10
c. 2x 6
d. x 3
Getting Ready for
the Next Lesson
53. any three: 2 and
11, 3 and 6, 4
and 7, 3 and 12,
7 and 10, 8 and
11
54. 1 and 4, 1
and 10, 5 and 2,
5 and 8, 9 and
6, 9 and 12
Speaking Ask students to call
out the classifications of selected
triangles from the book or on the
board by both angles and sides.
Provide angle measures, side
measures, congruency tick-marks,
or right angle symbols for figures
on the board. Label the triangles,
and encourage students to use
proper terminology to refer to
the triangle, its angles, and its
segments.
2. Subtraction Property
3. Addition Property
4. Division Property
46.
Reasons
1. Subtraction Property
2. Division Property
3. Given
4. Addition Property
y
yx2
BC
AC
RQ
PR
PQ
PREREQUISITE SKILL In the figure, A
B
㛳
, 㛳
, and 㛳
. Name the
indicated angles or pairs of angles.
x
O
(2, –2)
(To review angles formed by parallel lines and a transversal, see Lessons 3-1 and 3-2.)
53.
54.
55.
56.
57.
three pairs of alternate interior angles
six pairs of corresponding angles
all angles congruent to 3 6, 9, and 12
all angles congruent to 7 1, 4, and 10
all angles congruent to 11 2, 5, and 8
www.geometryonline.com/self_check_quiz
R
47.
9
C 8
6
Q
5
10
11
7
4
3 2
B
y
A
(3, 3)
12
1
xy2
P
x
O
Lesson 4-1 Classifying Triangles 183
Answers
43. Sample answer: Triangles are used in construction as structural support. Answers should
include the following.
• Triangles can be classified by sides and angles. If the measure of each angle is less than 90,
the triangle is acute. If the measure of one angle is greater than 90, the triangle is obtuse. If
one angle equals 90°, the triangle is right. If each angle has the same measure, the triangle
is equiangular. If no two sides are congruent, the triangle is scalene. If at least two sides are
congruent, it is isosceles. If all of the sides are congruent, the triangle is equilateral.
• Isosceles triangles seem to be used more often in architecture and construction.
48.
y
y7
x
O
(6, –2)
Lesson 4-1 Classifying Triangles 183
Geometry
Activity
A Preview of Lesson 4-2
Getting Started
A Preview of Lesson 4-2
Angles of Triangles
There are special relationships among the angles of a triangle.
scissors
Step 2
Step 3
Teach
• Advise students to label the
obtuse angle B when they are
first working through Activity 1.
They can also repeat Activity 1
using one of the acute angles as
angle B to further verify
concepts.
• As an extension, students can
repeat Activity 1 using an acute
triangle or a right triangle. They
can also cut along the segments
DF, FE, and DE and arrange
angles A, B, and C over their
congruent counterparts in
DEF to see that the angles
have equal measures.
Assess
In Exercises 1–12 students
determine angle measures of the
triangles used in this activity,
find relationships, and make
conjectures that will lead them to
the Angle Sum Theorem and the
Exterior Angle Theorem.
Step 4
Step 5
Step 6
E
C
Analyze the Model
Describe the relationship between each pair.
1. A and DFA congruent 2. B and DFE congruent 3. C and EFC congruent
4. What is the sum of the measures of DFA, DFE, and EFC? 180
5. What is the sum of the measures of A, B, and C? 180
6. Make a conjecture about the sum of the measures of the angles of any triangle. The sum of
the measures of the angles of any triangle is 180.
In the figure at the right, 4 is called an
exterior angle of the triangle. 1 and 2
are the remote interior angles of 4.
Step 1
Step 2
Step 3
Step 4
2
1
Find the relationship among the interior
and exterior angles of a triangle.
Trace ABC from Activity 1 onto a piece of
paper. Label the vertices.
Extend A
C
to draw an exterior angle at C.
Tear A and B off the triangle from Activity 1.
Place A and B over the exterior angle.
4
3
E
D
F
C
A
Materials
protractor
Step 1
Find the relationship among the measures
B
of the interior angles of a triangle.
D
Draw an obtuse triangle and cut it out. Label the
vertices A, B, and C.
F
A
Find the midpoint of A
B
by matching A to B. Label
this point D.
Find the midpoint of B
C
by matching B to C.
Label this point E.
Draw D
E
.
Fold ABC along DE
. Label the point where B touches A
C
as F.
Draw D
F
and F
E
. Measure each angle.
Analyze the Model 7. mA mB is the measure of the exterior angle at C.
7.
8.
9.
10.
11.
12.
Make a conjecture about the relationship of A, B, and the exterior angle at C.
Repeat the steps for the exterior angles of A and B. See students’ work.
Is your conjecture true for all exterior angles of a triangle? yes
Repeat Activity 2 with an acute triangle. 10–11. See students’ work.
Repeat Activity 2 with a right triangle.
Make a conjecture about the measure of an exterior angle and the sum of the measures of its
remote interior angles. The measure of an exterior angle is equal to the sum of measures of
184 Chapter 4 Congruent Triangles
the two remote interior angles.
Resource Manager
Study Notebook
Ask students to summarize what they
have learned about the relationships
among the measures of the interior
and exterior angles of triangles.
184 Chapter 4 Congruent Triangles
B
Objective Find the relationships
among the measures of the
interior and exterior angles of a
triangle.
Teaching Geometry with
Manipulatives
Glencoe Mathematics Classroom
Manipulative Kit
• p. 70 (student recording sheet)
• p. 16 (protractor)
• protractor
• scissors
Lesson
Notes
Angles of Triangles
• Apply the Angle Sum Theorem.
1 Focus
• Apply the Exterior Angle Theorem.
Vocabulary
•
•
•
•
exterior angle
remote interior angles
flow proof
corollary
are the angles of triangles
used to make kites?
5-Minute Check
Transparency 4-2 Use as a
quiz or review of Lesson 4-1.
The Drachen Foundation coordinates the annual
Miniature Kite Contest. This kite won second place in
the Most Beautiful Kite category in 2001. The overall
dimensions are 10.5 centimeters by 9.5 centimeters.
The wings of the beetle are triangular.
Mathematical Background notes
are available for this lesson on
p. 176C.
ANGLE SUM THEOREM If the measures of two of the angles of a triangle are
known, how can the measure of the third angle be determined? The Angle Sum
Theorem explains that the sum of the measures of the angles of any triangle is
always 180.
Theorem 4.1
Angle Sum Theorem The sum of the
X
measures of the angles of a triangle
is 180.
Example: mW mX mY 180
W
Proof
Study Tip
Angle Sum Theorem
Look Back
Given: ABC
Recall that sometimes
extra lines have to be
drawn to complete a
proof. These are called
auxiliary lines.
Prove: mC m2 mB 180
X
A
Y
1 2 3
C
Proof:
Statements
1. ABC
CB
2. Draw XY through A parallel to .
3. 1 and CAY form a linear pair.
4. 1 and CAY are supplementary.
5.
6.
7.
8.
9.
10.
Y
m1 mCAY 180
mCAY m2 m3
m1 m2 m3 180
1 C, 3 B
m1 mC, m3 mB
mC m2 mB 180
are the angles of
triangles used to
make kites?
Ask students:
• Assuming that the wings are
equal in size and the angle
between the two wings is 90°,
what type of triangle is formed
if you draw a line to connect
one wing-tip to the other
wing-tip? right isosceles triangle
• Are the wings of a real beetle
perfectly triangular in shape?
no
B
Reasons
1. Given
2. Parallel Postulate
3. Def. of a linear pair
4. If 2 form a linear pair, they are
supplementary.
5. Def. of suppl. 6. Angle Addition Postulate
7. Substitution
8. Alt. Int. Theorem
9. Def. of 10. Substitution
Lesson 4-2 Angles of Triangles 185
Courtesy The Drachen Foundation
Resource Manager
Workbook and Reproducible Masters
Chapter 4 Resource Masters
• Study Guide and Intervention, pp. 189–190
• Skills Practice, p. 191
• Practice, p. 192
• Reading to Learn Mathematics, p. 193
• Enrichment, p. 194
• Assessment, p. 239
Graphing Calculator and
Computer Masters, p. 23
Prerequisite Skills Workbook, pp. 81–84
Teaching Geometry With Manipulatives
Masters, p. 71
Transparencies
5-Minute Check Transparency 4-2
Answer Key Transparencies
Technology
Interactive Chalkboard
Lesson x-x Lesson Title 185
If we know the measures of two angles of a triangle, we can find the measure of
the third.
2 Teach
Example 1 Interior Angles
ANGLE SUM THEOREM
In-Class Example
Find the missing angle measures.
Find m1 first because the measures of two
angles of the triangle are known.
Power
Point®
m1 110 180
measures.
m1 70
79
1
74
28˚
m1 28 82 180 Angle Sum Theorem
1 Find the missing angle
43
82˚
1
2
Simplify.
3
1 and 2 are congruent vertical angles.
So m2 70.
2
3
m3 68 70 180
m3 138 180
m1 63; m2 63;
m3 38
m3 42
68˚
Subtract 110 from each side.
Angle Sum Theorem
Simplify.
Subtract 138 from each side.
Therefore, m1 70, m2 70, and m3 42.
The Angle Sum Theorem leads to a useful theorem about the angles in two triangles.
Theorem 4.2
Third Angle Theorem If two angles of one triangle are congruent to two angles
of a second triangle, then the third angles of the triangles are congruent.
C
A
B
D
F
C
A
E
B
D
F
E
3
remote
interior
angles
Example: If A F and C D, then B E.
You will prove this theorem in Exercise 44.
EXTERIOR ANGLE THEOREM
Study Tip
Reading Math
Remote means far away
and interior means
inside. The remote
interior angles are the
interior angles farthest
from the exterior angle.
Each angle of a triangle has an
exterior angle. An exterior angle is
formed by one side of a triangle and
the extension of another side. The
interior angles of the triangle not
adjacent to a given exterior angle are
called remote interior angles of the
exterior angle.
exterior
angle
2
1
Theorem 4.3
Exterior Angle Theorem The measure
of an exterior angle of a triangle is
equal to the sum of the measures of
the two remote interior angles.
Example: mYZP mX mY
Y
X
Z
P
186 Chapter 4 Congruent Triangles
Differentiated Instruction
Visual/Spatial Tell students that the Angle Sum Theorem and Exterior
Angle Theorem are both based on the idea that a straight angle
measures 180°. Show that if they cut the angles of any triangle and
place them right next to one another, they form a straight line. This
visually demonstrates why the sum of the interior angles of a triangle
measures 180°.
186
Chapter 4 Congruent Triangles
We will use a flow proof to prove this theorem. A flow proof organizes a series
of statements in logical order, starting with the given statements. Each statement is
written in a box with the reason verifying the statement written below the box.
Arrows are used to indicate how the statements relate to each other.
Proof
Exterior Angle Theorem
EXTERIOR ANGLE
THEOREM
In-Class Example
2 Find the measure of each
numbered angle in the figure.
C
Write a flow proof of the Exterior Angle Theorem.
Given: ABC
Prove: mCBD mA mC
5
Flow Proof:
D
ABC
CBD and ABC form a linear pair.
Given
Definition of linear pair
B
A
If 2 s form a
linear pair, they
are supplementary.
mA mABC C 180
mCBD mABC = 180
Definition of
supplementary
2
1
29
In Chapter 3, students used
angle relationships to find angle
measures. In this lesson, students
will apply their knowledge of
vertical angles, supplementary
angles, and complementary
angles along with the Angle Sum
Theorem and the Exterior Angle
Theorem to find angle measures
in figures.
Substitution Property
mA mC mCBD
Subtraction Property
Example 2 Exterior Angles
128
41 64
Building on Prior
Knowledge
mA mABC mC mCBD mABC
Find the measure of each numbered angle
in the figure.
Exterior Angle Theorem
m1 50 78
38
4 3 32
m1 70, m2 110,
m3 46, m4 102, and
m5 37.
CBD and ABC are supplementary.
Angle Sum
Theorem
Power
Point®
3
2
50˚
Simplify.
1
78˚
120˚
4
5
56˚
m1 m2 180 If 2 form a linear pair,
they are suppl.
128 m2 180 Substitution
m2 52
Subtract 128 from each side.
m2 m3 120 Exterior Angle Theorem
52 m3 120 Substitution
m3 68
120 m4 180
m4 60
m5 m4 56
Subtract 52 from each side.
If 2
form a linear pair, they are suppl.
Subtract 120 from each side.
Exterior Angle Theorem
60 56
Substitution
116
Simplify.
Therefore, m1 128, m2 52, m3 68, m4 60, and m5 116.
www.geometryonline.com/extra_examples
Lesson 4-2 Angles of Triangles
187
Lesson 4-2 Angles of Triangles 187
In-Class Example
A statement that can be easily proved using a theorem is often called a corollary
of that theorem. A corollary, just like a theorem, can be used as a reason in a proof.
Power
Point®
3 GARDENING The flower bed
Corollaries
shown is in the shape of a
right triangle. Find mA if
mC is 20.
B
4.1
C
4.2
The acute angles of a right
triangle are complementary.
G
There can be at most one right or
obtuse angle in a triangle.
P
K
acute
❀
❀ ❀ ❀ ❀ ❀ ❀ 20
✿ ✿ ✿ ✿ ✿ ✿ ✿
❀ ❀ ❀ ❀ ❀ ❀
✿ ✿ ✿ ✿
❀ ❀ ❀
✿
142˚
H
A
M
J
Q
L
R
acute
Example: mG mJ 90
mA 70
You will prove Corollaries 4.1 and 4.2 in Exercises 42 and 43.
Example 3 Right Angles
3 Practice/Apply
SKI JUMPING Ski jumper Simon
Ammann of Switzerland forms a right
triangle with his skis and his line of
sight. Find m2 if m1 is 27.
Use Corollary 4.1 to write an equation.
Study Notebook
Have students—
• add the definitions/examples of
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 4.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
Answer
1. Sample answer: 2 and 3 are
the remote interior angles of
exterior 1.
1
m1 m2 90
27 m2 90 Substitution
m2 63 Subtract 27 from each side.
Concept Check
2. Najee; the sum
of the measures of
the remote interior
angles is equal to
the measure of the
corresponding
exterior angle.
Guided Practice
GUIDED PRACTICE KEY
Exercises
Examples
3–4
5–7
8–10
1
2
3
2
1
2
3
1. OPEN ENDED Draw a triangle. Label one exterior angle and its remote interior
angles. See margin.
2. FIND THE ERROR Najee and Kara are discussing the Exterior Angle Theorem.
2
1
3
4
Najee
Kara
mΩ1 + mΩ2 = mΩ4
mΩ1 + mΩ2 + mΩ4 = 180
Who is correct? Explain your reasoning.
Find the missing angle measure.
43
3.
Toledo
85˚
Pittsburgh
62˚
52˚
Cincinnati
Find each measure.
5. m1 55
6. m2 33
7. m3 147
188
Chapter 4 Congruent Triangles
Adam Pretty/Getty Images
188
Chapter 4 Congruent Triangles
99
4.
32˚
23˚
1
2
3
22˚
19˚
Find each measure.
8. m1 65
9. m2 25
D
1
25˚
65˚
E
Application
FIND THE ERROR
Explain to
students that Kara’s
equation could only be true in
the special case when
m3 m4 90.
2
G
F
10. SKI JUMPING American ski jumper
Eric Bergoust forms a right angle with
his skis. If m2 70, find m1. 20
2
About the Exercises…
1
Organization by Objective
• Angle Sum Theorem: 11–17
• Exterior Angle Theorem:
18–38
★ indicates increased difficulty
Practice and Apply
For
Exercises
See
Examples
11–17
18–31
32–35
36–38
1
2
3
2
Find the missing angle measures.
11. 93
40˚
Odd/Even Assignments
Exercises 11–38 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
70.5, 70.5
12.
47˚
Assignment Guide
Extra Practice
See page 761.
39˚
65, 65
13.
63
14.
50˚
27˚
Find each measure.
15. m1 76
16. m2 76
17. m3 49
55˚
47˚
1
2
57˚
3
Find each measure if m4 m5.
19. m2 53
18. m1 64
20. m3 116
21. m4 32
22. m5 32
23. m6 44
24. m7 89
Find each measure.
25. m1 123
26. m2 28
27. m3 14
Basic: 11–35 odd, 41, 43, 45,
46–64
Average: 11–45 odd, 46–64
Advanced: 12–44 even, 45–58
(optional: 59–64)
95˚
7
63˚
2
69˚
47˚
1
5
6
136˚
3
4
109˚
1
2
33˚
3
24˚
Lesson 4-2 Angles of Triangles 189
Doug Pensinger/Getty Images
Lesson 4-2 Angles of Triangles 189
NAME ______________________________________________ DATE
SPEED SKATING For Exercises 28–31, use the following information.
Speed skater Catriona Lemay Doan of Canada forms at least two sets of triangles
and exterior angles as she
skates. Use the measures
of given angles to find each
measure.
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
4-2
Study
Guide and
p. 189
Angles(shown)
of Triangles and p. 190
Angle Sum Theorem If the measures of two angles of a triangle are known,
the measure of the third angle can always be found.
Angle Sum
Theorem
The sum of the measures of the angles of a triangle is 180.
In the figure at the right, mA mB mC 180.
B
A
Example 1
28.
29.
30.
31.
Find the missing
angle measures.
S
B
35
90
T
mR mS mT 180
25 35 mT 180
60 mT 180
mT 120
A
58
C
1
2
Angle Sum
Theorem
Substitution
Add.
Subtract 60
from each side.
D
3
108
E
m1 mA mB
m1 58 90
m1 148
m1
180
180
180
32
Angle Sum Theorem
Substitution
Add.
Subtract 148 from
each side.
m2 32
m3 m2 mE
m3 32 108
m3 140
m3
Vertical angles are
congruent.
180
180
180
40
Lesson 4-2
25
R
C
Example 2
Find mT.
m1
m2
m3
m4
1
54
53
137
103
126˚
2
73˚
3
43˚
34˚
4
Online Research
Data Update Use the Internet or other resource to find the
world record in speed skating. Visit www.geometryonline.com/data_update to
learn more.
Angle Sum Theorem
Substitution
Add.
Subtract 140 from
each side.
Exercises
Find the measure of each numbered angle.
1.
m1 28
M
62
90
1
P
60
W
U
1
30
T
4. M
3
66
1
58
m1 30,
m2 60
R
1 2
30
Q
6. A
50
Catriona Lemay Doan is
the first Canadian to win
a Gold medal in the same
event in two consecutive
Olympic games.
m1 8
20
152
1
G
D
NAME
______________________________________________
DATE
/M
G
Hill
189
Gl
m1 56,
m2 56,
m3 74
N
2
O
S
Speed Skating
R
P
T 60
W
Find each measure if mDGF 53
and mAGC 40.
33. m2 50
32. m1 37
34. m3 50
35. m4 40
1
2
5.
m1 120
S
30
Q
m1 30,
m2 60
3. V
2.
N
____________
Gl PERIOD
G _____
Skills
Practice,
p. 191 and
4-2
Practice
(Average)
Practice,
p. 192 (shown)
Angles of Triangles
18
72
40
★ 36. m1 53
★ 37. m2 129
★ 38. m3 153
55
Find the measure of each angle.
3
58
3. m1 97
1
2
B
HOUSING For Exercises 36–38, use
the following information.
The two braces for the roof of a house
form triangles. Find each measure.
85
2.
4
3
G
Source: www.catrionalemaydoan.
com
?
A
F
1
2
Find the missing angle measures.
1.
D
C
101˚
128˚
3
26˚
103˚
2
1
35
4. m2 83
39
5. m3 62
Find the measure of each angle.
★
5
2
6. m1 104
3
1
7. m4 45
70
36
118
6
4
65
68
39. flow proof
Given: FGI IGH
F
H
GI Prove: F H
82
8. m3 65
9. m2 79
10. m5 73
11. m6 147
Find the measure of each angle if BAD and
BDC are right angles and mABC 84.
B
For Exercises 39–44, write the specified type of proof. 39–44. See
PROOF
Given: ABCD is a quadrilateral.
Prove: mDAB mB mBCD mD 360
A
G
1
pp. 233A–233B.
★ 40. two-column
B
64 C
12. m1 26
2
A
D
13. m2 32
14. CONSTRUCTION The diagram shows an
example of the Pratt Truss used in bridge
construction. Use the diagram to find m1.
145
F
NAME
______________________________________________
DATE
/M
G
Hill
192
Gl
ELL
Mathematics,
p. 193
Angles of Triangles
How are the angles of triangles used to make kites?
The frame of the simplest kind of kite divides the kite into four triangles.
Describe these four triangles and how they are related to each other.
Sample answer: There are two pairs of right triangles that have
the same size and shape.
42. flow proof of Corollary 4.1
44. two-column proof of Theorem 4.2
m1 48, m2 60, m3 72
Reading the Lesson
1. Refer to the figure.
1
E
A
a. Name the three interior angles of the triangle. (Use three
letters to name each angle.) BAC, ABC, BCA
D
B
39
b. Name three exterior angles of the triangle. (Use three letters
to name each angle.) EAB, DBC, FCA
c. Name the remote interior angles of EAB. ABC, BCA
A
23
C
F
d. Find the measure of each angle without using a protractor.
ii. ABC 118
iii. ACF 157
190
Chapter 4 Congruent Triangles
Jed Jacobsohn/Getty Images
iv. EAB 141
2. Indicate whether each statement is true or false. If the statement is false, replace the
underlined word or number with a word or number that will make the statement true.
a. The acute angles of a right triangle are supplementary. false; complementary
b. The sum of the measures of the angles of any triangle is 100. false; 180
c. A triangle can have at most one right angle or acute angle. false; obtuse
d. If two angles of one triangle are congruent to two angles of another triangle, then the
third angles of the triangles are congruent. true
e. The measure of an exterior angle of a triangle is equal to the difference of the
measures of the two remote interior angles. false; sum
NAME ______________________________________________ DATE
4-2
Enrichment
Enrichment,
____________ PERIOD _____
p. 194
Finding Angle Measures in Triangles
You can use algebra to solve problems involving triangles.
f. If the measures of two angles of a triangle are 62 and 93, then the measure of the
third angle is 35. false; 25
Example
In triangle ABC, mA, is twice mB, and mC
is 8 more than mB. What is the measure of each angle?
g. An exterior angle of a triangle forms a linear pair with an interior angle of the
triangle. true
Write and solve an equation. Let x mB.
Helping You Remember
3. Many students remember mathematical ideas and facts more easily if they see them
demonstrated visually rather than having them stated in words. Describe a visual way
to demonstrate the Angle Sum Theorem.
Sample answer: Cut off the angles of a triangle and place them
side-by-side on one side of a line so that their vertices meet at a common
point. The result will show three angles whose measures add up to 180.
190
D
H
and BC
are opposite
45. CRITICAL THINKING BA
rays. The measures of 1, 2, and 3 are in a
4:5:6 ratio. Find the measure of each angle.
Read the introduction to Lesson 4-2 at the top of page 185 in your textbook.
i. DBC 62
I
41. two-column proof of Theorem 4.3
43. paragraph proof of Corollary 4.2
____________
Gl PERIOD
G _____
Reading
4-2
Readingto
to Learn
Learn Mathematics
Pre-Activity
C
1
55
Chapter 4 Congruent Triangles
mA mB mC 180
2x x (x 8) 180
4x 8 180
4x 172
x 43
So, m A 2(43) or 86, mB 43, and mC 43 8 or 51.
Solve each problem.
1. In triangle DEF, mE is three times
mD, and mF is 9 less than mE.
What is the measure of each angle?
2. In triangle RST, mT is 5 more than
mR, and mS is 10 less than mT.
What is the measure of each angle?
2
B
3
C
46. WRITING IN MATH
Answer the question that was posed at the beginning
of the lesson. See margin.
How are the angles of triangles used to make kites?
4 Assess
Include the following in your answer:
• if two angles of two triangles are congruent, how you can find the measure of
the third angle, and
• if one angle measures 90, describe the other two angles.
Standardized
Test Practice
47. In the triangle, what is the measure of Z? A
A 18
B 24
C
72
D
X
2a˚
90
a˚
2
Y
Z
48. ALGEBRA The measure of the second angle of a triangle is three times the
measure of the first, and the measure of the third angle is 25 more than the
measure of the first. Find the measure of each angle. B
A 25, 85, 70
B 31, 93, 56
C 39, 87, 54
D 42, 54, 84
Identify the indicated type of triangle if
AD
EB
EC
AC
B
C
, , bisects B
D
,
and mAED 125. (Lesson 4-1)
49. scalene triangles AED
50. obtuse triangles BEC, AED
51. isosceles triangles BEC
B
C
E
A
125˚
D
53. 20
units
Find the distance between each pair of parallel lines. (Lesson 3-6)
52. y x 6, y x 10 128
units 53. y 2x 3, y 2x 7
54. 4x y 20, 4x y 3 17
units 55. 2x 3y 9, 2x 3y 6
117 units
13
13
Find x, y, and z in each figure. (Lesson 3-2)
56.
z˚
57.
58.
(5z 2)˚
3x˚
42˚
(4x 6)˚
(2y 8)˚
Getting Ready for
the Next Lesson
142˚
x 34, y 15, z 142
68˚
y˚
x 112, y 28, z 22
48˚
z˚
x 16, y 90,
z 42
PREREQUISITE SKILL Name the property of congruence that justifies each
statement. (To review properties of congruence, see Lessons 2-5 and 2-6.)
59. 1 1 and AB
AB
. reflexive
60. If A
B
X
Y, then X
Y
A
B. symmetric
61. If 1 2, then 2 1. symmetric
62. If 2 3 and 3 4, then 2 4. transitive
63. If P
Q
X
Y and X
Y
H
K, then P
Q
H
K. transitive
64. If A
B
C
D, C
D
P
Q, and P
Q
X
Y, then A
B
X
Y. transitive
www.geometryonline.com/self_check_quiz
Prerequisite Skill Students will
learn about congruent triangles
in Lesson 4-3. They will use
properties of congruent
segments and angles to identify
corresponding parts of
congruent triangles and to prove
congruency between a triangle
and its transformed image. Use
Exercises 59–64 to determine
your students’ familiarity with
properties of congruence for
segments and angles.
Assessment Options
x˚
4y˚
Writing Draw an acute triangle
with two angles that measure 44°
and 56°, an obtuse triangle with
angles 110° and 40°, and an
isosceles triangle sitting on a line
with two angles measuring 75°.
Ask students to use the theorems
in this lesson to find the missing
angle measures in each triangle
and then write a paragraph
summarizing how they found
the measures.
Getting Ready for
Lesson 4-3
Maintain Your Skills
Mixed Review
Open-Ended Assessment
Lesson 4-2 Angles of Triangles
Quiz (Lessons 4-1 and 4-2) is
available on p. 239 of the Chapter 4
Resource Masters.
Answer
191
46. Sample answer: The shape of a
kite is symmetric. If triangles are
used on one side of the kite,
congruent triangles are used on
the opposite side. The wings of
this kite are made from congruent
right triangles. Answers should
include the following.
• By the Third Angle Theorem, if
two angles of two congruent
triangles are congruent, then
the third angles of each triangle
are congruent.
• If one angle measures 90, the
other two angles are both acute.
Lesson 4-2 Angles of Triangles 191
Lesson
Notes
Congruent Triangles
• Name and label corresponding parts of congruent triangles.
1 Focus
5-Minute Check
Transparency 4-3 Use as a
quiz or review of Lesson 4-2.
• Identify congruence transformations.
Vocabulary
• congruent triangles
• congruence
transformations
Mathematical Background notes
are available for this lesson on
p. 176C.
are triangles used in
bridges?
Ask students:
• What types of triangles do you
notice in the construction of
the bridge? acute triangles
• What do you notice about the
size and shape of the triangles?
The triangles appear to be the
same size and shape.
are triangles
used in bridges?
In 1930, construction started on
the West End Bridge in Pittsburgh,
Pennsylvania. The arch of the bridge
is trussed, not solid. Steel rods are
arranged in a triangular web that lends
structure and stability to the bridge.
CORRESPONDING PARTS OF CONGRUENT TRIANGLES Triangles
that are the same size and shape are congruent triangles . Each triangle has three
angles and three sides. If all six of the corresponding parts of two triangles are
congruent, then the triangles are congruent.
B
F
C
G
A
E
←
←
If ABC is congruent to EFG, the vertices of the two triangles correspond in the
same order as the letters naming the triangles.
←
←
←
←
ABC EFG
This correspondence of vertices can be used to name the corresponding congruent
sides and angles of the two triangles.
A E
B F
C G
EF
AB
B
C
FG
A
C
EG
The corresponding sides and angles can be determined from any congruence
statement by following the order of the letters.
Definition of Congruent Triangles (CPCTC)
Study Tip
Congruent Parts
In congruent triangles,
congruent sides are
opposite congruent
angles.
192
Two triangles are congruent if and only if their corresponding parts are congruent.
CPCTC stands for corresponding parts of congruent triangles are congruent. “If and
only if” is used to show that both the conditional and its converse are true.
Chapter 4 Congruent Triangles
Aaron Haupt
Resource Manager
Workbook and Reproducible Masters
Chapter 4 Resource Masters
• Study Guide and Intervention, pp. 195–196
• Skills Practice, p. 197
• Practice, p. 198
• Reading to Learn Mathematics, p. 199
• Enrichment, p. 200
School-to-Career Masters, p. 7
Teaching Geometry With Manipulatives
Masters, pp. 72, 73
Transparencies
5-Minute Check Transparency 4-3
Answer Key Transparencies
Technology
Interactive Chalkboard
Example 1 Corresponding Congruent Parts
FURNITURE DESIGN The seat and legs of this
stool form two triangles. Suppose the measures
in inches are QR 12, RS 23, QS 24, RT 12,
TV 24, and RV 23.
a. Name the corresponding congruent angles and
sides.
Q T
QRS TRV
S V
QR
T
R
R
S
RV
Q
T
R
2 Teach
CORRESPONDING PARTS
OF CONGRUENT TRIANGLES
In-Class Example
S
Q
S
TV
V
Teaching Tip
Students can use
tick marks on sides and angles
to help visually organize the
corresponding parts of
congruent triangles.
b. Name the congruent triangles.
QRS TRV
1 ARCHITECTURE A tower roof
Like congruence of segments and angles, congruence of triangles is reflexive,
symmetric, and transitive.
is composed of congruent
triangles all converging
toward a point at the top.
Properties of Triangle Congruence
Theorem 4.4
Power
Point®
I
Congruence of triangles is reflexive, symmetric, and transitive.
Reflexive
JKL JKL
K
Symmetric
Transitive
If JKL PQR,
then PQR JKL.
If JKL PQR, and
PQR XYZ, then
JKL XYZ
K
L
J
K
L
J
Q
K
L
J
Q
R
Y
L
P
J
R
P
H
Z
Theorem 4.4 (Transitive)
Given: ABC DEF
DEF GHI
Prove: ABC GHI
A
Proof:
Statements
1. ABC DEF
b. Name the congruent
triangles. HIJ LIK
I
F
C
G
D
a. Name the corresponding
congruent angles and sides of
HIJ and LIK.
HJI LKI; ILK IHJ;
HIJ LIK; HI LI ; H
J LK
;
JI KI
H
E
B
L
K
X
You will prove the symmetric and reflexive parts of Theorem 4.4 in Exercises 33 and 35, respectively.
Proof
J
Reasons
1. Given
2. A D, B E, C F
2. CPCTC
AB
DE
EF
AC
DF
, B
C
, 3. DEF GHI
3. Given
4. D G, E H, F I
4. CPCTC
DE
GH
HI, D
GI
, E
F
F
5. A G, B H, C I
5. Congruence of angles is transitive.
6. AB
GH
BC
HI, A
GI
, C
6. Congruence of segments is transitive.
7. ABC GHI
7. Def. of s
www.geometryonline.com/extra_examples
Lesson 4-3 Congruent Triangles
193
Private Collection/Bridgeman Art Library
Differentiated Instruction
Auditory/Musical Explain to students that congruency can be appealing
to both the eyes and the ears. Point out that if students use beats to model
two congruent equilateral triangles, they could use three equally-spaced
drum beats for the first and then repeat the exact same rhythm for the
second. An isosceles beat could consist of two quick beats and one slow
beat or vice versa. Tell students that often in music, a “congruent” rhythm
is used throughout a song. A popular example is the song, “Louie, Louie.”
Lesson 4-3 Congruent Triangles 193
IDENTIFY CONGRUENCE
TRANSFORMATIONS
In-Class Example
Power
Point®
Teaching Tip
In Chapter 9,
students will be introduced to
the formal names for the slide,
flip, and turn transformations.
Study Tip
Naming
Congruent
Triangles
IDENTIFY CONGRUENCE TRANSFORMATIONS In the figures below,
ABC is congruent to DEF. If you slide DEF up and to the right, DEF is still
congruent to ABC.
E'
B
There are six ways to
name each pair of
congruent triangles.
E
slide
A
D'
D
F'
C
F
The congruency does not change whether you turn DEF or flip DEF. ABC is
still congruent to DEF.
2 COORDINATE GEOMETRY
S y
E
D'
Study Tip
T
R
O
E
D
Transformations
R
x
T
S
a. Verify that RST RST.
RS RS 34
ST ST 17
TR TR 17
b. Name the congruence
transformation for RST and
RST. turn
3 Practice/Apply
Not all of the
transformations preserve
congruence. Only
transformations that do
not change the size or
shape of the triangle are
congruence
transformations.
F
D
F'
F
F'
Have students—
• add the definitions/examples of
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 4.
• include an example from each
theorem introduced in this lesson.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
194
Chapter 4 Congruent Triangles
E'
E'
If you slide, flip, or turn a triangle, the size and shape do not change. These three
transformations are called congruence transformations .
Example 2 Transformations in the Coordinate Plane
COORDINATE GEOMETRY The vertices of CDE
are C(5, 7), D(8, 6), and E(3, 3). The vertices
of CDE are C(5, 7), D(8, 6), and E(3, 3).
D
a. Verify that CDE CDE.
Use the Distance Formula to find the length of
each side in the triangles.
–8
[8 (5)]2
(6
7)2
DC 9 1 or 10
DE [8 (3)]2
(6
3)2
25 9 or 34
Study Notebook
flip
D'
turn
CE [5
(3)]2
(7
3)2
4 16 or 20
C
8
y
C'
D'
4
E
–4
E'
O
4
8x
DC (8 5
)2 (6
7)2
9 1 or 10
DE (8 3
)2 (6
3)2
25 9 or 34
2 (7
CE (5
3)
3)2
4 16 or 20
By the definition of congruence, D
DC
DE
DE
CE
C
, , and C
E
.
Use a protractor to measure the angles of the triangles. You will find that the
measures are the same.
DC
DE
CE
In conclusion, because DC
, D
E
, and C
E
, D D,
C C, and E E, DCE DCE.
b. Name the congruence transformation for CDE and CDE.
CDE is a flip of CDE.
194 Chapter 4 Congruent Triangles
Concept Check
Guided Practice
GUIDED PRACTICE KEY
Exercises
Examples
3–6, 8
7
1
2
1. Explain how slides, flips, and turns preserve congruence. See margin.
2. OPEN ENDED Draw a pair of congruent triangles and label the congruent sides
and angles. See margin.
Identify the congruent triangles in each figure.
D
AFC DFB
4. H
HJT TKH
3. A
K
F
C
5. W S,
X T, Z J,
W
X ST, XZ TJ,
W
Z SJ
B
J
T
5. If WXZ STJ, name the congruent angles and congruent sides.
B
6. QUILTING In the quilt design, assume that angles and
segments that appear to be congruent are congruent.
Indicate which triangles are congruent. See margin.
8. GARDENING This garden lattice will be covered with
morning glories in the summer. Wesley wants to save
two triangular areas for artwork. If GHJ KLP,
name the corresponding congruent angles and sides.
A
G
E
7. The coordinates of the vertices of QRT and QRT
are Q(4, 3), Q(4, 3), R(4, 2), R(4, 2), T(1, 2),
and T(1, 2). Verify that QRT QRT. Then name
the congruence transformation. See margin.
Application
M
L
F
C
K
D
H
G
L
J
K
9–22,
27–35
23–26
1
Answers
Identify the congruent triangles in each figure.
9. CFH JKL
10.
K
S
RSV TSV
L
F
2
1. The sides and the angles of the
triangle are not affected by a
congruence transformation, so
congruence is preserved.
2. Sample answer:
J
Extra Practice
See page 761.
V
H
C
11.
R
WPZ QVS
P
T
12.
F
EFH GHF
E
Z
S
W
G
Q
V
13– 16. See margin.
Basic: 9–19 odd, 23, 27–33,
36–51
Average: 9–35 odd, 36–51
Advanced: 10–36 even, 37–48
(optional: 49–51)
All: Quiz 1 (1–5)
P
Practice and Apply
See
Examples
Odd/Even Assignments
Exercises 9–35 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
Assignment Guide
H
★ indicates increased difficulty
For
Exercises
Organization by Objective
• Corresponding Parts of
Congruent Triangles: 9–22,
27–35
• Identify Congruence
Transformations: 23–26
N
J
G K, H L, J P, G
H
KL, H
J LP,
J KP
G
About the Exercises…
H
Name the congruent angles and sides for each pair of congruent triangles.
13. TUV XYZ
14. CDG RSW
15. BCF DGH
16. ADG HKL
Lesson 4-3 Congruent Triangles 195
6. BME, ANG, DKH, CLF;
EMJ, GNJ, HKJ, FLJ;
BAJ, ADJ, DCJ, CBJ;
BCD, ADC, CBA, DAB;
BLJ, AMJ, JND, JKC,
BMJ, ANJ, JKD, JLC
7. QR 5, QR 5, RT 3,
RT 3, QT 34, and
QT 34. Use a protractor to
confirm that the corresponding
angles are congruent; flip.
13. T X, U Y, V Z,
TU
XY
, UV
YZ, TV
XZ
14. C R, D S, G W,
CD
RS
, DG
SW
, CG
R
W
15. B D, C G, F H,
BC
DG
, CF GH
, BF DH
16. A H, D K, G L,
AD
HK
, DG
KL, AG
HL
Lesson 4-3 Congruent Triangles 195
Answers
22. Flip; PQ 2, PQ 2, QV 4,
QV 4, PV 20, and
PV 20. Use a protractor to
confirm that the corresponding
angles are congruent.
23. Flip; MN 8, MN 8, NP 2,
NP 2, MP 68, and
MP 68. Use a protractor to
confirm that the corresponding
angles are congruent.
24. Slide; GH 10, GH 10,
HF 8, HF 8,
GF 10, and GF 10. Use
a protractor to confirm that the
corresponding angles are
congruent.
25. Turn; JK 40, JK 40,
KL 29, KL 29,
JL 17, and JL 17. Use
a protractor to confirm that the
corresponding angles are
congruent.
26. False; A X, B Y, and
C Z but the corresponding
sides are not congruent.
Y
B
X
Z
A
C
27.
D
18. s 1–4, s 5–12,
s 13–20
19. s 1, 5, 6, and
11, s 3, 8, 10,
and 12, s 2, 4,
7, and 9
Assume that segments and angles that appear to be congruent in the numbered
triangles are congruent. Indicate which triangles are congruent.
17.
18.
19.
5
9
2
1
21. MOSAICS
1
9
8
6
10
10
7
3
11
8
4
12
2
12
7
3
8
6
11
4
10
14
13
17
18
16
15
19
20
9
5
S
V
B
U
F
E
T
A
C
D
The picture at the left is the center of a Roman mosaic. Because the
Verify that each of the following preserves congruence and name the congruence
transformation. 22–25. See margin for verification.
22. PQV PQV flip
23. MNP MNP flip
P
y Q'
Q
y
M
N
P'
P
V
Mosaics
A mosaic is composed of
glass, marble, or ceramic
pieces often arranged in
a pattern. The pieces, or
tesserae, are set in cement.
Mosaics are used to
decorate walls, floors,
and gardens.
x
O
V'
P'
M'
x
O
★
★
24. GHF GHF slide
y
H
N'
25. JKL JKL turn
y
J
H'
L
L'
F
F'
K'
x
O
F
x
O
G'
K
J'
C
12
H
J
S
Determine whether each statement is true or false. Draw an example or
counterexample for each.
26. Two triangles with corresponding congruent angles are congruent. false
27. Two triangles with angles and sides congruent are congruent. true
26–27. See margin
for drawings.
6
10
12
R
G
★
6
10
Q
31.
K
36
80
D
E
L
36
Chapter 4 Congruent Triangles
North Carolina Museum of Art, Raleigh. Gift of Mr. & Mrs. Gordon Hanes
64
80
196
28. UMBRELLAS Umbrellas usually have eight congruent
triangular sections with ribs of equal length. Are the
statements JAD IAE and JAD EAI both
correct? Explain.
Both statements are correct because the spokes are the
same length, EA IA, and AE AI.
64
F
35. Given: DEF
Prove: DEF DEF
Proof:
DEF
Given
E
D
F
DE DE, EF EF,
DF DF
Congruence of
segments is reflexive.
D D, E E,
F F
Congruence of is reflexive.
DEF DEF
Def. of s
196
1
2
in another triangle. What else do you need to know to conclude that the four
triangles are congruent?
G
29.
J
7
★ four triangles connect to a square, they have at least one side congruent to a side
A
B
6
21. We need to know ★ 20. All of the small triangles in the figure at the right
are congruent. Name three larger congruent
that all of the corretriangles. UFS, TDV, ACB
sponding angles are
congruent and that the
other corresponding
sides are congruent.
Source: www.dimosaico.com
E
5
4
3
Chapter 4 Congruent Triangles
F
D
B
A
J
I
E
C
G
ALGEBRA For Exercises 29 and 30, use the following information.
QRS GHJ, RS 12, QR 10, QS 6, and HJ 2x 4.
29. Draw and label a figure to show the congruent triangles. See margin.
30. Find x. 8
NAME ______________________________________________ DATE
p. 195
(shown)
Congruent
Triangles and p. 196
Corresponding Parts of Congruent Triangles
X R, Y S, Z T,
XY RS, YZ ST, XZ RT
RS XY, ST YZ, RT XZ
2.
K
?
34.
a. Given
b. Given
c. Congruence of
segments is
symmetric.
d. Given
e. Def. of lines
f. Given
g. Def. of lines
h. All right are .
i. Given
j. Alt. int. are .
k. Given
l. Alt. int. are .
m. Def. of s
XYZ RST
?
D
3. K
L
J
M
C
ABC DCB
G L
5. B
K
JKM LMK
6. R
D
U
E
J
A
E J; F K;
G L; EF
JK
;
EG
JL
; FG
KL
?
B
A
Gl
C
T
A D;
ABC DCB;
ACB DBC;
AB
DC
; AC
DB
;
B
C
CB
R T;
RSU TSU;
RUS TUS;
RU
TU
; RS
TS
;
U
S
SU
NAME
______________________________________________
DATE
/M
G
Hill
195
Skills
Practice,
4-3
Practice
(Average)
?
S
Lesson 4-3
L
C
4. F
Y
RST XYZ
T
Name the corresponding congruent angles and sides for the congruent triangles.
Z
R X, S Y, T Z,
R
Z
B
33. PROOF The statements below can be used to prove that congruence of triangles
is symmetric. Use the statements to construct a correct flow proof. Provide the
reasons for each statement. See p. 233B.
T
Y
S
X
Exercises
1.
A
X
T
A
Identify the congruent triangles in each figure.
ABC JKL
S
R
C
Example
If XYZ RST, name the pairs of
congruent angles and congruent sides.
X R, Y S, Z T
XY
RS
, XZ
RT
, YZ
ST
J
R
S
B
Triangles that have the same size and same shape are
congruent triangles. Two triangles are congruent if and
only if all three pairs of corresponding angles are congruent
and all three pairs of corresponding sides are congruent. In
the figure, ABC RST.
ALGEBRA For Exercises 31 and 32, use the following information.
JKL DEF, mJ 36, mE 64, and mF 3x 52.
31. Draw and label a figure to show the congruent triangles. See margin.
28
32. Find x. 3
Given: RST XYZ
Prove: XYZ RST
Flow Proof:
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
4-3
Study
Guide and
____________
Gl PERIOD
G _____
p. 197 and
Practice,
198 (shown)
Congruent p.
Triangles
Identify the congruent triangles in each figure.
34. PROOF
1.
Copy the flow proof and provide the reasons for each statement.
2.
B
CD
CB
DC
Given: AB
, A
D
, A
D
,
BC
CD
BC
AB
, A
D
㛳
, A
B
㛳
Prove: ACD CAB
B
C
4
S
LMN QPN
3. GKP LMN
2
G L, K M, P N; GK
LM
, KP
MN
, GP
LN
1
4. ANC RBV
D
A R, N B, C V; AN
RB
, NC
BV
, AC
RV
Verify that each of the following transformations preserves congruence, and name
the congruence transformation.
5. PST PST
a. ?
b. ?
Q
L
D
Name the congruent angles and sides for each pair of congruent triangles.
Flow Proof:
AD CB
P
N
C
ABC DRS
3
A
AB CD
M
R
A
AC CA
c. ?
AD DC
d. ?
AB BC
AD BC
f. ?
i. ?
AB CD
k. ?
O
B is a rt. .
1 4
2 3
e. ?
g. ?
j. ?
l. ?
L
N
O
x
T T
P
y
M
S
S
D is a rt. .
6. LMN LMN
y
x
L
P
N
M
D B
h. ?
PS 13
, P S 13
,
LM 22
, LM 22
,
, ST 5
, PT 10
,
ST 5
MN 29
, MN 29
,
, P P ,
PT 10
LN 7, LN 7, L L,
S S, T T; flip
M M , N N; flip
QUILTING For Exercises 7 and 8, refer to the quilt design.
A
C
D
E
G
F
7. Indicate the triangles that appear to be congruent.
ABI EBF, CBD HBG
ACD CAB
8. Name the congruent angles and congruent sides of a pair of
congruent triangles.
m. ?
I
H
Sample answer: A E, ABI EBF, I F;
EB
, B
I BF
, AI EF
AB
NAME
______________________________________________
DATE
/M
G
Hill
198
Gl
★ 35.
B
____________
Gl PERIOD
G _____
Reading
4-3
Readingto
to Learn
Learn Mathematics
Write a flow proof to prove Congruence of triangles is reflexive.
(Theorem 4.4) See margin.
PROOF
Mathematics,
p. 199
Congruent Triangles
Pre-Activity
ELL
Why are triangles used in bridges?
Read the introduction to Lesson 4-3 at the top of page 192 in your textbook.
36. CRITICAL THINKING RST is isosceles with RS = RT,
M, N, and P are midpoints of their sides, S MPS,
MP
and NP
. What else do you need to know to prove
that SMP TNP? SMP TNP, MPS NPT S
R
diagonal braces make the structure stronger and prevent it
from being deformed when it has to withstand a heavy load.
M
N
Reading the Lesson
1. If RST UWV, complete each pair of congruent parts.
P
T
R U
S W
V
U
W
S
R
U
RT
T V
T
S
WV
2. Identify the congruent triangles in each diagram.
Lesson 4-3 Congruent Triangles 197
a.
B
ABC ADC
b.
PQS RQS
Q
C
A
S
D
P
c. M
NAME ______________________________________________ DATE
4-3
Enrichment
Enrichment,
Q
____________ PERIOD _____
p. 200
T
V
N
O
S
P
MNO QPO
U
RTV USV
3. Determine whether each statement says that congruence of triangles is reflexive,
symmetric, or transitive.
Transformations in The Coordinate Plane
The following statement tells one way to map preimage
points to image points in the coordinate plane.
a. If the first of two triangles is congruent to the second triangle, then the second
triangle is congruent to the first. symmetric
(x, y) → (x 6, y 3)
y
(x, y) → (x 6, y 3)
This can be read, “The point with coordinates (x, y) is
mapped to the point with coordinates ( x 6, y 3).”
With this transformation, for example, (3, 5) is mapped to
(3 6, 5 3) or (9, 2). The figure shows how the triangle
ABC is mapped to triangle XYZ.
R
d. R
B
b. If there are three triangles for which the first is congruent to the second and the second
is congruent to the third, then the first triangle is congruent to the third. transitive
Y
c. Every triangle is congruent to itself. reflexive
A
O
C
x
Helping You Remember
X
Z
1. Does the transformation above appear to be a congruence transformation? Explain your
answer. Yes; the transformation slides the figure to the lower right without
changing its size or shape.
4. A good way to remember something is to explain it to someone else. Your classmate Ben is
having trouble writing congruence statements for triangles because he thinks he has to
match up three pairs of sides and three pairs of angles. How can you help him understand
how to write correct congruence statements more easily? Sample answer: Write the
three vertices of one triangle in any order. Then write the corresponding
vertices of the second triangle in the same order. If the angles are written
in the correct correspondence, the sides will automatically be in the
correct correspondence also.
Draw the transformation image for each figure. Then tell whether the
Lesson 4-3 Congruent Triangles 197
Lesson 4-3
www.geometryonline.com/self_check_quiz
In the bridge shown in the photograph in your textbook, diagonal braces
were used to divide squares into two isosceles right triangles. Why do you
think these braces are used on the bridge? Sample answer: The
37. WRITING IN MATH
Answer the question that was posed at the beginning
of the lesson. See margin.
Why are triangles used in bridges?
4 Assess
Include the following in your answer:
• whether the shape of the triangle matters, and
• whether the triangles appear congruent.
Open-Ended Assessment
Modeling Ask students to name
examples of how congruent
triangles are modeled in objects
or in nature. Then they can name
the corresponding congruent
angles and sides and determine
the congruence transformations
applied to the triangles. The
umbrella for Exercise 28 on
p. 196 is an example of rotated
triangles, and students can name
corresponding parts.
Standardized
Test Practice
38. Determine which statement is true given ABC XYZ. B
A
39. ALGEBRA
A
Assessment Options
Practice Quiz 1 The quiz
provides students with a brief
review of the concepts and skills
in Lessons 4-1 through 4-3.
Lesson numbers are given to the
right of the exercises or
instruction lines so students can
review concepts not yet mastered.
Answers
37. Sample answer: Triangles are
used in bridge design for structure
and support. Answers should
include the following.
• The shape of the triangle does
not matter.
• Some of the triangles used in
the bridge supports seem to be
congruent.
198 Chapter 4 Congruent Triangles
B
A
C
XZ
A
B
YZ
C
D
cannot be
determined
D
185
Find the length of DF
if D(5, 4) and F(3, 7). D
5
B
13
57
C
Maintain Your Skills
Mixed Review
Find x. (Lesson 4-2)
40.
75
41. 58
42. 75
x˚
x˚
40˚
Getting Ready for
Lesson 4-4
Prerequisite Skill Students will
learn about proving congruence
using SSS and SAS in Lesson 4-4.
They will use the Distance
Formula to find side lengths of
triangles in a coordinate plane.
Use Exercises 49–51 to determine
your students’ familiarity with
the Distance Formula.
B
C
ZX
x˚
43. x 3, BC 10,
CD 10, BD 5
x˚
30˚
115˚
100˚
42˚
Find x and the measure of each side of the triangle. (Lesson 4-1)
43. BCD is isosceles with BC
CD
, BC 2x 4, BD x 2, and CD 10.
44. Triangle HKT is equilateral with HK x 7 and HT 4x 8.
x 5; HK 12, HT 12, KT 12
Write an equation in slope-intercept form for the line that satisfies the given
conditions. (Lesson 3-4) 45. y 3x 3
3
3
2
45. contains (0, 3) and (4, 3)
46. m , y-intercept 8 y x 8
4
4
47. parallel to y 4x 1;
48. m 4, contains (3, 2)
y 4x 10
contains (3, 1)
y 4x 11
Getting Ready for
the Next Lesson
PREREQUISITE SKILL Find the distance between each pair of points.
(To review the Distance Formula, see Lesson 1-4.)
49. (1, 7), (1, 6)
5
50. (8, 2), (4, 2)
32
P ractice Quiz 1
51. (3, 5), (5, 2)
Lessons 4-1 through 4-3
1. Identify the isosceles triangles in the figure, if F
H and D
G are congruent
perpendicular bisectors. (Lesson 4-1) DFJ, GJF, HJG, DJH,
ALGEBRA ABC is equilateral with AB 2x, BC 4x 7,
and AC x 3.5. (Lesson 4-1)
2. Find x. x 3.5
3. Find the measure of each side.
F
D
m1 60, m2 110, m3 49
(Lesson 4-2)
J
H
AB BC AC 7
4. Find the measure of each numbered angle.
1 3
50˚
5. M J, N K, P L; M
N
JK, N
P KL, M
P JL
2
21˚
70˚
5. If MNP JKL, name the corresponding congruent angles and sides. (Lesson 4-3)
198 Chapter 4 Congruent Triangles
13
G
Reading
Mathematics
Making Concept Maps
Getting Started
When studying a chapter, it is wise to record the main topics and
vocabulary you encounter. In this chapter, some of the new vocabulary
words were triangle, acute triangle, obtuse triangle, right triangle, equiangular
triangle, scalene triangle, isosceles triangle, and equilateral triangle. The
triangles are all related by the size of the angles or the number of
congruent sides.
Advise students to create their
own version of this concept map
and place it in their study
notebooks. Encourage them to use
colored pencils or highlighters to
group related items.
A graphic organizer called a concept map is a convenient way to show
these relationships. A concept map is shown below for the different
types of triangles. The main ideas are in boxes. Any information that
describes how to move from one box to the next is placed along the
arrows.
Teach
Making Concept Maps
Students should discern from the
concept map that if all the angle
measures of a triangle are equal,
then the triangle must also be
acute, equilateral, and isosceles.
Similarly, if all sides of a triangle
are congruent, then the triangle
must also be acute, equiangular,
and isosceles.
Classifying Triangles
Classify by
angle measure.
Classify by the number
of congruent sides.
Angles
Sides
Measure of
one angle
is 90.
Measure of one
angle is greater
than 90.
Measures of all
angles are less
than 90.
At least 2
sides congruent
No sides
congruent
Right
Obtuse
Acute
Isosceles
Scalene
3 congruent
angles
3 sides
congruent
Equiangular
Equilateral
Assess
Study Notebook
Ask students to summarize what
they have learned about using
concept maps to review chapter
material and enhance their
knowledge of chapter concepts.
Reading to Learn
1. Describe how to use the concept map to classify triangles by their side
lengths. See margin.
2. In ABC, mA 48, mB 41, and mC 91. Use the concept map
to classify ABC. obtuse
3. Identify the type of triangle that is linked to both classifications.
equiangular or equilateral
Reading Mathematics Making Concept Maps 199
ELL English Language
Learners may benefit from
writing key concepts from this
activity in their Study Notebooks
in their native language and then
in English.
Answer
1. Sample answer: If side lengths
are given, determine the number
of congruent sides and name the
triangle. Some isosceles triangles
are equilateral triangles.
Reading Mathematics Making Concept Maps 199
Lesson
Notes
Proving Congruence—SSS, SAS
• Use the SSS Postulate to test for triangle congruence.
1 Focus
5-Minute Check
Transparency 4-4 Use as a
quiz or review of Lesson 4-3.
• Use the SAS Postulate to test for triangle congruence.
do land surveyors use
congruent triangles?
Vocabulary
• included angle
Land surveyors mark and establish property
boundaries. To check a measurement, they mark
out a right triangle and then mark a second
triangle that is congruent to the first.
Mathematical Background notes
are available for this lesson on
p. 176D.
SSS POSTULATE Is it always necessary to show that all of the corresponding
parts of two triangles are congruent to prove that the triangles are congruent? In this
lesson, we will explore two other methods to prove that triangles are congruent.
do land surveyors use
congruent triangles?
Ask students:
• What does it mean for two
triangles to be congruent? All
three corresponding sides and all
three corresponding angles are
congruent.
• Would two congruent triangles
have the same perimeter?
Explain. Yes, the three
corresponding sides are congruent
so the sum of the measures of
the sides of the triangles would
be equal.
Congruent Triangles Using Sides
1
Draw a triangle and label the
vertices X, Y, and Z.
2
Use a straightedge to draw
any line and select a point
R. Use a compass to construct
R
S
on such that R
S
X
Z
.
3
Using R as the center, draw
an arc with radius equal
to XY.
Y
Z
X
4
R
Using S as the center, draw an
arc with radius equal to YZ.
5
S
Let T be the point of
intersection of the two
arcs. Draw R
T
and S
T
to
form RST.
R
6
S
Cut out RST and place
it over XYZ. How does
RST compare to XYZ?
RST XYZ
T
R
S
R
S
If the corresponding sides of two triangles are congruent, then the triangles are
congruent. This is the Side-Side-Side Postulate, and is written as SSS.
200
Chapter 4 Congruent Triangles
Paul Conklin/PhotoEdit
Resource Manager
Workbook and Reproducible Masters
Chapter 4 Resource Masters
• Study Guide and Intervention, pp. 201–202
• Skills Practice, p. 203
• Practice, p. 204
• Reading to Learn Mathematics, p. 205
• Enrichment, p. 206
• Assessment, pp. 239, 241
Prerequisite Skills Workbook, pp. 1–2
Teaching Geometry With Manipulatives
Masters, pp. 8, 75
Transparencies
5-Minute Check Transparency 4-4
Answer Key Transparencies
Technology
Interactive Chalkboard
Postulate 4.1
Side-Side-Side Congruence If the sides of
one triangle are congruent to the sides of
a second triangle, then the triangles are
congruent.
Abbreviation: SSS
2 Teach
Z
SSS POSTULATE
B
A
In-Class Examples
C
ABC ZXY
Y
Teaching Tip
Explain to
students that when naming
congruent triangles, it is
customary to use triangle names
in the same order as their
congruent parts. BYA CYA
uses appropriate order to signify
the corresponding sides and
angles that are congruent in the
two triangles. It would be
incorrect to write BAY CYA.
Example 1 Use SSS in Proofs
B
MARINE BIOLOGY The tail of an orca whale
can be viewed as two triangles that share
a common side. Write a two-column proof
to prove that BYA CYA if A
B
A
C
and
Y
C
Y.
B
C
Y
A
AC
BY
CY
Given: AB
; Prove: BYA CYA
Proof:
Statements
Reasons
1. A
AC
CY
B
; B
Y
AY
2. A
Y
3. BYA CYA
1. Given
2. Reflexive Property
3. SSS
1 ENTOMOLOGY The wings of
one type of moth form two
triangles. Write a two-column
proof to prove that FEG FH
, FE
HI,
HIG if EI and G is the midpoint of
I and F
H
.
both E
Example 2 SSS on the Coordinate Plane
COORDINATE GEOMETRY Determine
whether RTZ JKL for R(2, 5), Z(1, 1),
T(5, 2), L(3, 0), K(7, 1), and J(4, 4). Explain.
Use the Distance Formula to show that the
corresponding sides are congruent.
y
F
R
I
J
T
K
G
Z
L
O
x
E
(2 5
)2 (5
2)2
RT 9 9 or 18
1)2
16 1 or 17
1
16 or 17
KL [7 (3)]2
(1
0)2
16 1 or 17
RZ (2 (5
1)2
1)2
H
Statements (Reasons)
1. FE
HI; G is the midpoint of
EI; G is the midpoint of F
H. (Given)
2. FG
HG
; EG
I
G. (Midpoint
Theorem)
3. FEG HIG. (SSS)
JK [4 (7)]2
(4
1)2
9 9 or 18
TZ (5 (2
1)2
Power
Point®
X
JL [4 (3)]2
(4
0)2
1
16 or 17
COORDINATE GEOMETRY
Determine whether WDV
2 MLP. Explain.
RT JK, TZ KL, and RZ JL. By definition of congruent segments, all
corresponding segments are congruent. Therefore, RTZ JKL by SSS.
y
SAS POSTULATE
Suppose you are given the measures of two sides and the
angle they form, called the included angle . These conditions describe a unique
triangle. Two triangles in which corresponding sides and the included pairs of
angles are congruent provide another way to show that triangles are congruent.
www.geometryonline.com/extra_examples
D
O
P
x
V
W
Lesson 4-4 Proving Congruence—SSS, SAS 201
Jeffrey Rich/Pictor International/PictureQuest
L
M
Differentiated Instruction
Logical/Mathematical Students can use a systematic approach to write
the proofs for problems and examples in this lesson. Have students start
by looking for possible methods of proof using SSS or SAS. Then they
should examine the problem to determine how much necessary
information is given and how they can find any other information that they
need for the proof. Finally, they can draw on prior knowledge of midpoints,
distances, angle relationships, and so on, to extract any other necessary
information and compile the facts for the final proof.
WD ML, DV LP, and
VW PM. By definition of
congruent segments, all
corresponding segments are
congruent. Therefore,
WDV MLP by SSS.
Lesson 4-4 Proving Congruence—SSS, SAS
201
SAS POSTULATE
Postulate 4.2
In-Class Examples
Side-Angle-Side Congruence If two sides and
Power
Point®
B
3 Write a proof for the
following.
F
Abbreviation: SAS
R
D
the included angle of one triangle are congruent
to two sides and the included angle of another
triangle, then the triangles are congruent.
A
S
E
C
ABC FDE
You can also construct congruent triangles given two sides and the included angle.
Q
T
Q
|| T
S
Given: R
Q
R
TS
Prove: QRT STR
Proof:
Statements (Reasons)
1. RQ
|| TS
, RQ
TS
(Given)
2. QRT STR (Alt. int. are .)
3. RT
TR
(Reflexive Property)
4. QRT STR (SAS)
Congruent Triangles using Two Sides and the Included Angle
1
Draw a triangle and
label its vertices A,
B, and C.
Select a point K on
line m . Use a
compass to
construct K
L on m
such that KL B
C
.
2
a.
AB
that JK
.
Draw JL
to
complete JKL.
J
m
B
5
K
C
L
m
m
K
K
L
Cut out JKL and place it over ABC. How does JKL compare to ABC? JKL ABC
SAS
Study Tip
b.
4 Construct JK
such
A
4 Determine which postulate
can be used to prove that the
triangles are congruent. If it
is not possible to prove that
they are congruent, write not
possible.
Construct an angle
congruent to B
as a side
using KL
of the angle and
point K as the
vertex.
3
SSS
Example 3 Use SAS in Proofs
Flow Proofs
Write a flow proof.
Flow proofs can be written
vertically or horizontally.
Given: X is the midpoint of BD
.
X is the midpoint of A
C
.
D
A
Prove: DXC BXA
X
C
Flow Proof:
B
X is the midpoint of DB.
DX BX
Given
Midpoint Theorem
X is the midpoint of AC.
CX AX
DXC BXA
Given
Midpoint Theorem
SAS
DXC BXA
Vertical s are .
202 Chapter 4 Congruent Triangles
Unlocking Misconceptions
Figures Point out that figures will not always be marked and that it is
up to students to draw on their knowledge of geometric concepts to
prove congruence. Stress the importance of using only information that
is given and not forming any assumptions about two figures just
because they appear to be congruent.
202
Chapter 4 Congruent Triangles
L
Example 4 Identify Congruent Triangles
3 Practice/Apply
Determine which postulate can be used to prove that the triangles are
congruent. If it is not possible to prove that they are congruent, write
not possible.
a.
b.
Study Notebook
Each pair of corresponding sides
are congruent. The triangles are
congruent by the SSS Postulate.
Concept Check
The triangles have three pairs of
corresponding angles congruent. This
does not match the SSS Postulate or
the SAS Postulate. It is not possible to
prove the triangles congruent.
1. OPEN ENDED Draw a triangle and label the vertices. Name two sides and
the included angle. See margin.
Have students—
• add the definitions/examples of
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 4.
• include a simple example of a proof
using SSS and a proof using SAS.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
2. FIND THE ERROR Carmelita and Jonathan are trying to determine whether
ABC is congruent to DEF.
A
Jonathan
Congruence
cannot be
determined.
Carmelita
πABC πDEF
by SAS
D
2
78˚
C
1.5
B
2
F
48˚
1.5
E
Who is correct and why? Jonathan; the measure of DEF is needed to use SAS.
Guided Practice
GUIDED PRACTICE KEY
Exercises
Examples
3–4
5–6
7–8
9
2
3
4
1
Determine whether EFG MNP given the coordinates of the vertices. Explain.
3. E(4, 3), F(2, 1), G(2, 3), M(4, 3), N(2, 1), P(2, 3) 3–4. See margin.
4. E(2, 2), F(4, 6), G(3, 1), M(2, 2), N(4, 6), P(3, 1)
5. Write a flow proof.
B
E and B
Given: D
C
bisect each
other.
Prove: DGB EGC
6. Write a two-column proof.
Given: KM
㛳 JL
, K
M
JL
Prove: JKM MLJ
K
D
G
E
J
M
About the Exercises…
C
Exercise 5
Organization by Objective
• SSS Postulate: 10–13, 20–21,
28–29
• SAS Postulate: 14–19
L
Exercise 6
5 – 6. See p. 233B.
Determine which postulate can be used to prove that the triangles are congruent.
If it is not possible to prove that they are congruent, write not possible.
SAS
8. SSS
7.
Lesson 4-4 Proving Congruence—SSS, SAS
203
Answer
1. Sample answer: In QRS,
R is the included angle
of the sides
QR
and R
S
.
R
Q
S
FIND THE ERROR
Even though the
figures look congruent
and E appears to have the
same measure as B, this cannot
be assumed based on the
information given in the figures.
Carmelita’s answer would have
been correct if it could be shown
that mE 78.
3. EG 2, MP 2, FG 4, NP 4, EF 20
, and
MN 20. The corresponding sides have the same
measure and are congruent. EFG MNP by SSS.
4. EG 10, FG 26, EF 68, MP 2,
NP 26, and MN 20. The corresponding sides
are not congruent, so the triangles are not congruent.
Odd/Even Assignments
Exercises 10–27 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
Assignment Guide
Basic: 11–25 odd, 28–47
Average: 11–27 odd, 28–47
Advanced: 10–26 even, 28–43
(optional: 44–47)
Lesson 4-4 Proving Congruence—SSS, SAS
203
Application
Answers
9. Given: T is the midpoint of 苶
SQ
苶.
苶R
苶⬵苶
QR
苶
S
Prove: 䉭SRT ⬵ 䉭QRT
R
S
T
Q
Proof:
Statements (Reasons)
1.T is the midpoint of 苶
SQ
苶. (Given)
2.S
苶T苶 ⬵ 苶TQ
苶 (Def. of midpoint)
3.S
苶R
苶⬵苶
QR
苶 (Given)
4.R
苶T苶 ⬵ 苶
RT苶 (Reflexive Prop.)
5.䉭SRT ⬵ 䉭QRT (SSS)
10. JK ⫽兹苶
20, KL ⫽ 兹61
苶,
JL ⫽ 兹53
苶, FG ⫽ 兹苶
20,
GH ⫽ 兹61
苶, and FH ⫽ 兹53
苶. Each
pair of corresponding sides have
the same measure so they are
congruent. 䉭JKL ⬵ 䉭FGH by SSS.
11. JK ⫽ 兹10
苶, KL ⫽ 兹10
苶,
20, FG ⫽ 兹苶2, GH ⫽ 兹苶
50,
JL ⫽ 兹苶
and FH ⫽ 6. The corresponding
sides are not congruent so 䉭JKL
is not congruent to 䉭FGH.
12. JK ⫽ 兹苶
50, KL ⫽ 兹苶
13, JL ⫽ 5,
FG ⫽ 兹苶8, GH ⫽ 兹苶
13, and
FH ⫽ 5. The corresponding sides
are not congruent so 䉭JKL is not
congruent to 䉭FGH.
13. JK ⫽ 兹苶
10, KL ⫽ 兹苶
10,
JL ⫽ 兹苶
20, FG ⫽ 兹苶
10,
GH ⫽ 兹苶
10, and FH ⫽ 兹苶
20. Each
pair of corresponding sides have
the same measure so they are
congruent. 䉭JKL ⬵ 䉭FGH by SSS.
9. PRECISION FLIGHT The United States
Navy Flight Demonstration Squadron,
the Blue Angels, fly in a formation
that can be viewed as two triangles
with a common side. Write a
two-column proof to prove that
䉭SRT ⬵ 䉭QRT if T is the midpoint
QR
of 苶
SQ
苶 and S
苶R
苶⬵苶
苶. See margin.
R
T
S
Q
★ indicates increased difficulty
Practice and Apply
For
Exercises
See
Examples
10–13
14–19
20–21,
28–29
22–27
2
3
1
4
Extra Practice
See page 761.
Determine whether 䉭JKL ⬵ 䉭FGH given the coordinates of the vertices. Explain.
10. J(⫺3, 2), K(⫺7, 4), L(⫺1, 9), F(2, 3), G(4, 7), H(9, 1)
11. J(⫺1, 1), K(⫺2, ⫺2), L(⫺5, ⫺1), F(2, ⫺1), G(3, ⫺2), H(2, 5)
12. J(⫺1, ⫺1), K(0, 6), L(2, 3), F(3, 1), G(5, 3), H(8, 1)
13. J(3, 9), K(4, 6), L(1, 5), F(1, 7), G(2, 4), H(⫺1, 3)
10–13. See margin.
Write a flow proof. 14–19. See p. 233C.
14. Given: 苶
AE
FC
BC
15. Given: 苶
RQ
TQ
WQ
苶⬵苶
苶, A
苶B
苶⬵苶
苶,
苶⬵苶
苶⬵Y
苶Q
苶⬵苶
苶
B苶
E⬵苶
B苶
F
⬔RQY ⬵ ⬔WQT
苶
Prove: 䉭AFB ⬵ 䉭CEB
Prove: 䉭QWT ⬵ 䉭QYR
R
B
Y
Q
C
F
E
A
W
Write a two-column proof.
16. Given: 䉭CDE is isosceles.
G is the midpoint of 苶
CE
苶.
Prove: 䉭CDG ⬵ 䉭EDG
D
T
17. Given: 䉭MRN ⬵ 䉭QRP
⬔MNP ⬵ ⬔QPN
Prove: 䉭MNP ⬵ 䉭QPN
M
Q
R
C
G
N
E
18. Given: 苶
AC
GC
苶⬵苶
苶
EC
苶
苶 bisects A
苶G
苶.
Prove: 䉭GEC ⬵ 䉭AEC
P
★ 19. Given: 䉭GHJ ⬵ 䉭LKJ
Prove: 䉭GHL ⬵ 䉭LKG
H
K
A
J
E
C
G
G
204
Chapter 4 Congruent Triangles
Elaine Thompson/AP/Wide World Photos
29. Sample answer: The properties of congruent triangles help land surveyors double check
measurements. Answers should include the following.
• If each pair of corresponding angles and sides are congruent, the triangles are
congruent by definition. If two pairs of corresponding sides and the included angle are
congruent, the triangles are congruent by SAS. If each pair of corresponding sides are
congruent, the triangles are congruent by SSS.
• Sample answer: Architects also use congruent triangles when designing buildings.
204 Chapter 4 Congruent Triangles
L
20–21. See p. 233C.
20. CATS A cat’s ear is triangular in
shape. Write a two-column proof to
PN
prove RST PNM if RS
,
MP
RT
, S N, and
T M.
21. GEESE This photograph shows a
flock of geese flying in formation.
Write a two-column proof to prove
that EFG HFG, if EF
HF
and
G is the midpoint of E
H
.
NAME ______________________________________________ DATE
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
4-4
Study
Guide and
p. 201
andSAS
p. 202
Proving(shown)
Congruence—SSS,
SSS Postulate You know that two triangles are congruent if corresponding sides are
congruent and corresponding angles are congruent. The Side-Side-Side (SSS) Postulate lets
you show that two triangles are congruent if you know only that the sides of one triangle
are congruent to the sides of the second triangle.
If the sides of one triangle are congruent to the sides of a second triangle,
then the triangles are congruent.
SSS Postulate
P
R
Example
Write a two-column proof.
Given: A
B
DB
and C is the midpoint of A
D
.
Prove: ABC DBC
E
S N
B
A
M
G
F
T
H
Statements
Reasons
B
DB
1. A
1. Given
2. C is the midpoint of A
D
.
2. Given
D
C
DC
3. AC
3. Definition of midpoint
BC
4. BC
4. Reflexive Property of 5. ABC DBC
5. SSS Postulate
Exercises
Write a two-column proof.
Determine which postulate can be used to prove that the triangles are congruent.
If it is not possible to prove that they are congruent, write not possible.
SSS
23.
not possible
22.
not possible
Z
X
T
U
R
S
XY
, AC
XZ
, B
C
YZ
Given: AB
Prove: ABC XYZ
UT
, RT
US
Given: RS
Prove: RST UTS
Statements
Statements
Reasons
Reasons
1.R
S
UT
1. Given
US
RT
2.S
T
TS
2. Refl. Prop.
3.RST UTS 3. SSS Post.
NAME
______________________________________________
DATE
/M
G
Hill
201
____________
Gl PERIOD
G _____
Skills
Practice,
4-4
Practice
(Average)
SSS or SAS
25.
2.
Y
C
1.A
B
XY
1.Given
XZ
AC
C
B
YZ
2.ABC XYZ 2. SSS Post.
Gl
24.
B
A
Lesson 4-4
1.
p. 203 and
Practice,
p. 204 (shown)
Proving Congruence—SSS,
SAS
Determine whether DEF PQR given the coordinates of the vertices. Explain.
1. D(6, 1), E(1, 2), F(1, 4), P(0, 5), Q(7, 6), R(5, 0)
DE 52
, PQ 52
, EF 210
, QR 210
, DF 52
, PR 52
.
DEF PQR by SSS since corresponding sides have the same
measure and are congruent.
2. D(7, 3), E(4, 1), F(2, 5), P(2, 2), Q(5, 4), R(0, 5)
DE 13
, PQ 13,
EF 25
, QR 26
, DF 29
, PR 13
.
Corresponding sides are not congruent, so DEF is not congruent
to PQR.
3. Write a flow proof.
Given: R
S
TS
V is the midpoint of RT
.
Prove: RSV TSV
Baseball
The infield is a square
90 feet on each side.
A
B
SV SV
Reflexive
Property
RS TS
Given
RSV TSV
V is the
midpoint of RT.
Given
SSS
RV VT
Definition
of midpoint
Determine which postulate can be used to prove that the triangles are congruent.
If it is not possible to prove that they are congruent, write not possible.
4.
F
5.
not possible
6.
SAS or SSS
SSS
7. INDIRECT MEASUREMENT To measure the width of a sinkhole on
his property, Harmon marked off congruent triangles as shown in the
diagram. How does he know that the lengths AB and AB are equal?
C
NAME
______________________________________________
DATE
/M
G
Hill
204
Gl
Answer the question that was posed at the beginning of
the lesson. See margin.
How do land surveyors use congruent triangles?
Include the following in your answer:
• description of three methods to prove triangles congruent, and
• another example of a career that uses properties of congruent triangles.
Mathematics,
p. 205
Proving Congruence—SSS, SAS
Pre-Activity
Lesson 4-4 Proving Congruence—SSS, SAS 205
(tl)G.K. & Vikki Hart/PhotoDisc, (tr)Chase Swift/CORBIS, (b)Index Stock
NAME ______________________________________________ DATE
Read the introduction to Lesson 4-4 at the top of page 200 in your textbook.
Why do you think that land surveyors would use congruent right triangles
rather than other congruent triangles to establish property boundaries?
Sample answer: Land is usually divided into rectangular lots,
so their boundaries meet at right angles.
Reading the Lesson
1. Refer to the figure.
N
a. Name the sides of LMN for which L is the included angle.
M
L
, L
N
M
____________ PERIOD _____
p. 206
L
N
, NM
L
M
, M
N
2. Determine whether you have enough information to prove that the two triangles in each
figure are congruent. If so, write a congruence statement and name the congruence
postulate that you would use. If not, write not possible.
b.
B
E
D
D
C
Congruent Parts of Regular Polygonal Regions
Congruent figures are figures that have exactly the same size and shape. There are many
ways to divide regular polygonal regions into congruent parts. Three ways to divide an
equilateral triangular region are shown. You can verify that the parts are congruent by
tracing one part, then rotating, sliding, or reflecting that part on top of the other parts.
L
c. Name the sides of LMN for which M is the included angle.
a. A
4-4
Enrichment
Enrichment,
ELL
How do land surveyors use congruent triangles?
b. Name the sides of LMN for which N is the included angle.
www.geometryonline.com/self_check_quiz
B
C
____________
Gl PERIOD
G _____
Reading
4-4
Readingto
to Learn
Learn Mathematics
29. WRITING IN MATH
A
Since ACB and ACB are vertical angles, they are
A
B
AC
and BC
B C. So
congruent. In the figure, AC
ABC ABC by SAS. By CPCTC, the lengths AB and AB are equal.
E
D
Source: www.mlb.com
S
T
Proof:
26–27. See pp. 233C–233D.
28. CRITICAL THINKING Devise a plan and
write a two-column proof for the following.
E
F
B, A
E
F
C, See p. 233D.
Given: D
A
DB
DB
E
, C
F
Prove: ABD CDB
R
V
ABD CBD; SAS
c. E
H
and D
G
bisect each other.
not possible
d.
R
G
E
U
F
D
F
G
H
DEF GHF; SAS
S
Lesson 4-4
BASEBALL For Exercises 26 and 27, use the following information.
A baseball diamond is a square with four right angles and all sides congruent.
★ 26. Write a two-column proof to prove that the distance from first base to third base
is the same as the distance from home plate to second base.
★ 27. Write a two-column proof to prove that the angle formed by second base, home
plate, and third base is the same as the angle formed by second base, home
plate, and first base.
T
RSU TSU; SSS
Helping You Remember
1. Divide each square into four congruent parts. Use three
different ways. Sample answers are shown.
3. Find three words that explain what it means to say that two triangles are congruent and
that can help you recall the meaning of the SSS Postulate.
Sample answer: Congruent triangles are triangles that are the same size
and shape, and the SSS Postulate ensures that two triangles with three
corresponding sides congruent will be the same size and shape.
Lesson 4-4 Proving Congruence—SSS, SAS
205
4 Assess
Standardized
Test Practice
Open-Ended Assessment
Speaking Ask students to
explain in their own words how
they can use SSS and SAS to
prove triangle congruence. Then
students can explain how they
would approach proving triangle
congruence for some of the
exercises in Lesson 4-4.
30. Which of the following statements about the figure is true? C
A 90 a b
B a b 90
b˚
a˚
C a b 90
D a b
31. Classify the triangle with the measures of the angles in the ratio 3:6:7. B
A isosceles
B acute
C obtuse
D right
Maintain Your Skills
Mixed Review Identify the congruent triangles in each figure.
32. B
E
Getting Ready for
Lesson 4-5
33.
(Lesson 4-3)
X
34. M
N
L
P
C
W
Z
Y
WXZ YXZ
Prerequisite Skill Students will
learn about proving congruence
with ASA and AAS in Lesson 4-5.
They will apply the Angle Sum
Theorem and concepts of angle
and segment bisection toward
proving triangle congruence. Use
Exercises 44–47 to determine
your students’ familiarity with
bisectors of segments and angles.
A
D
Find each measure if P
QR
Q
.
(Lesson 4-2)
35. m2 78
37. m5 68
39. m1 59
LMP NPM
ACB DCE
P
Q
1
36. m3 102
38. m4 22
40. m6 34
56˚
43˚
T
For Exercises 41–43, use the graphic
at the right. (Lesson 3-3)
41. Find the rate of change from
first quarter to the second
quarter. 1
42. Find the rate of change from
the second quarter to the third
quarter. 1.4
43. Compare the rate of change
from the first quarter to the
second, and the second quarter
to the third. Which had the
greater rate of change?
Assessment Options
Quiz (Lessons 4-3 and 4-4) is
available on p. 239 of the Chapter 4
Resource Masters.
Mid-Chapter Test (Lessons 4-1
through 4-4) is available on
p. 241 of the Chapter 4 Resource
Masters.
There is a steeper rate of
decline from the second quarter
to the third.
4
3
5
78˚
2
6
R
USA TODAY Snapshots®
GDP slides in 2001
Gross domestic product in private industries,
which generate 88% of GDP, slowed to 4.1% in
2000 from 4.8% in 1999.
1.3%
Percentage changes for the
first three quarters of 2001:
0.3%
0%
–1.1%
First
quarter
Second
quarter
Third
quarter
Source: The Bureau of Economic Analysis
By Shannon Reilly and Suzy Parker, USA TODAY
Getting Ready for
the Next Lesson
៮៬ and AE
៮៬ are angle bisectors
PREREQUISITE SKILL BD
and segment bisectors. Name the indicated segments
and angles.
B
44.
45.
46.
47.
a segment congruent to EC
BE
an angle congruent to ABD CBD
an angle congruent to BDC BDA
a segment congruent to A
D CD
206 Chapter 4 Congruent Triangles
206 Chapter 4 Congruent Triangles
E
X
(To review bisectors of segments and angles, see Lessons 1-5 and 1-6.)
A
D
C
Proving Congruence—ASA, AAS
Lesson
Notes
• Use the ASA Postulate to test for triangle congruence.
1 Focus
• Use the AAS Theorem to test for triangle congruence.
Vocabulary
are congruent triangles
used in construction?
• included side
5-Minute Check
Transparency 4-5 Use as a
quiz or review of Lesson 4-4.
The Bank of China Tower in Hong Kong has triangular
trusses for structural support. These trusses form
congruent triangles. In this lesson, we will explore two
additional methods of proving triangles congruent.
Mathematical Background notes
are available for this lesson on
p. 176D.
ASA POSTULATE Suppose you were given the measures of two angles of a
triangle and the side between them, the included side . Do these measures form a
unique triangle?
Congruent Triangles Using Two Angles and Included Side
1 Draw a triangle and
label its vertices A, B,
and C.
2 Draw any line
3 Construct an angle
m
and select a point L.
Construct L
K
such
that L
K
C
B
.
congruent to C at L
as a side of
using LK
the angle.
4 Construct an angle
congruent to B at
as a side
K using LK
of the angle. Label the
point where the new
sides of the angles
meet J.
J
A
C
B
m
L
K
m
m
L
L
K
5 Cut out JKL and place it over ABC. How does JKL compare to ABC?
are congruent triangles
used in construction?
Ask students:
• How do the congruent triangles
in the trusses contribute to the
appearance of the structure?
Sample answer: They make the
structure visually appealing.
• How would using congruent
triangles make the structure
easier to assemble? The triangles
could be manufactured in bulk and
construction workers could place
any triangle in any location.
K
JKL ABC
This construction leads to the Angle-Side-Angle Postulate, written as ASA.
Study Tip
Postulate 4.3
Reading Math
Angle-Side-Angle Congruence If two
angles and the included side of one
triangle are congruent to two angles
and the included side of another
triangle, then the triangles are
congruent.
The included side refers to
the side that each of the
angles share.
C04-174C
T
C
W
H
G
R
RTW CGH
Abbreviation: ASA
Lesson 4-5 Proving Congruence—ASA, AAS 207
Sylvain Grandadam/Photo Researchers
Resource Manager
Workbook and Reproducible Masters
Chapter 4 Resource Masters
• Study Guide and Intervention, pp. 207–208
• Skills Practice, p. 209
• Practice, p. 210
• Reading to Learn Mathematics, p. 211
• Enrichment, p. 212
Teaching Geometry With Manipulatives
Masters, pp. 8, 16, 17, 77
Transparencies
5-Minute Check Transparency 4-5
Real-World Transparency 4
Answer Key Transparencies
Technology
GeomPASS: Tutorial Plus, Lesson 10
Interactive Chalkboard
Multimedia Applications: Virtual Activities
Lesson x-x Lesson Title 207
Example 1 Use ASA in Proofs
2 Teach
D
Proof: W E because
alternate interior angles are
congruent. By the Midpoint
Theorem, WL EL. Since
vertical angles are congruent,
WLR ELD. WRL EDL
by ASA.
P
CP
CP
Since CP
bisects BCR and BPR, BCP RCP and BPC RPC. by the Reflexive Property. By ASA, BCP RCP.
Given: L is the midpoint
.
of WE
R
|| E
D
W
Prove: WRL EDL
E
B
Proof:
Power
Point®
1 Write a paragraph proof.
W
BCP RCP
Prove:
In-Class Examples
L
R
Write a paragraph proof.
Given: CP
bisects BCR and BPR.
ASA POSTULATE
R
C
AAS THEOREM Suppose you are given the measures of two angles and a
nonincluded side. Is this information sufficient to prove two triangles congruent?
Angle-Angle-Side Congruence
Model
1. Draw a triangle on a piece of
patty paper. Label the vertices
A, B, and C.
2. Copy AB
, B, and C on
another piece of patty paper
and cut them out.
B
B
A
A
C
B
B
A
3. Assemble them to form a
triangle in which the side is
not the included side of the
angles.
B
B
C
C
B
C
Analyze 1. They are congruent.
1. Place the original ABC over the assembled figure. How do the two triangles compare?
2. Make a conjecture about two triangles with two angles and the nonincluded side of one triangle
congruent to two angles and the nonincluded side of the other triangle. The triangles are congruent.
This activity leads to the Angle-Angle-Side Theorem, written as AAS.
Theorem 4.5
Angle-Angle-Side Congruence If two angles and
a nonincluded side of one triangle are congruent to
the corresponding two angles and side of a second
triangle, then the two triangles are congruent.
J
Abbreviation: AAS
Example: JKL CAB
Proof
K
A
L
C
B
Theorem 4.5
T
Given: M S, J R, M
P
S
T
M
R
Prove: JMP RST
Proof:
Statements
1. M S, J R, M
P
S
T
2. P T
3. JMP RST
J
Reasons
1. Given
2. Third Angle Theorem
3. ASA
208 Chapter 4 Congruent Triangles
Geometry Activity
Materials: patty paper, straightedge, scissors
• When students are copying B and C, tell them to extend the sides so
that they are longer than the sides in the original triangle. Explain that these
sides represent unknown side lengths.
• In Step 2, make sure students copy C and not A.
208
Chapter 4 Congruent Triangles
S
P
Study Tip
Overlapping
Triangles
When triangles overlap,
it is a good idea to draw
each triangle separately
and label the congruent
parts.
Example 2 Use AAS in Proofs
AAS THEOREM
B
Write a flow proof.
Given: EAD EBC
BC
AD
A
A
E
BE
Prove:
Flow Proof:
In-Class Examples
D
C
Power
Point®
2 Write a proof.
E
Given: NKL NJM
KL
JM
Prove: LN
MN
EAD EBC
Given
AD BC
ADE BCE
AE BE
Given
AAS
CPCTC
J
K
M
L
E E
Reflexive Property
N
Proof:
Statements (Reasons)
1. N N (Reflex. Prop. of )
2. NKL NJM (Given)
3. KL
JM
(Given)
4. JNM KNL (AAS)
5. LN
MN
(CPCTC)
You have learned several methods for proving triangle congruence. The Concept
Summary lists ways to help you determine which method to use.
Methods to Prove Triangle Congruence
Definition of
Congruent Triangles
All corresponding parts of one triangle are congruent to
the corresponding parts of the other triangle.
SSS
The three sides of one triangle must be congruent to the
three sides of the other triangle.
SAS
Two sides and the included angle of one triangle must be
congruent to two sides and the included angle of the other
triangle.
ASA
Two angles and the included side of one triangle must be
congruent to two angles and the included side of the other
triangle.
AAS
Two angles and a nonincluded side of one triangle must be
congruent to two angles and side of the
other triangle.
3 STANCES When Ms. Gomez
puts her hands on her hips,
she forms two triangles with
her upper body and arms.
Suppose her arm lengths AB
and DE measure 9 inches, and
AC and EF measure 11 inches.
Also suppose that you are
C
DF
.
given that B
Determine whether ABC EDF. Justify your answer.
Example 3 Determine if Triangles Are Congruent
Architect
About 28% of architects
are self-employed.
Architects design a variety
of buildings including
offices, retail spaces,
and schools.
ARCHITECTURE This glass chapel was designed
by Frank Lloyd Wright’s son, Lloyd Wright.
TV
Suppose the redwood supports, T
U
and ,
measure 3 feet, TY 1.6 feet, and mU and
mV are 31. Determine whether TYU TYV.
Justify your answer.
V
Y
D
B
U
Explore
We are given three measurements of
each triangle. We need to determine
whether the two triangles are
congruent.
Plan
Since mU mV, U V.
TV
TY
Likewise, TU TV so TU
, and TY TY so T
Y
. Check each
possibility using the five methods you know.
Solve
We are given information about side-side-angle (SSA). This is not a
method to prove two triangles congruent.
Online Research
For information about
a career as an
architect, visit:
www.geometryonline.
com/careers
T
A
E
C
F
With B
C
DF, you could use
SSS to prove ABC EDF.
(continued on the next page)
www.geometryonline.com/extra_examples
Lesson 4-5 Proving Congruence—ASA, AAS 209
(l)Dennis MacDonald/PhotoEdit, (r)Michael Newman/PhotoEdit
Differentiated Instruction
Intrapersonal Ask students to study the proofs for the examples in this
lesson and note the properties that recur, such as the reflexive properties
of angles and segments, bisectors, midpoints, and so on. Students can
start a list of helpful tools and things to watch for when they are working
proofs and include recurring properties, theorems, formulas and methods
that they can refer to in later lessons. They can also look at the order of
the steps in paragraph proofs, flow proofs, and two-column proofs for
similarities and differences.
Lesson 4-5 Proving Congruence—ASA, AAS 209
Intervention
A student may
ask about
proving
congruence
with AAA. Explain that while
congruent triangles do share
three congruent angles, AAA is
not a possible tool for proving
congruence of triangles because
two triangles with three
corresponding congruent
angles can be similar but not
congruent. Provide students
with an example of two
different-sized similar triangles.
Examine Use a compass, protractor, and ruler to draw
a triangle with the given measurements.
1.6 cm
For simplicity of measurement, we will
use centimeters instead of feet, so the
measurements of the construction and those
31°
of the support beams will be proportional.
3.0 cm
• Draw a segment 3.0 centimeters long.
• At one end, draw an angle of 31°. Extend
the line longer than 3.0 centimeters.
• At the other end of the segment, draw an arc with a radius of
1.6 centimeters such that it intersects the line.
New
Notice that there are two possible segments that could determine
the triangle. Since the given measurements do not lead to a unique
triangle, we cannot show that the triangles are congruent.
Concept Check
1–2. See margin.
3 Practice/Apply
Third Angle Theorem. Postulates are accepted as true without proof.
Guided Practice
GUIDED PRACTICE KEY
Study Notebook
Have students—
• add the definitions/examples of
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 4.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
1. Find a counterexample to show why AAA (Angle-Angle-Angle) cannot be used
to prove triangle congruence.
2. OPEN ENDED Draw a triangle and label the vertices. Name two angles and the
included side.
3. Explain why AAS is a theorem, not a postulate. AAS can be proven using the
Exercises
Examples
4, 6
5, 7
8
1
2
3
Write a flow proof. 4–5. See p. 233D.
KJ, G
HJ
4. Given: GH
㛳
K
㛳
Prove: GJK JGH
G
YZ
5. Given: XW
㛳
, X Z
Prove: WXY YZW
X
H
K
Y
W
J
Z
Write a paragraph proof. 6–7. See p. 233D.
6. Given: QS
bisects RST; R T. 7. Given: E K, DGH DHG
KH
Prove: QRS QTS
EG
Prove:
EGD
KHD
R
D
Q
Answers
S
T
1. Two triangles can have
corresponding congruent angles
without corresponding congruent
sides. A D, B E, and
C F. However, AB
DE, so
ABC DEF.
Application
E
8. PARACHUTES Suppose S
T
and M
L
each measure
R and M
7 feet, S
K
each measure 5.5 feet, and
mT mL 49. Determine whether SRT MKL.
Justify your answer. See margin.
D
A
C
C
A
210
210 Chapter 4 Congruent Triangles
F
2. Sample answer: In ABC, A
B
is
the included side of A and B.
B
Chapter 4 Congruent Triangles
H
K
R
8. This cannot be determined. The
information given cannot be used
with any of the triangle congruence
postulates, theorems or the
definition of congruent triangles. By
construction, two different triangles
can be shown with the given
information. Therefore, it cannot be
determined if SRT MKL.
5.5
49
7
5.5
K
M
S
T
B
E
G
L
★ indicates increased difficulty
Practice and Apply
For
Exercises
See
Examples
9, 11, 14,
15–18
10, 12,
13, 19,
20
21–28
2
1
F
Organization by Objective
• ASA Postulate: 10, 12, 13, 19,
20
• AAS Theorem: 9, 11, 14–18,
21–28
J
E
D
3
K
Odd/Even Assignments
Exercises 9–28 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
G
Extra Practice
See page 762.
About the Exercises…
Write a flow proof. 9–14. See pp. 233D– 233E.
9. Given: EF
GH
10. Given: DE
DK
GH
㛳
, E
F
㛳 JK
, bisects JE
.
K
K
H
Prove: EGD JGK
Prove: E
K
H
G
E
12. Given: EJ 㛳 EF
GH
FK
KH
, JG
㛳
, Prove: EJG FKH
11. Given: V S, TV
QS
Prove: V
SR
R
T
J
S
1
K
Assignment Guide
Basic: 9–17 odd, 21–29 odd,
30–41
Average: 9–29 odd, 30–41
Advanced: 10–28 even, 29–38
(optional: 39–41)
2
R
V
E
Q
PQ
13. Given: MN
, M Q
2 3
Prove: MLP QLN
F
G
H
14. Given: Z is the midpoint of CT
.
Y㛳
E
C
T
Prove: Y
Z
E
Z
L
E
Answers
C
15. Given: NOM POR, NM
⊥
MR
,
P
R
⊥
MR
, NM
PR
Prove: MO
OR
Z
1
M
2
N
3
4
P
T
Q
Y
Write a paragraph proof. 15–18. See margin.
15. Given: NOM POR,
16. Given: DL
bisects B
N
,
N
M
M
R
XLN XDB
R
M
R, N
M
P
R
Prove: L
DB
P
N
Prove: M
O
O
R
B
N
M
D
X
O
L
R
F
T
B
D
X
L
C
J
R
X
H
G
E
N
SY
18. Given: TX
㛳
TXY TSY
Prove: TSY YXT
17. Given: F J, E H
E
GH
C
F
HJ
Prove: E
O
Proof: Since N
M
⊥M
R
and
PR
⊥
MR
, M and R are right
angles. M R because all
right angles are congruent. We
know that NOM POR and
NM
PR
. By AAS, NMO PRO. M
O
OR
by CPCTC.
16. Given: DL bisects B
N
.
XLN XDB
Prove: LN
DB
P
M
P
N
S
Y
Lesson 4-5 Proving Congruence—ASA, AAS
17. Given: F J, E H
F
EC
GH
H
G
Prove: EF HJ
C
E
Proof: We are given that
J
F J, E H, and
EC
GH
. By the Reflexive Property, CG
CG
. Segment
addition results in EG EC CG and CH CG GH. By the
definition of congruence, EC GH and CG CG. Substitute to
find EG CH. By AAS, EFG HJC. By CPCTC, EF HJ.
211
N
Proof: Since D
L bisects B
N
,
BX
XN
. XLN XDB .
LXN DXB because vertical
angles are congruent. LXN DXB by AAS. LN
DB
by CPCTC.
18. Given: TX
|| S
Y
X
T
TXY TSY
Prove: TSY YXT
Proof: Since TX
|| S
Y
,
S
Y
YTX TYS by Alternate
Interior Angles Theorem. TY
TY
by the Reflexive Property.
Given TXY TSY, TSY YXT by AAS.
Lesson 4-5 Proving Congruence—ASA, AAS 211
NAME ______________________________________________ DATE
Write a two-column proof. 19–20. See pp. 233E–233F.
★ 20. Given: BMI KMT
★ 19. Given: MYT NYT
PT
MTY NTY
IP
Prove: RYM RYN
Prove: IPK TPB
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
4-5
Study
Guide and
p. 207
andAAS
p. 208
Proving(shown)
Congruence—ASA,
ASA Postulate
The Angle-Side-Angle (ASA) Postulate lets you show that two triangles
are congruent.
If two angles and the included side of one triangle are congruent to two angles
and the included side of another triangle, then the triangles are congruent.
ASA Postulate
M
Example
Find the missing congruent parts so that the triangles can be
proved congruent by the ASA Postulate. Then write the triangle congruence.
a.
B
B
I
P
E
A
C D
M
F
Two pairs of corresponding angles are congruent, A D and C F. If the
and D
F
are congruent, then ABC DEF by the ASA Postulate.
included sides AC
b. S
R
X
R
T
W
T
Y
T
Y
XY
. If S X, then RST YXW by the ASA Postulate.
R Y and SR
What corresponding parts must be congruent in order to prove that the triangles
are congruent by the ASA Postulate? Write the triangle congruence statement.
1.
2.
C
D
W
A
Z
W
Y
WY
;
XYW ZYW;
WXY WZY
5.
B
GARDENING For Exercises 21 and 22, use
the following information.
Beth is planning a garden. She wants the
triangular sections, CFD and HFG, to be
congruent. F is the midpoint of DG
, and
DG 16 feet. 21–22. See p. 233E.
B
Y
A
DC
BC
;
CDE CBA
4. A
3.
X
B
E
E
D
C
ABE CBD;
ABE CBD
6.
V
R
B
D
T
C
BD
DB
;
ADB CBD;
ABD CDB
A
S
T
VT
;
RST UVT
C
E
Lesson 4-5
D
S
ACB E;
ABC CDE
NAME
______________________________________________
DATE
/M
G
Hill
207
____________
Gl PERIOD
G _____
Skills
Practice,
4-5
Practice
(Average)
p. 209 and
Practice,
p. 210 (shown)
Proving Congruence—ASA,
AAS
1. Write a flow proof.
T
.
Given: S is the midpoint of Q
Q
R
|| T
U
Prove: QSR TSU
R
U
Sample proof:
S is the
midpoint of QT.
QS TS
Def.of midpoint
Given
QR || TU
Q T
Alt. Int. are .
Given
QSR TSU
ASA
QSR TSU
Vertical are .
Kites
2. Write a paragraph proof.
Given: D F
bisects DEF.
GE
Prove: D
G
FG
D
The largest kite ever flown
was 210 feet long and 72
feet wide.
E
G
F
Proof: Since it is given that GE
bisects DEF, DEG FEG by the
definition of an angle bisector. It is given that D F. By the
Reflexive Property, GE
GE
. So DEG FEG by AAS. Therefore
DG
FG
by CPCTC.
ARCHITECTURE For Exercises 3 and 4, use the following
information.
An architect used the window design in the diagram when remodeling
an art studio. AB
and C
B
each measure 3 feet.
Source: Guinness Book of
World Records
B
A
D
C
3. Suppose D is the midpoint of A
C
. Determine whether ABD CBD.
Justify your answer.
CD
Since D is the midpoint of A
C
, AD
by the definition of midpoint.
A
B
CB
by the definition of congruent segments. By the Reflexive
Property, BD
BD
. So ABD CBD by SSS.
4. Suppose A C. Determine whether ABD CBD. Justify your answer.
We are given A
B
CB
and A C. BD
BD
by the Reflexive
Property. Since SSA cannot be used to prove that triangles are
congruent, we cannot say whether ABD CBD.
NAME
______________________________________________
DATE
/M
G
Hill
210
Gl
ELL
Mathematics,
p. 211
Proving Congruence—ASA, AAS
How are congruent triangles used in construction?
Read the introduction to Lesson 4-5 at the top of page 207 in your textbook.
Which of the triangles in the photograph in your textbook appear to be
congruent? Sample answer: The four right triangles are congruent
to each other. The two obtuse isosceles triangles are congruent
to each other.
Reading the Lesson
1. Explain in your own words the difference between how the ASA Postulate and the AAS
Theorem are used to prove that two triangles are congruent.
Sample answer: In ASA, you use two pairs of congruent angles and the
included congruent sides. In AAS, you use two pairs of congruent angles
and a pair of nonincluded congruent sides.
B, D, E, G, H
2. Which of the following conditions are sufficient to prove that two triangles are congruent?
A. Two sides of one triangle are congruent to two sides of the other triangle.
B. The three sides of one triangles are congruent to the three sides of the other triangle.
C. The three angles of one triangle are congruent to the three angles of the other triangle.
D. All six corresponding parts of two triangles are congruent.
E. Two angles and the included side of one triangle are congruent to two sides and the
included angle of the other triangle.
F. Two sides and a nonincluded angle of one triangle are congruent to two sides and a
nonincluded angle of the other triangle.
G. Two angles and a nonincluded side of one triangle are congruent to two angles and
the corresponding nonincluded side of the other triangle.
H. Two sides and the included angle of one triangle are congruent to two sides and the
included angle of the other triangle.
I. Two angles and a nonincluded side of one triangle are congruent to two angles and a
nonincluded side of the other triangle.
3. Determine whether you have enough information to prove that the two triangles in each
figure are congruent. If so, write a congruence statement and name the congruence
postulate or theorem that you would use. If not, write not possible.
AEB DEC; AAS
E
a.
b. T is the midpoint of R
U
.
U
S
T
A
B
C
RST UVT;
ASA
D
R
V
Helping You Remember
4. A good way to remember mathematical ideas is to summarize them in a general statement.
If you want to prove triangles congruent by using three pairs of corresponding parts,
what is a good way to remember which combinations of parts will work?
Sample answer: At least one pair of corresponding parts must be sides. If
you use two pairs of sides and one pair of angles, the angles must be the
included angles. If you use two pairs of angles and one pair of sides,
then the sides must both be included by the angles or must both be
corresponding nonincluded sides.
212
F
D
H
K
Chapter 4 Congruent Triangles
J
23. If N is the midpoint of JL
and K
M
JL
, determine
whether JKN LKN. Justify your answer.
M
L
M and NJM NLM, determine whether
24. If J
JNM LNM. Justify your answer.
Complete each congruence statement and the
postulate or theorem that applies.
M
R
V and 2 5, then
25. If I
INM ? by ? . VNR, AAS or ASA
M
R
V and I
R
M
V, then
26. If I
M
IRN ? by ? . VMN, ASA or AAS
V and R
M bisect each other, then
27. If I
RVN ? by ? . MIN, SAS
28. If MIR RVM and 1 6, then
MRV ? by ? . RMI, AAS or ASA
212
M
I
4
R
3
1
2
N
5
6
7
8
V
See margin.
Chapter 4 Congruent Triangles
Courtesy Peter Lynn Kites
NAME ______________________________________________ DATE
4-5
Enrichment
Enrichment,
____________ PERIOD _____
p. 212
Congruent Triangles in the Coordinate Plane
If you know the coordinates of the vertices of two triangles in the coordinate
plane, you can often decide whether the two triangles are congruent. There
may be more than one way to do this.
1. Consider ABD and CDB whose vertices have coordinates A(0, 0),
B(2, 5), C(9, 5), and D(7, 0). Briefly describe how you can use what you
know about congruent triangles and the coordinate plane to show that
ABD CDB. You may wish to make a sketch to help get you started.
Sample answer: Show that the slopes of A
B
and C
D
are
BC
equal and that the slopes of A
D
and are equal. Conclude
B
C
D and B
C
A
D . Use the angle relationships for
that A
parallel lines and a transversal and the fact that BD
is a common side for the triangles to conclude that
ABD CDB by ASA.
L
N
29. CRITICAL THINKING Aiko wants to estimate
the distance between herself and a duck. She
adjusts the visor of her cap so that it is in line
with her line of sight to the duck. She keeps her
neck stiff and turns her body to establish a line
of sight to a point on the ground. Then she
paces out the distance to the new point. Is the
distance from the duck the same as the distance
she just paced out? Explain your reasoning.
____________
Gl PERIOD
G _____
Reading
4-5
Readingto
to Learn
Learn Mathematics
Pre-Activity
G
KITES For Exercises 23 and 24, use the following information.
Austin is building a kite. Suppose JL is 2 feet, JM is 2.7 feet,
and the measure of NJM is 68. 23–24. See p. 233F.
T
S
Q
C
21. Suppose C
D
and G
H
each measure 4 feet
and the measure of CFD is 29. Determine
whether CFD HFG. Justify your answer.
H, and C
H
D
G.
22. Suppose F is the midpoint of C
Determine whether CFD HFG. Justify your answer.
U
Gl
K
N
Exercises
Answer
29. Since Aiko is perpendicular to the ground,
two right angles are formed and right
angles are congruent. The angles of sight
are the same and her height is the same for
each triangle. The triangles are congruent
by ASA. By CPCTC, the distances are the
same. The method is valid.
30. WRITING IN MATH
Answer the question that was posed at the beginning
of the lesson. See margin.
How are congruent triangles used in construction?
4 Assess
Include the following in your answer:
• explain how to determine whether the triangles are congruent, and
• why it is important that triangles used for structural support are congruent.
Standardized
Test Practice
31. In ABC, AD
and D
C
are angle bisectors and
mB 76. What is the measure of ADC? D
A 26
B 52
C 76
D 128
Open-Ended Assessment
Writing Have students practice
writing different versions of
proofs for each example. For
Example 1, students can write a
flow proof and a two-column
proof.
B
D
A
C
32. ALGEBRA For a positive integer x,
1 percent of x percent of 10,000 equals A
A x.
B 10x.
C 100x.
D
Getting Ready for
Lesson 4-6
1000x.
Prerequisite Skill Students will
learn about isosceles triangles in
Lesson 4-6. They will use
congruence postulates and
theorems when writing proofs.
Use Exercises 39–41 to determine
your students’ familiarity with
the classification of triangles by
sides.
Maintain Your Skills
Mixed Review
Write a flow proof. (Lesson 4-4) 33–34. See p. 233F.
33. Given: BA
DE
BE
34. Given: XZ
WY
, D
A
Prove: BEA DAE
X
Z
bisects W
Y
.
Prove:
WZX
YZX
B
D
W
C
Z
A
E
Answers
Y
X
35–36. See margin.
30. Sample answer: The triangular
trusses support the structure.
Answers should include the
following.
• To determine whether two
triangles are congruent,
information is needed about
consecutive side-angle-side,
side-side-side, angle-sideangle, angle-angle-side, or
about each angle and each side.
• Triangles that are congruent
will support weight better
because the pressure will be
evenly divided.
35. Turn; RS 2, RS 2,
ST 1, ST 1, RT 1,
RT 1. Use a protractor to
confirm that the corresponding
angles are congruent.
36. Flip; MP 2, MP 2, MN 3,
MN 3, NP 13,
NP 13. Use a protractor to
confirm that the corresponding
angles are congruent.
Verify that each of the following preserves congruence and name the
congruence transformation. (Lesson 4-3)
y
y
35.
36.
T
S
O S'
T'
x
M
Getting Ready for
the Next Lesson
P P'
M'
x
O
R'
37. If people are
happy, then they
rarely correct their
faults.
N'
N
R
Write each statement in if-then form. (Lesson 2-3)
37. Happy people rarely correct their faults.
38. A champion is afraid of losing. If a person is a champion, then he or she is
afraid of losing.
PREREQUISITE SKILL Classify each triangle according to its sides.
(To review classification by sides, see Lesson 4-1.)
39.
40.
equilateral
41.
isosceles
isosceles
www.geometryonline.com/self_check_quiz
Lesson 4-5 Proving Congruence—ASA, AAS
213
Teacher to Teacher
Karyn S. Cummins, Franklin Central High School
Indianapolis, IN
After proofs are introduced I write up several simple proofs on card stock.
I then cut the statements and reasons apart, give them to the students and
have them reconstruct them.
Lesson 4-5 Proving Congruence—ASA, AAS 213
Geometry
Activity
A Follow-Up of Lesson 4-5
Getting Started
Objective Explore congruence in
right triangles.
Materials
ruler protractor
Teach
• Explain that right triangles are
unique and typically have
special relationships. Students
will want to check for these
relationships when they are
working on proofs.
• Remind students that the right
triangle theorems do not work
for acute or obtuse triangles,
but only for right triangles.
Assess
Exercises 1–3 guide students
through SAS, ASA, and AAS,
and introduce LL, HA, and LA.
Exercises 4–6 demonstrate that
SSA works with right triangles
and forms the HL Postulate.
Exercises 7–11 use the right
triangle congruence theorems
and postulate in proofs.
A Follow-Up of Lesson 4-5
Congruence in Right Triangles
In Lessons 4-4 and 4-5, you learned theorems and postulates to prove triangles
congruent. Do these theorems and postulates apply to right triangles?
Activity 1
Triangle Congruence
Model
Study each pair of right triangles.
a.
b.
c.
Analyze
1. Is each pair of triangles congruent? If so, which congruence theorem or
postulate applies? yes; a. SAS, b. ASA, c. AAS
2. Rewrite the congruence rules from Exercise 1 using leg, (L), or hypotenuse, (H),
to replace side. Omit the A for any right angle since we know that all right
triangles contain a right angle and all right angles are congruent. a. LL, b. LA, c. HA
3. Make a conjecture If you know that the corresponding legs of two right None; two pairs of
triangles are congruent, what other information do you need to declare legs congruent is
the triangles congruent? Explain.
sufficient for prov-
ing right triangles
congruent.
In Lesson 4-5, you learned that SSA is not a valid test for determining
triangle congruence. Can SSA be used to prove right triangles congruent?
Activity 2
SSA and Right Triangles
Make a Model
How many right triangles exist that have a hypotenuse of 10 centimeters and
a leg of 7 centimeters?
Draw X
Y
so
that XY 7 centimeters.
Use a
protractor to draw a
ray from Y that is
perpendicular to X
Y
.
Open your
compass to a width of
10 centimeters. Place
the point at X and draw
a long arc to intersect
the ray.
Label the
intersection Z and
draw X
Z
to complete
XYZ.
Z
10 cm
Study Notebook
Ask students to summarize what they
have learned about congruence in
right triangles. Tell students to list
each method with a brief description.
X
Y
X
Y
X
Y
X
214 Investigating Slope-Intercept Form
214 Chapter 4 Congruent Triangles
Resource Manager
214 Chapter 4 Congruent Triangles
Teaching Geometry with
Manipulatives
Glencoe Mathematics Classroom
Manipulative Kit
• p. 78 (student recording sheet)
• p. 16 (protractor)
• p. 17 (ruler)
• protractor
• ruler
7 cm
Y
A Follow-Up of Lesson 4-5
Answers
Analyze
4. Does the model yield a unique triangle? yes
5. Can you use the lengths of the hypotenuse and a leg to show right triangles are
7. Given: DEF and RST are right
triangles.
E and S are right angles.
EF ST
ED
SR
Prove: DEF RST
congruent? yes
6. Make a conjecture about the case of SSA that exists for right triangles.
SSA is a valid test of congruence for right triangles.
The two activities provide evidence for four ways to prove right triangles congruent.
F
T
E R
S
Right Triangle Congruence
Theorem
Abbreviation
4.6
Leg-Leg Congruence If the legs of one
right triangle are congruent to the
corresponding legs of another right
triangle, then the triangles are congruent.
LL
4.7
Hypotenuse-Angle Congruence If the
hypotenuse and acute angle of one
right triangle are congruent to the
hypotenuse and corresponding
acute angle of another right triangle,
then the two triangles are congruent.
HA
4.8
Leg-Angle Congruence If one leg
and an acute angle of one right triangle
are congruent to the corresponding leg
and acute angle of another right triangle,
then the triangles are congruent.
LA
D
Example
Proof: We are given that EF ST,
ED
SR
, and E and S are
right angles. Since all right
angles are congruent, E S.
Therefore, by SAS, DEF RST.
8. Given: ABC and XYZ are right
triangles.
A and X are right angles.
BC
YZ
B Y
Prove: ABC XYZ
Postulate
4.4
PROOF
Hypotenuse-Leg Congruence If the
hypotenuse and a leg of one right
triangle are congruent to the hypotenuse
and corresponding leg of another right
triangle, then the triangles are congruent.
C
Z
A
B X
Y
Proof: We are given that ABC
and XYZ are right triangles with
right angles A and X, BC
YZ,
and B Y. Since all right
angles are congruent, A X.
Therefore, ABC XYZ by AAS.
HL
Write a paragraph proof of each theorem. 7–9. See margin.
7. Theorem 4.6
8. Theorem 4.7
9. Theorem 4.8 (Hint: There are two possible cases.)
Use the figure to write a two-column proof. 10–11. See p. 233F.
10. Given: ML
MK
KM
11. Given: JK
KM
, JM
KL
, JK
J L
ML
JK
Prove: JM
KL
Prove: ML
JK
J
M
L
K
Geometry Activity Congruence in Right Triangles 215
9. Case 1:
C
F
Given: ABC and DEF
are right triangles.
A
B D
E
AC
DF, C F
Prove: ABC DEF
Proof: It is given that ABC and DEF are right triangles,
AC
DF, C F. By the definition of right triangles,
A and D are right angles. Thus, A D since all
right angles are congruent. ABC DEF by ASA.
Case 2:
C
F
Given: ABC and DEF
are right triangles.
A
B D
E
C
A
DF, B E
Prove: ABC DEF
Proof: If is given that ABC and DEF are right triangles, AC
DF,
and B E. By the definition of right triangle, A and D are
right angles. Thus, A D since all right angles are congruent.
ABC DEF by AAS.
Geometry Activity Congruence in Right Triangles 215
Lesson
Notes
Isosceles Triangles
• Use properties of isosceles triangles.
1 Focus
5-Minute Check
Transparency 4-6 Use as a
quiz or review of Lesson 4-5.
• Use properties of equilateral triangles.
are triangles used in art?
Vocabulary
• vertex angle
• base angles
Mathematical Background notes
are available for this lesson on
p. 176D.
are triangles used
in art?
Ask students:
• How would the painting’s
overall appearance change if
you removed or covered the
triangles? The painting would
appear more blended and based on
curves, circles, and ovals without
the stark triangles.
• Describe where isosceles
triangles appear in the art.
Accept all reasonable answers.
The art of Lois Mailou Jones, a
twentieth-century artist, includes
paintings and textile design, as well as
book illustration. Notice the isosceles
triangles in this painting, Damballah.
PROPERTIES OF ISOSCELES TRIANGLES In Lesson 4-1, you learned that
isosceles triangles have two congruent sides. Like the right triangle, the parts of an
isosceles triangle have special names.
The angle formed by
the congruent sides is
called the vertex angle. leg
leg
base
The two angles formed by
the base and one of the
congruent sides are called
base angles.
In this activity, you will investigate the relationship of the base angles and legs of
an isosceles triangle.
Isosceles Triangles
C
Model
• Draw an acute triangle on patty paper with A
C
B
C
.
• Fold the triangle through C so that A and B coincide.
Analyze 2, 3. They are congruent.
A
1. What do you observe about A and B? A B
2. Draw an obtuse isosceles triangle. Compare the base angles.
3. Draw a right isosceles triangle. Compare the base angles.
B
The results of the Geometry Activity suggest Theorem 4.9.
Theorem 4.9
Isosceles Triangle Theorem If two sides of a
triangle are congruent, then the angles opposite
those sides are congruent.
Example: If AB
CB
, then A C.
216
B
A
Chapter 4 Congruent Triangles
Marvin T. Jones
Resource Manager
Workbook and Reproducible Masters
Chapter 4 Resource Masters
• Study Guide and Intervention, pp. 213–214
• Skills Practice, p. 215
• Practice, p. 216
• Reading to Learn Mathematics, p. 217
• Enrichment, p. 218
• Assessment, p. 240
School-to-Career Masters, p. 8
Teaching Geometry With Manipulatives
Masters, pp. 8, 80, 81
Transparencies
5-Minute Check Transparency 4-6
Answer Key Transparencies
Technology
Interactive Chalkboard
C
Example 1 Proof of Theorem
Write a two-column proof of the
Isosceles Triangle Theorem.
Q R
Q
Given: PQR, P
Prove: PR
R
Proof:
Statements
1. Let S be the midpoint of P
R
.
2. Draw an auxiliary segment Q
S
.
3. P
S
R
S
4. Q
S
Q
S
5. P
Q
R
Q
6. PQS RQS
7. PR
S
2 Teach
P
PROPERTIES OF
ISOSCELES TRIANGLES
Q
Reasons
1. Every segment has exactly one midpoint.
2. Two points determine a line.
3. Midpoint Theorem
4. Congruence of segments is reflexive.
5. Given
6. SSS
7. CPCTC
Diagrams Label the
diagram with the given
information. Use your
drawing to plan the next
step in solving the
problem.
Given: AB CB BD,
ACB BCD
Prove: A D
D
B
Multiple-Choice Test Item
A
If GH
HJ JK
H
K
, , and mGJK100, what is the
measure of HGK?
A 10
B 15
C 20
D 25
K
G
Test-Taking Tip
Power
Point®
1 Write a two-column proof.
C
Example 2 Find the Measure of a Missing Angle
Standardized
Test Practice
In-Class Examples
H
J
Read the Test Item
GHK is isosceles with base GK
. Likewise, HJK is isosceles with base H
K
.
Solve the Test Item
Step 1
The base angles of HJK are congruent. Let x mKHJ mHKJ.
mKHJ mHKJmHJK 180 Angle Sum Theorem
x x 100 180 Substitution
2 If DE
CD
, BC
AC
, and
2x 100 180 Add.
2x80
x 40
Step 2
Subtract 100 from each side.
So, mKHJ mHKJ 40.
mCDE 120, what is the
measure of BAC? D
D
GHK and KHJ form a linear pair. Solve for mGHK.
mKHJmGHK 180 Linear pairs are supplementary.
40 mGHK 180 Substitution
mGHK 140
Step 3
Proof:
Statements (Reasons)
1. AB CB BD (Given)
2. A
B
C
B
B
D (Def. of seg.)
3. ABC and BCD are
isosceles. (Def. of isos. )
4. A ACB,
BCD D (Isos. Th.)
5. ACB BCD (Given)
6. A D (Transitive Prop.)
C
B
Subtract 40 from each side.
The base angles of GHK are congruent. Let y represent mHGK
and mGKH.
mGHK mHGK mGKH 180 Angle Sum Theorem
140 y y 180 Substitution
140 2y180 Add.
2y 40 Subtract 140 from each side.
y 20 Divide each side by 2.
E
A
A 45.5
C 68.5
B 57.5
D 75
The measure of HGK is 20. Choice C is correct.
www.geometryonline.com/extra_examples
Lesson 4-6 Isosceles Triangles 217
Geometry Activity
Materials: paper, scissors, ruler
• You may wish to provide students with rectangular dot paper to help them
draw accurate isosceles triangles.
• Ask students to name the legs, base, vertex angle, and base angles of their
triangles.
• Have students repeat the activity starting with a line segment and two
congruent angles drawn at each end of the segment.
Lesson 4-6 Isosceles Triangles 217
In-Class Example
3
The converse of the Isosceles Triangle Theorem is also true.
Power
Point®
Theorem 4.10
M
If two angles of a triangle are congruent, then the
sides opposite those angles are congruent.
P
D
Abbreviation: Conv. of Isos. Th.
L
Example: If D F, then DE
F
E
.
N
a. Name two congruent angles.
MLN and MNL
F
You will prove Theorem 4.10 in Exercise 33.
b. Name two congruent
L and P
M
segments. P
Example 3 Congruent Segments and Angles
PROPERTIES OF
EQUILATERAL TRIANGLES
In-Class Example
E
Power
Point®
4 Copy the figure in Example 4
and then draw EJ so that EJ
bisects 2, and J lies on F
G
.
You can use properties of
triangles to prove Thales
of Miletus’ important
geometric ideas. Visit
www.geometryonline.
com/webquest to
continue work on your
WebQuest project.
a. Name two congruent angles.
AFC is opposite AC
and ACF is opposite A
F
,
so AFCACF.
C
A
B
H
F
b. Name two congruent segments.
By the converse of the Isosceles Triangle Theorem, the sides opposite congruent
angles are congruent. So, BC
B
F
.
PROPERTIES OF EQUILATERAL TRIANGLES Recall that an equilateral
triangle has three congruent sides. The Isosceles Triangle Theorem also applies to
equilateral triangles. This leads to two corollaries about the angles of an equilateral
triangle.
a. Find mHEJ and mEJH.
15; 75
Corollaries
b. Find mEJG. 105
4.3
4.4
A triangle is equilateral if and
only if it is equiangular.
Each angle of an equilateral
triangle measures 60°.
60˚
60˚
60˚
You will prove Corollaries 4.3 and 4.4 in Exercises 31 and 32.
Example 4 Use Properties of Equilateral Triangles
EFG is equilateral, and EH
bisects E.
a. Find m1 and m2.
Each angle of an equilateral triangle measures 60°.
So, m1 m2 60. Since the angle was bisected,
m1 m2. Thus, m1m230.
b. ALGEBRA Find x.
mEFHm1mEHF180
60 30 15x 180
90 15x180
15x 90
x 6
E
1 2
15x˚
F
Angle Sum Theorem
mEFH 60, m1 30, mEFH 15x
Add.
Subtract 90 from each side.
Divide each side by 15.
218 Chapter 4 Congruent Triangles
Differentiated Instruction
Interpersonal Have groups of students work Exercise 7 on p. 219.
Some group members can provide the statements, and other group
members can provide corresponding reasons. Encourage groups to
discuss the properties of isosceles and equilateral triangles while they
are figuring out the proofs.
218
Chapter 4 Congruent Triangles
H
G
Concept Check
3 Practice/Apply
1. Explain how many angles in an isosceles triangle must be given to find the
measures of the other angles. 1–3. See margin.
2. Name the congruent sides and angles of isosceles WXZ with base WZ
.
3. OPEN ENDED
Guided Practice
GUIDED PRACTICE KEY
Exercises
Examples
4–5
6
7
8
3
4
1
2
Study Notebook
Describe a method to construct an equilateral triangle.
Refer to the figure.
D
4. If AD
A
H
, name two congruent angles. ADH AHD
5. If BDH BHD, name two congruent segments.
BH
BD
C
B
Have students—
• add the definitions/examples of
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 4.
• include a concept map with all
methods of proving triangle
congruence accompanied by
helpful theorems, postulates,
properties, and formulas.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
A
H
6. ALGEBRA Triangle GHF is equilateral with mF3x 4, mG6y, and
6 , 10, 3
mH19z3. Find x, y, and z. 5
3
Write a two-column proof.
7. Given: CTE is isosceles with vertex C.
mT60
Prove: CTE is equilateral.
T
60˚
C
See p. 233F.
E
Standardized
Test Practice
8. If P
QS
Q
R R
Q
, S
, and mPRS 72, what is the
measure of QPS? A
A 27
B 54
C 63
D 72
S
R
Q
P
★ indicates increased difficulty
About the Exercises…
Practice and Apply
For
Exercises
See
Examples
9–14
15–22,
27–28,
34–37
23–26,
38–39
29–33
3
4
2
1
Extra Practice
See page 762.
Refer to the figure.
9. If L
T
L
R
, name two congruent angles. LTR LRT
10. If L
X L
W, name two congruent angles. LXW LWX
11. If S
L Q
L, name two congruent angles. LSQ LQS
T
12. If LXY LYX, name two congruent segments. LXLY
13. If LSR LRS, name two congruent segments. LSLR
X
14. If LYW LWY, name two congruent segments. LYLW
KLN and LMN are isosceles and mJKN 130.
Find each measure.
16. mM 140
15. mLNM 20
17. mLKN 81
18. mJ 106
L
S
Y
R
Q
Z
W
K
N
J
18˚
25˚
M
20˚
F
28˚
D
Odd/Even Assignments
Exercises 9–39 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
Assignment Guide
L
DFG and FGH are isosceles, mFDH 28 and
FG
FH
DG
. Find each measure.
20. mDGF 124
19. mDFG 28
21. mFGH 56
22. mGFH 68
Organization by Objective
• Properties of Isosceles
Triangles: 9–14, 23–26,
29–33, 38–39
• Properties of Equilateral
Triangles: 15–22, 27–28,
34–37
G
H
Lesson 4-6 Isosceles Triangles 219
Basic: 9–39 odd, 40–54
Average: 9–39 odd, 40–54
Advanced: 10–40 even, 41–51
(optional: 52–54)
All: Quiz 2 (1–5)
Answers
1. The measure of only one angle must be given in an isosceles
triangle to determine the measures of the other two angles.
2. W
X
ZX
, W Z
3. Sample answer: Draw a line segment. Set your compass to the
length of the line segment and draw an arc from each endpoint.
Draw segments from the intersection of the arcs to each endpoint.
Lesson 4-6 Isosceles Triangles 219
NAME ______________________________________________ DATE
In the figure, JM
ML
P
M
and P
L
.
23. If mPLJ34, find mJPM. 36.5
24. If mPLJ58, find mPJL. 30.5
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
4-6
Study
Guide and
Properties of Isosceles Triangles
An isosceles triangle has two congruent sides.
The angle formed by these sides is called the vertex angle. The other two angles are called
base angles. You can prove a theorem and its converse about isosceles triangles.
A
• If two sides of a triangle are congruent, then the angles opposite
those sides are congruent. (Isosceles Triangle Theorem)
• If two angles of a triangle are congruent, then the sides opposite
those angles are congruent.
B
C
Lesson 4-6
p. 213
(shown)
Isosceles
Triangles and p. 214
Example 1
Example 2
Find x.
B
R
BC BA, so
mA mC.
5x 10 4x 5
x 10 5
x 15
Substitution
Subtract 4x from each side.
Add 10 to each side.
G
Triangle LMN is equilateral, and M
P bisects L
N
.
27. Find x and y. x3; y18
28. Find the measure of each side of LMN. 10
Converse of Isos. Thm.
Substitution
Add 13 to each side.
Subtract 2x from each side.
Exercises
K
35
R
P
40
2x 2. S
2x 6
T
3x 6
12
3.
5y ˚
P
L
15
W
4x 2
3x 1
N
V
Q
4. D
P
K
T (6x 6)
2x 12
5. G
3x Y
20
B
Z
6.
PROOF
36
T
30
L
D
x
R
S
B
A
1
3
D
2
C
E
Statements
Reasons
1.1 2
1.Given
2.2 3
2.Vertical angles are congruent.
3.1 3
3.Transitive Property of B
CB
4.A
4.If two angles of a triangle are , then
the sides opposite the angles are .
Write a two-column proof. 29–33. See pp. 233F–233G.
29. Given: XKF is equilateral.
XJ bisects X.
Prove: J is the midpoint of K
F
.
3x 3x Q
7. Write a two-column proof.
Given: 1 2
Prove: AB
CB
Gl
J
M
Find x.
1.
L
T
2x
mS mT, so
SR TR.
3x 13 2x
3x 2x 13
x 13
Isos. Triangle Theorem
M
H
3x 13
(5x 10)
A
Find x.
S
(4x 5)
J
In the figure, G
K G
H and H
K K
J.
25. If mHGK28, find mHJK. 38
26. If mHGK42, find mHJK. 34.5
If AB
CB
, then A C.
If A C, then A
B
CB
.
C
P
★ 30. Given: MLP is isosceles.
N is the midpoint of M
P
.
Prove: L
MP
N
X
NAME
______________________________________________
DATE
/M
G
Hill
213
Skills
Practice,
4-6
Practice
(Average)
1
____________
Gl PERIOD
G _____
p. 215 and
Practice,
p. 216 (shown)
Isosceles Triangles
2
K
Refer to the figure.
R
1. If R
V
RT
, name two congruent angles. RTV RVT
L
J
F
31. Corollary 4.3
M
32. Corollary 4.4
N
P
33. Theorem 4.10
S
V
2. If R
S
SV
, name two congruent angles. SVR SRV
T
U
34. DESIGN The basic structure covering Spaceship Earth at the Epcot Center in
Orlando, Florida, is a triangle. Describe the minimum requirement to show that
these triangles are equilateral. The minimum requirement is that two angles
3. If SRT STR, name two congruent segments. S
T
SR
4. If STV SVT, name two congruent segments. S
T
SV
Triangles GHM and HJM are isosceles, with G
H
M
H
and H
J
M
J
. Triangle KLM is equilateral, and mHMK 50.
Find each measure.
J
measure 60°.
L
K
M
H
5. mKML 60
6. mHMG 70
7. mGHM 40
ALGEBRA
35.
G
8. If mHJM 145, find mMHJ. 17.5
9. If mG 67, find mGHM. 46
2x 5
10. Write a two-column proof.
E
|| B
C
Given: DE
1 2
Prove: AB
AC
2
3
C
D
4
B
Spaceship Earth is a
completely spherical
geodesic dome
that is covered with
11,324 triangular aluminum
and plastic alloy panels.
Proof:
Statements
1. D
E
Reasons
|| B
C
1. Given
2. 1 4
2 3
2. Corr. are .
3. 1 2
3. Given
4. 3 4
4. Congruence of is transitive.
5. AB
AC
5. If 2 of a are , then the sides opposite
those are .
11. SPORTS A pennant for the sports teams at Lincoln High
School is in the shape of an isosceles triangle. If the measure
of the vertex angle is 18, find the measure of each base angle.
Lin
81, 81
NAME
______________________________________________
DATE
/M
G
Hill
216
Gl
coln
Haw
Source: disneyworld.disney.go.com
ks
ELL
37.
30
symmetry is pleasing to the eye.
• Why might isosceles right triangles be used in art? Sample answer:
Two congruent isosceles right triangles can be placed
together to form a square.
m217, m326, m417,
m518
Reading the Lesson
R
a. What kind of triangle is QRS? isosceles
(2x 25)˚
(x 5)˚
C
• Why do you think that isosceles and equilateral triangles are used more
often than scalene triangles in art? Sample answer: Their
S
1 2 3 4 5
42˚
A
77˚
D
F
G
E
Q
d. Name the vertex angle of QRS. S
e. Name the base angles of QRS. Q, R
220
Chapter 4 Congruent Triangles
Dallas & John Heaton/Stock Boston
C04-251C-829637
13 2 7
2. Determine whether each statement is always, sometimes, or never true.
a. If a triangle has three congruent sides, then it has three congruent angles. always
b. If a triangle is isosceles, then it is equilateral. sometimes
c. If a right triangle is isosceles, then it is equilateral. never
d. The largest angle of an isosceles triangle is obtuse. sometimes
e. If a right triangle has a 45° angle, then it is isosceles. always
f. If an isosceles triangle has three acute angles, then it is equilateral. sometimes
g. The vertex angle of an isosceles triangle is the largest angle of the triangle.
sometimes
3. Give the measures of the three angles of each triangle.
a. an equilateral triangle 60, 60, 60
b. an isosceles right triangle 45, 45, 90
c. an isosceles triangle in which the measure of the vertex angle is 70 70, 55, 55
d. an isosceles triangle in which the measure of a base angle is 70 70, 70, 40
e. an isosceles triangle in which the measure of the vertex angle is twice the measure of
one of the base angles 90, 45, 45
Helping You Remember
NAME ______________________________________________ DATE
4-6
Enrichment
Enrichment,
____________ PERIOD _____
p. 218
Triangle Challenges
Some problems include diagrams. If you are not sure how to solve the
problem, begin by using the given information. Find the measures of as many
angles as you can, writing each measure on the diagram. This may give you
more clues to the solution.
1. Given: BE BF, BFG BEF BED, mBFE 82 and
ABFG and BCDE each have
opposite sides parallel and
congruent.
Find m ABC. 148
A
4. If a theorem and its converse are both true, you can often remember them most easily by
combining them into an “if-and-only-if” statement. Write such a statement for the Isosceles
Triangle Theorem and its converse. Sample answer: Two sides of a triangle are
B
2. Given: AC AD, and A
B
B
D
,
mDAC 44 and
C
E
bisects ACD.
Find mDEC. 78
A
C
congruent if and only if the angles opposite those sides are congruent.
E
B
G
220
(3x 8)˚ 18
40. CRITICAL THINKING In the figure, ABC
is isosceles, DCE is equilateral, and FCG
is isosceles. Find the measures of the five
numbered angles at vertex C. m118,
Read the introduction to Lesson 4-6 at the top of page 216 in your textbook.
c. Name the base of QRS. Q
R
36.
38. Trace the figure. Identify and draw one isosceles
triangle from each set in the sign.
39. Describe the similarities between the different triangles.
How are triangles used in art?
b. Name the legs of QRS. Q
S
, R
S
18
ARTISANS For Exercises 38 and 39, use the
following information.
This geometric sign from the Grassfields area
in Western Cameroon (Western Africa) uses
approximations of isosceles triangles within
and around two circles. 38–39. See p. 233G.
____________
Gl PERIOD
G _____
Mathematics,
p. 217
Isosceles Triangles
1. Refer to the figure.
3x 13
(2x 20)˚
Reading
4-6
Readingto
to Learn
Learn Mathematics
Pre-Activity
60˚
Design
A
1
Find x.
Chapter 4 Congruent Triangles
D
B
41. WRITING IN MATH
Answer the question that was posed at the beginning
of the lesson. See margin.
How are triangles used in art?
4 Assess
Include the following in your answer:
• at least three other geometric shapes frequently used in art, and
• a description of how isosceles triangles are used in the painting.
Standardized
Test Practice
42. Given right triangle XYZ with hypotenuse X
Y
, YP is equal
to YZ. If mPYZ 26, find mXZP. A
A 13
B 26
C 32
D 64
Open-Ended Assessment
PX
Y
Z
43. ALGEBRA A segment is drawn from (3, 5) to (9, 13). What are the coordinates
of the midpoint of this segment? D
A (3, 4)
B (12, 18)
C (6, 8)
D (6, 9)
Maintain Your Skills
Mixed Review
Write a paragraph proof. (Lesson 4-5) 44–45. See p. 233G.
44. Given: ND, GI,
45. Given: VR
RS
UT
SU
, A
N
S
D
Prove: ANG SDI
G
R
S
U
S
Prove: VRS TUS
N
V
R
Getting Ready for
Lesson 4-7
S
S
A
D
Prerequisite Skill Students will
learn about triangles and
coordinate proof in Lesson 4-7.
They will prove congruency
using distances and midpoints in
coordinate planes. Use Exercises
52–54 to determine your
students’ familiarity with finding
midpoints.
I
U
T
Determine whether QRS EGH given the coordinates of the vertices.
Explain. (Lesson 4-4) 46–47. See margin.
46. Q(3, 1), R(1, 2), S(1, 2), E(6, 2), G(2, 3), H(4, 1)
47. Q(1, 5), R(5, 1), S(4, 0), E(4, 3), G(1, 2), H(2, 1)
48–51. See p. 233G.
Getting Ready for
the Next Lesson
Construct a truth table for each compound statement. (Lesson 2-2)
48. a and b
49. p or q
50. k and m
51. y or z
PREREQUISITE SKILL Find the coordinates of the midpoint of the segment
with the given endpoints. (To review finding midpoints, see Lesson 1-5.)
52. A(2, 15), B(7, 9)
53. C(4, 6), D(2, 12)
54. E(3, 2.5), F(7.5, 4)
(4.5, 12)
(1, 3)
P ractice Quiz 2
Assessment Options
(5.25, 3.25)
Lessons 4-4 through 4-6
1. Determine whether JMLBDG given that J(4, 5), M(2, 6),
A
L(1, 1), B(3, 4), D(4, 2), and G(1, 1). (Lesson 4-4) See p. 233G.
E
2. Write a two-column proof to prove that A
J E
H
, given AH,
AEJHJE. (Lesson 4-5) See p. 233G.
J
WXY and XYZ are isosceles and mXYZ128. Find each
measure. (Lesson 4-6)
3. mXWY 52
4. mWXY 76
5. mYZX 26
H
X
128˚
W
www.geometryonline.com/self_check_quiz
Speaking Have students come up
with examples of how isosceles
and equilateral triangles are used
in paintings, ceramics, and
decorative architecture. Students
can talk about where the base
and legs of isosceles triangles are
typically situated in architecture,
and they can discuss how
geometry can help builders
determine how much material
they will need. They can also
discuss the visual effect that
isosceles and equilateral triangles
have in different art forms.
Y
Z
Practice Quiz 2 The quiz
provides students with a brief
review of the concepts and skills
in Lessons 4-4 through 4-6.
Lesson numbers are given to the
right of the exercises or
instruction lines so students can
review concepts not yet mastered.
Quiz (Lessons 4-5 and 4-6) is
available on p. 240 of the Chapter 4
Resource Masters.
Lesson 4-6 Isosceles Triangles 221
Answer
41. Sample answer: Artists use angles, lines and shapes to
create visual images. Answers should include the following.
• Rectangle, squares, rhombi, and other polygons are
used in many works of art.
• There are two rows of isosceles triangles in the painting.
One row has three congruent isosceles triangles. The
other row has six congruent isosceles triangles.
46. QR 17
, RS 20
, QS 13
, EG 17
, GH 20
, and
EH 13. Each pair of corresponding sides have the same
measure so they are congruent. QRS EGH by SSS.
47. QR 52
, RS 2, QS 34
, EG 34
, GH 10
, and
EH 52. The corresponding sides are not congruent so QRS
is not congruent to EGH.
Lesson 4-6 Isosceles Triangles 221
Lesson
Notes
Triangles and Coordinate Proof
• Position and label triangles for use in coordinate proofs.
1 Focus
• Write coordinate proofs.
5-Minute Check
Transparency 4-7 Use as a
quiz or review of Lesson 4-6.
Vocabulary
• coordinate proof
Mathematical Background notes
are available for this lesson on
p. 176D.
can the coordinate plane
be useful in proofs?
Ask students:
• If d 3 and BC 5, what are
the coordinates of B and C?
Classify ABC. (6, 0); (3, 4);
isosceles
• If you draw C
D
where D is the
B, what method(s)
midpoint of A
could you use to prove
ACD BCD? the Distance
Formula and SSS
Study Tip
Placement of
Figures
The guidelines apply to
any polygon placed on
the coordinate plane.
y
x
A (0, 0) O
B (2d , 0 )
Coordinate proof uses figures in
the coordinate plane and algebra to prove geometric concepts. The first step in
writing a coordinate proof is the placement of the figure on the coordinate plane.
Placing Figures on the Coordinate Plane
1. Use the origin as a vertex or center of the figure.
2. Place at least one side of a polygon on an axis.
3. Keep the figure within the first quadrant if possible.
Example 1 Position and Label a Triangle
Position and label isosceles triangle JKL on a coordinate plane so that base JK
is
a units long.
• Use the origin as vertex J of the triangle.
y
L ( a2 , b )
• Place the base of the triangle along the positive x-axis.
• Position the triangle in the first quadrant.
• Since K is on the x-axis, its y-coordinate is 0. Its x-coordinate
x
O J (0, 0)
is a because the base of the triangle is a units long.
K (a , 0)
• Since JKL is isosceles, the x-coordinate of L is halfway
a
between 0 and a or . We cannot determine the y-coordinate
2
in terms of a, so call it b.
Power
Point®
1 Position and label right
Z
triangle XYZ with leg X
d units long on the
coordinate plane.
Example 2 Find the Missing Coordinates
Name the missing coordinates of isosceles right EFG.
y
y
E (0, a )
Vertex F is positioned at the origin; its coordinates are (0, 0).
Vertex E is on the y-axis, and vertex G is on the x-axis. So
EFG is a right angle. Since EFG is isosceles, EF
G
F
.
The distance from E to F is a units. The distance from F to
F (?, ?) O
G (?, ? )
G must be the same. So, the coordinates of G are (a, 0).
Y(0, b)
Z(d, 0) x
222 Chapter 4 Congruent Triangles
Resource Manager
Workbook and Reproducible Masters
Chapter 4 Resource Masters
• Study Guide and Intervention, pp. 219–220
• Skills Practice, p. 221
• Practice, p. 222
• Reading to Learn Mathematics, p. 223
• Enrichment, p. 224
• Assessment, p. 240
C (d , e )
4. Use coordinates that make computations as simple as possible.
POSITION AND LABEL
TRIANGLES
X(0, 0)
In this chapter, we have used several methods of
proof. You have also used the coordinate plane
to identify characteristics of a triangle. We can
combine what we know about triangles in the
coordinate plane with algebra in a new method
of proof called coordinate proof.
POSITION AND LABEL TRIANGLES
2 Teach
In-Class Example
can the coordinate plane be useful in proofs?
Teaching Geometry With Manipulatives
Masters, pp. 1, 2, 8
Transparencies
5-Minute Check Transparency 4-7
Answer Key Transparencies
Technology
Interactive Chalkboard
WRITE COORDINATE PROOFS After the figure has been placed on the
coordinate plane and labeled, we can use coordinate proof to verify properties and
to prove theorems. The Distance Formula, Slope Formula, and Midpoint Formula
are often used in coordinate proof.
In-Class Example
Power
Point®
2 Name the missing coordinates
of isosceles right QRS.
y
Example 3 Coordinate Proof
S(?, ?)
Write a coordinate proof to prove that the measure of the segment that joins the
vertex of the right angle in a right triangle to the midpoint of the hypotenuse is
one-half the measure of the hypotenuse.
y
The first step is to position and label a right triangle on the
B (0, 2b )
coordinate plane. Place the right angle at the origin and label
it A. Use coordinates that are multiples of 2 because the
P
Midpoint Formula takes half the sum of the coordinates.
Given: right ABC with right BAC
A (0, 0) O
Q(?, ?)
Q(0, 0); S(c, c)
C (2c , 0 ) x
WRITE COORDINATE
PROOFS
P is the midpoint of BC
.
1
2
Prove: AP BC
Proof:
In-Class Examples
0 2c 2b 0
2
2
By the Midpoint Formula, the coordinates of P are , or (c, b).
Use the Distance Formula to find AP and BC.
AP (c 0
)2 (b
0)2
prove that the segment that
joins the vertex angle of an
isosceles triangle to the
midpoint of its base is
perpendicular to the base.
2 4b
2 b2
BC 4c
2 or 2 c
1
BC 2
2b2
c
1
2
Therefore, AP BC.
y
Example 4 Classify Triangles
Vertex Angle
Remember from the
Geometry Activity on
page 216 that an isosceles
triangle can be folded in
half. Thus, the x-coordinate
of the vertex angle is the
same as the x-coordinate
of the midpoint of the
base.
ARROWHEADS Write a coordinate proof to prove
that this arrowhead is shaped like an isosceles triangle.
The arrowhead is 3 inches long and 1.5 inches wide.
The first step is to label the coordinates of each vertex.
Q is at the origin, and T is at (1.5, 0). The y-coordinate
of R is 3. The x-coordinate is halfway between 0 and
1.5 or 0.75. So, the coordinates of R are (0.75, 3).
y
Z (2a, 0) x
Midpoint of XZ is (a, 0). Slope of
Y
W is undefined. Slope of XZ is 0.
So, YW
⊥
XZ.
Teaching Tip
Advise students
that they may want to place a
figure using numeric coordinates
first and then translate to variable
coordinates to write their proofs.
Q
DRAFTING Write a coordinate
Tx
4 proof to prove that this
drafter’s tool is shaped like a
right triangle. The length of
one side is 10 inches and the
length of another side is
5.75 inches.
RT (1.5
0.75)2
(0 3)2
0.5625
9 or 9.5625
Since each leg is the same length, QRT is isosceles. The arrowhead is shaped like
an isosceles triangle.
www.geometryonline.com/extra_examples
W
X(0, 0)
O
0.5625
9 or 9.5625
Y(a, b)
R
If the legs of the triangle are the same length, the
triangle is isosceles. Use the Distance Formula to
determine the lengths of QR and RT.
0)2 (3 0
)2
QR (0.75
Power
Point®
3 Write a coordinate proof to
BC (2c 0)2 (0 2
b)2
2 b2
c
Study Tip
R(c, 0) x
y
E
Lesson 4-7 Triangles and Coordinate Proof 223
(0, 10)
Francois Gohier/Photo Researchers
Differentiated Instruction
D
Kinesthetic You can mark a coordinate plane on a corkboard. You can
demonstrate for students, or volunteers can practice placing different
figures on the coordinate plane using pushpins for vertices and string for
sides.
F
(5.75, 0)
Slope of ED
is undefined.
Slope of DF is 0. ED
⊥
DF
DEF is a right triangle. The
drafter’s tool is shaped like a
right triangle.
Lesson 4-7 Triangles and Coordinate Proofs 223
Concept Check
Groups of students can work
together writing coordinate
proofs with one student placing
the triangle, another student
labeling the coordinates, and so
on. Tell groups to discuss
labeling variables with multiples
of two to make midpoint
computation easier and using
points on the coordinate axes as
much as possible to simplify
formulas. Groups can also
discuss limitations of coordinate
proofs such as the difficulty of
placing and labeling an
equilateral triangle, and so on.
Concept Check
2. OPEN ENDED Draw a scalene right triangle on the coordinate plane for use in
a coordinate proof. Label the coordinates of each vertex. 1–2. See margin.
Guided Practice
GUIDED PRACTICE KEY
Exercises
Examples
3–4
5–7
8
9
1
2
3
4
Application
y
P (0, c)
P (?, ?)
Q (a, 0)
N (?, ?)
x
Q (?, ?) O
x
N (2a, 0)
R (–a, 0) O
Q (?, ?)
9. TEPEES Write a coordinate proof to prove that
the tepee is shaped like an isosceles triangle.
Suppose the tepee is 8 feet tall and 4 feet wide.
Practice and Apply
For
Exercises
See
Examples
10–15
16–24
25–29
30–33
1
2
3
4
About the Exercises…
Position and label each triangle on the coordinate plane.
10. isosceles 䉭QRT with base Q
苶R
苶 that is b units long 10–15. See p. 233H.
11. equilateral 䉭MNP with sides 2a units long
M and legs c units long
12. isosceles right 䉭JML with hypotenuse J苶苶
1
2
13. equilateral 䉭WXZ with sides ᎏᎏb units long
14. isosceles 䉭PWY with a base P
苶W
苶 that is (a ⫹ b) units long
XY
15. right 䉭XYZ with hypotenuse X
苶Z
苶, ZY ⫽2(XY), and 苶
苶 b units long
Find the missing coordinates of each triangle.
y
y
16.
17.
Organization by Objective
• Position and Label
Triangles: 10–24
• Write Coordinate Proofs:
25–33
224 Chapter 4 Congruent Triangles
N(0, b), Q(a, 0)
See p. 233H.
See page 762.
Basic: 11–37 odd, 38–47
Average: 11–37 odd, 38–47
Advanced: 10–36 even, 37–47
7.
8. Write a coordinate proof for the following statement.
The midpoint of the hypotenuse of a right triangle is equidistant from each of
the vertices. See p. 233H.
Extra Practice
Assignment Guide
Find the missing coordinates of each triangle.
y
y Q(⫺2a, 0)
P(0, b) 6.
5.
x
Study Notebook
Odd/Even Assignments
Exercises 10–33 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
Position and label each triangle on the coordinate plane. 3–4. See margin.
3. isosceles 䉭FGH with base 苶
FH
苶 that is 2b units long
4. equilateral 䉭CDE with sides a units long
O R ( 0, 0)
3 Practice/Apply
Have students—
• add the definitions/examples of
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 4.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
1. Explain how to position a triangle on the coordinate plane to simplify a proof.
18.
R (?, b)
N (?, ?)
Q (a, ?)
x
O P ( 0, 0)
Q (2a, 0)
x
O L ( 0, 0) P (?, ?)
R(a, b)
224
Answers
2. Sample answer:
y
C(0, b)
A(0, 0)
x
Q(a, a), P(a, 0)
Chapter 4 Congruent Triangles
John Elk III/Stock Boston
1. Place one vertex at the
origin, place one side of the
triangle on the positive xaxis. Label the coordinates
with expressions that will
simplify the computations.
y
B(a, 0) x
O J ( 0, 0)
K (2a, 0)
N(0, 2a)
19.
20.
F (b, b √3)
y
21.
y
y
NAME ______________________________________________ DATE
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
4-7
Study
Guide and
P (?, ?)
E (?, ?)
p. 219
(shown)
and
p. 220
Triangles
and Coordinate
Proof
Position and Label Triangles A coordinate proof uses points, distances, and slopes to
prove geometric properties. The first step in writing a coordinate proof is to place a figure on
the coordinate plane and label the vertices. Use the following guidelines.
O C ( 0, 0)
B (?, ?) O
D (?, ?)
D(2b, 0)
22.
M (–2b, 0) O
B(a, 0), E(0, b)
23.
y
y
1.
2.
3.
4.
x
x
C (a , 0 )
N (?, ?)
Example
Position an equilateral triangle on the
coordinate plane so that its sides are a units long and
one side is on the positive x-axis.
Start with R(0, 0). If RT is a, then another vertex is T(a, 0).
P(0, c), N(2b, 0)
24.
J (?, ?)
Use the origin as a vertex or center of the figure.
Place at least one side of the polygon on an axis.
Keep the figure in the first quadrant if possible.
Use coordinates that make the computations as simple as possible.
y
P (a , a )
G (?, ?)
y
a
2
For vertex S, the x-coordinate is . Use b for the y-coordinate,
so the vertex is S, b.
a
2
S a–2, b
Lesson 4-7
x
T (a, 0) x
R (0, 0)
Exercises
Find the missing coordinates of each triangle.
1.
J (–b, 0) O
x
H (?, ?)
x
L (2c, 0)
O K ( 0, 0)
y
x
Q (?, ?)
O N ( 0, 0)
2.
C (?, q)
J(c, b)
y
F (?, b)
T (?, ?)
A(0, 0) B(2p, 0) x
R(0, 0) S(2a, 0) x
C(p, q)
G(0, c), H(b, 0)
3.
y
G(2g, 0) x
E(?, ?)
T(2a, 2a)
E(2g, 0); F(0, b)
Q(a, 0)
Sample answers
Position and label each triangle on the coordinate plane. are given.
Write a coordinate proof for each statement. 25–28. See pp. 233H–233I.
25. The segments joining the vertices to the midpoints of the legs of an isosceles
triangle are congruent.
y
Gl
S(4a, 0) x
D(0, 0)
Q(0, a)
E(e, 0) x
I (b, 0) x
E(–b, 0)
NAME
______________________________________________
DATE
/M
G
Hill
219
____________
Gl PERIOD
G _____
p. 221 and
Practice,
p. Coordinate
222 (shown)
Triangles and
Proof
1. equilateral SWY with
1
sides a long
2. isosceles BLP with
base BL
3b units long
4
y
y
Y 1–8a, b
W 1–4a, 0 x
S(0, 0)
29. STEEPLECHASE Write a coordinate
proof to prove that triangles ABD and
is perpendicular
FBD are congruent. BD
, and B is the midpoint of the upper
to AF
bar of the hurdle. See p. 233I.
y
F (e, e)
Sample answers
are given.
Position and label each triangle on the coordinate plane.
28. If a line segment joins the midpoints of two sides of a triangle, then its length is
equal to one-half the length of the third side.
Source: www.steeplechasetimes.
com
6. equilateral triangle EQI
with vertex Q(0, a) and
sides 2b units long
Skills
Practice,
4-7
Practice
(Average)
27. If a line segment joins the midpoints of two sides of a triangle, then it is parallel
to the third side.
The Steeplechase is a
horse race two to four
miles long that focuses
on jumping hurdles.
The rails of the fences
vary in height.
y
T(2a, b)
R(0, 0)
26. The three segments joining the midpoints of the sides of an isosceles triangle
form another isosceles triangle.
Steeplechase
5. isosceles right DEF
with legs e units long
4. isosceles triangle
RST with base R
S
4a units long
3. isosceles right DGJ
with hypotenuse D
J
and
legs 2a units long
y
P 3–2b, c
D (0, 2a)
L(3b, 0) x
B(0, 0)
G(0, 0) J (2a, 0)
x
Find the missing coordinates of each triangle.
B
4.
y
5.
S (?, ?)
6.
y
y
M (0, ?)
E (0, ?)
4 ft
A
F
D
R 1–3b, 0 x
J (0, 0)
16
B (–3a, 0)
S b, c
1 ft
6 ft
C (?, 0) x
P (2b, 0) x
N (?, 0)
C(3a, 0), E(0, c)
M(0, c), N(2b, 0)
NEIGHBORHOODS For Exercises 7 and 8, use the following information.
Karina lives 6 miles east and 4 miles north of her high school. After school she works part
time at the mall in a music store. The mall is 2 miles west and 3 miles north of the school.
7. Write a coordinate proof to prove that Karina’s high school, her home, and the mall are
at the vertices of a right triangle.
NAVIGATION For Exercises 30 and 31, use the following information.
A motor boat is located 800 yards east of the port. There is a ship 800 yards to the
east, and another ship 800 yards to the north of the motor boat.
30. Write a coordinate proof to prove that the port, motor boat, and the ship to the
north form an isosceles right triangle.
Given: SKM
Prove: SKM is a right triangle.
y
K (6, 4)
M (–2, 3)
Proof:
40
60
30
2 0
2
3
Slope of SK or x
S (0, 0)
3
2
Slope of SM or Since the slope of S
M
is the negative reciprocal of the
slope of SK
, SM
⊥
SK
. Therefore, SKM is right triangle.
31. Write a coordinate proof to prove that the distance between the two ships is the
same as the distance from the port to the northern ship. 30–31. See p. 233I.
8. Find the distance between the mall and Karina’s home.
KM (2 6)2 (3 4)2 64 1
65
or 8.1 miles
NAME
______________________________________________
DATE
/M
G
Hill
222
Gl
____________
Gl PERIOD
G _____
Reading
4-7
Readingto
to Learn
Learn Mathematics
Mathematics,
p. 223
Triangles and Coordinate Proof
HIKING For Exercises 32 and 33, use the following information.
Tami and Juan are hiking. Tami hikes 300 feet east of the camp and then hikes
500 feet north. Juan hikes 500 feet west of the camp and then 300 feet north.
32. Write a coordinate proof to prove that Juan, Tami, and the camp form a right
triangle. See p. 233I.
ELL
How can the coordinate plane be useful in proofs?
Read the introduction to Lesson 4-7 at the top of page 222 in your textbook.
From the coordinates of A, B, and C in the drawing in your textbook, what
do you know about ABC? Sample answer: ABC is isosceles
with C as the vertex angle.
Reading the Lesson
Lesson 4-7
33. Find the distance between Tami and Juan.
Pre-Activity
1. Find the missing coordinates of each triangle.
680,00
0 or about 824.6 ft
a.
b.
y
R (?, b)
y
F (?, ?)
E (?, a)
T(a, ?)
www.geometryonline.com/self_check_quiz
S (?, ?)
Lesson 4-7 Triangles and Coordinate Proof 225
x
D (?, ?)
x
b
2
R(0, b), S(0, 0), T a, Christopher Morrow/Stock Boston
D(0, 0), E(0, a), F(a, a)
2. Refer to the figure.
y
R
and the slope of S
T
. 1; 1
a. Find the slope of S
NAME ______________________________________________ DATE
Answers
3. Sample answer:
y
G(b, c)
4-7
Enrichment
Enrichment,
4. Sample answer:
y
E(a2 , b)
____________ PERIOD _____
c. What does your answer from part b tell you about RST ?
p. 224
O (0, 0) T (a, 0) x
SR 2a 2 or a2
; ST 2a 2 or a2
; SR
ST
e. What does your answer from part d tell you about RST?
Sample answer: RST is isosceles with RST as the vertex angle.
Each puzzle below contains many triangles. Count them carefully.
Some triangles overlap other triangles.
f. Combine your answers from parts c and e to describe RST as completely as possible.
Sample answer: RST is an isosceles right triangle. RST is the right
angle and is also the vertex angle.
How many triangles are there in each figure?
8
R (–a, 0)
Sample answer: RST is a right triangle with S as the right angle.
d. Find SR and ST. What does this tell you about S
R
and S
T
?
How Many Triangles?
1.
S (0, a)
b. Find the product of the slopes of S
R
and S
T
. What
does this tell you about SR
and ST
? 1; SR
⊥
ST
2.
40
3.
35
g. Find mSRT and mSTR. 45; 45
h. Find mOSR and mOST. 45; 45
Helping You Remember
4.
F(0, 0) H(2b, 0) x
5
5.
13
6.
C(0, 0) D (a, 0) x
27
3. Many students find it easier to remember mathematical formulas if they can put them
into words in a compact way. How can you use this approach to remember the slope and
midpoint formulas easily?
Sample answer: Slope Formula: change in y over change in x;
Midpoint Formula: average of x-coordinates, average of y-coordinates
How many triangles can you form by joining points on each circle?
Lesson 4-7 Triangles and Coordinate Proofs 225
4 Assess
Open-Ended Assessment
Speaking Have students speak
about how they would place
certain figures in a coordinate
plane and how they would label
the vertices. Students can discuss
different ideas about placement
and how they can simplify
coordinate proofs by using the
origin and simple labeling
techniques.
34. (a, 0) or (0, b)
35. (2a, 0)
Find the coordinates of point Z so XYZ is the indicated type of triangle. Point X
has coordinates (0, 0) and Y has coordinates (a, b).
34. right triangle
35. isosceles triangle
36. scalene triangle
Sample answer: (c, 0)
with right angle Z
with base XZ
37. CRITICAL THINKING Classify ABC by its angles
y
C (0, 2a)
and its sides. Explain. See margin.
38. WRITING IN MATH
Answer the question that was
posed at the beginning of the
lesson. See margin.
How can the coordinate plane be useful in proofs?
x
A (–2a, 0) O
B (2a, 0)
Include the following in your answer:
• types of proof, and
• a theorem from this chapter that could be proved using a coordinate proof.
Standardized
Test Practice
Assessment Options
39. What is the length of the segment whose endpoints are at (1,2) and (3, 1)? C
A 3
B 4
C 5
D 6
40. ALGEBRA What are the coordinates of the midpoint of the line segment whose
endpoints are (5, 4) and (2, 1)? B
A (3, 3)
B (3.5, 1.5)
C (1.5, 2.5)
D (3.5, 2.5)
Quiz (Lesson 4-7) is available
on p. 240 of the Chapter 4
Resource Masters.
Maintain Your Skills
Answers
37. AB 4a
AC (0 (
2a))2 (2a 0)2
Mixed Review
4a 2 4a 2 or 8a 2
Write a two-column proof. (Lessons 4-5 and 4-6) 41–44. See pp. 233I–233J.
41. Given: 3 4
42. Given: isosceles triangle JKN
LM
with vertex N, JK
Prove: QR
Q
S
Prove: NML is isosceles.
Q
K
CB (0 2
a)2 (2a 0)2
2
4a 2 4a 2 or 8a 2
3
2a 0
0 (2a)
2a 0
slope of C
B
or 1.
0 2a
Slope of AC
or 1;
AB
⊥
CB
and A
C
CB
, so ABC
is a right isosceles triangle.
38. Sample answer: Placing the
figures on the coordinate plane is
useful in proofs. We can use
coordinate geometry to prove
theorems and verify properties.
Answers should include the
following.
• flow proof, two-column proofs,
paragraph proofs, informal
proofs, and coordinate proofs
• Sample answer: The Isosceles
Triangle Theorem can be proved
using coordinate proof.
226 Chapter 4 Congruent Triangles
1
4
4
2
N
3
S
R
M
1
J
CE
43. Given: AD
AD
C
E
, Prove: ABDEBC
L
44. Given: W
X
X
Y
, VZ
Prove: WV
Y
Z
A
W
C
V
B
X
Z
D
Y
E
AD
45. BC
; if alt. int.
are , lines are 46. h j; Sample
answer: cons. int.
suppl.
47. ᐉ m; 2 lines to
same line are .
State which lines, if any, are parallel. State the postulate or theorem that justifies
your answer. (Lesson 3-5)
g
j
B
C
h
45.
46.
47.
k
f
24˚
71˚
111˚
j
24˚
A
226 Chapter 4 Congruent Triangles
D
69˚
m
Study Guide
and Review
Vocabulary and Concept Check
acute triangle (p. 178)
base angles (p. 216)
congruence transformations (p. 194)
congruent triangles (p. 192)
coordinate proof (p. 222)
corollary (p. 188)
equiangular triangle (p. 178)
equilateral triangle (p. 179)
exterior angle (p. 186)
flow proof (p. 187)
included angle (p. 201)
included side (p. 207)
Vocabulary and
Concept Check
isosceles triangle (p. 179)
obtuse triangle (p. 178)
remote interior angles (p. 186)
right triangle (p. 178)
scalene triangle (p. 179)
vertex angle (p. 216)
A complete list of theorems and postulates can be found on pages R1–R8.
Exercises Choose the letter of the word or phrase that best matches each statement.
1. A triangle with an angle whose measure is greater than 90 is a(n) ?
a. acute
triangle. h
b. AAS Theorem
2. A triangle with exactly two congruent sides is a(n) ? triangle. g
c. ASA Theorem
3. The ? states that the sum of the measures of the angles of a
d. Angle Sum Theorem
triangle is 180. d
e. equilateral
4. If BE, AB
D
E
, and B
C
E
F
, then ABCDEF by ? . j
f. exterior
5. In an equiangular triangle, all angles are ? angles. a
g. isosceles
6. If two angles of a triangle and their included side are congruent to
h. obtuse
two angles and the included side of another triangle, this is called
the ? . c
i. right
7. If AF, BG, and AC
F
H
, then ABCFGH, by ? . b
j. SAS Theorem
8. A(n) ? angle of a triangle has a measure equal to the measures of
k. SSS Theorem
the two remote interior angles of the triangle. f
4-1 Classifying Triangles
See pages
178–183.
Example
y
U
V
(2)]2
[4
(
2)]2
TU [5
9 36 or 45
UV [3 (
5)]2 (1 4)2
O
x
ELL The Vocabulary PuzzleMaker
software improves students’ mathematics
vocabulary using four puzzle formats—
crossword, scramble, word search using a
word list, and word search using clues.
Students can work on a computer screen
or from a printed handout.
MindJogger
Videoquizzes
VT (2
3)2 (2 1)2
25 9 or 34
Since none of the side measures are equal, TUV is scalene.
Chapter 4 Study Guide and Review 227
TM
For more information
about Foldables, see
Teaching Mathematics
with Foldables.
For each lesson,
• the main ideas are
summarized,
• additional examples review
concepts, and
• practice exercises are provided.
T
64 9 or 73
www.geometryonline.com/vocabulary_review
Lesson-by-Lesson
Review
Vocabulary
PuzzleMaker
Concept Summary
• Triangles can be classified by their angles as acute, obtuse, or right.
• Triangles can be classified by their sides as scalene, isosceles, or equilateral.
Find the measures of the sides of TUV. Classify
the triangle by sides.
Use the Distance Formula to find the measure of each side.
• This alphabetical list of
vocabulary terms in Chapter 4
includes a page reference
where each term was
introduced.
• Assessment A vocabulary
test/review for Chapter 4 is
available on p. 238 of the
Chapter 4 Resource Masters.
Have students look through the chapter to make sure they have
included notes and examples in their Foldables for each lesson of
Chapter 4.
Encourage students to refer to their Foldables while completing
the Study Guide and Review and to use them in preparing for the
Chapter Test.
ELL MindJogger Videoquizzes
provide an alternative review of concepts
presented in this chapter. Students work
in teams in a game show format to gain
points for correct answers. The questions
are presented in three rounds.
Round 1 Concepts (5 questions)
Round 2 Skills (4 questions)
Round 3 Problem Solving (4 questions)
Chapter 4 Study Guide and Review 227
Study Guide and Review
Chapter 4 Study Guide and Review
Exercises Classify each triangle by its angles and
by its sides if mABC 100. See Examples 1 and 2 on
Answers
15. E D, F C, G B,
EF DC
, FG
CB
, GE BD
16. FGC DLC, GCF LCD,
GFC LDC, GC
LC
,
C
F CD
, FG
DL
17. KNC RKE, NCK KER,
CKN ERK, NC
KE,
C
K
ER
, KN
RK
B
pages 178 and 179.
9. ABC
10. BDP
11. BPQ
60˚
A
P
D
Q
C
9. obtuse, isosceles 10. right, scalene 11. equiangular, equilateral
4-2 Angles of Triangles
See pages
185–191.
Example
Concept Summary
• The sum of the measures of the angles of a triangle is 180.
• The measure of an exterior angle is equal to the sum of the measures of the
two remote interior angles.
If T
UV
UV
VW
U
and , find m1.
m1 72 mTVW 180 Angle Sum Theorem
U
T
2
m1 72 (90 27) 180 mTVW 9027
m1 135 180 Simplify.
m1 45
Exercises
Subtract 135 from each side.
1
27˚
72˚
V
Find each measure.
13. m2 25
W
2
See Example 1 on page 186.
12. m1 85
S
45˚
14. m3 95
70˚
1 3
40˚
4-3 Congruent Triangles
See pages
192–198.
Example
Concept Summary
• Two triangles are congruent when all of their corresponding parts are congruent.
If EFGJKL, name the corresponding congruent angles and sides.
EJ, FK, GL, EF
FG
KL
JK
, , and E
G
JL
.
Exercises Name the corresponding angles and sides for each pair of congruent
triangles. See Example 1 on page 193. 15–17. See margin.
15. EFGDCB
16. LCD GCF
17. NCK KER
4-4 Proving Congruence—SSS, SAS
See pages
200–206.
Concept Summary
• If all of the corresponding sides of two triangles are congruent, then the
triangles are congruent (SSS).
• If two corresponding sides of two triangles and the included angle are
congruent, then the triangles are congruent (SAS).
228 Chapter 4 Congruent Triangles
228
Chapter 4 Congruent Triangles
Chapter 4 Study Guide and Review
Example
Determine whether ABC TUV. Explain.
[1 (2)]2
(1
0)2
AB TU (3
4
)2 (
1 0)2
1 1 or 2
BC [0
(
1)]2 (1
1)2
B
UV (2
3
)2 [1
(
1)]2
1 4 or 5
V
T
x
O
A
C
1 4 or 5
CA (2 0)2 [0 (
1)]2
Answers
y
1 1 or 2
U
VT (4 2
)2 (0
1)2
4 1 or 5
4 1 or 5
Exercises Determine whether MNP QRS given the coordinates of the
vertices. Explain. See Example 2 on page 201. 18 – 19. See margin.
18. M(0, 3), N(4, 3), P(4, 6), Q(5, 6), R(2, 6), S(2, 2)
19. M(3, 2), N(7, 4), P(6, 6), Q(2, 3), R(4, 7), S(6, 6)
4-5 Proving Congruence—ASA, AAS
Example
D
Concept Summary
• If two pairs of corresponding angles and the included sides of two triangles
are congruent, then the triangles are congruent (ASA).
• If two pairs of corresponding angles and a pair of corresponding nonincluded
sides of two triangles are congruent, then the triangles are congruent (AAS).
Write a proof.
KM
MN
Given: JK
; L is the midpoint of .
Prove:
J
C
K
JK || MN
JKL LMN
Given
Alt. int. s are .
L is the midpoint of KM.
KL ML
JLK NLM
Given
Midpoint Theorem
ASA
N
JLK NLM
Vertical s are .
Exercises For Exercises 20 and 21, use the figure. Write a
two-column proof for each of the following. See Example 2
on page 209.
D
20–21. See margin.
20. Given: D
F
bisects CDE.
DF
CE
Prove: DGC DGE
21. Given: DGC DGE
GCF GEF
Prove: DFC DFE
G
C
E
Proof:
Statements (Reasons)
L
Flow proof:
G
F
M
JLKNLM
18. MN 4, NP 3, MP 5, QR 3,
RS 4, and QS 5. Each pair of
corresponding sides does not have
the same measure. Therefore,
MNP is not congruent to QRS.
MNP is congruent to SRQ.
19. MN 20, NP 5, MP 5,
QR 20, RS 5, and
QS 5. Each pair of
corresponding sides has the same
measure. Therefore,
MNP QRS by SSS.
20. Given: DF bisects CDE, CE ⊥ DF.
Prove: DGC DGE
By the definition of congruent segments, all corresponding sides are congruent.
Therefore, ABCTUV by SSS.
See pages
207–213.
Study Guide and Review
E
1. D
F bisects CDE, CE ⊥ DF.
(Given)
2. DG
DG
(Reflexive Prop.)
3. CDF EDF (Def. of bisector)
4. DGC is a rt. ; DGE is a
rt. (Def. of ⊥ segments)
5. DGC DGE (All rt. are .)
6. DGC DGE (ASA)
21. Given: DGC DGE,
GCF GEF
Prove: DFC DFE
D
F
Chapter 4 Study Guide and Review 229
C
G
E
F
Proof:
Statements (Reasons)
1. DGC DGE, GCF GEF (Given)
2. CDG EDG, CD
ED
, and
CFD EFD (CPCTC)
3. DFC DFE (AAS)
Chapter 4 Study Guide and Review 229
• Extra Practice, see pages 760–762.
• Mixed Problem Solving, see page 785.
Study Guide and Review
4-6 Isosceles Triangles
Answers
See pages
216–221.
26. Sample answer:
y
R(2a, b)
Example
I(4a, 0) x
T(0, 0)
GJ, GJ JH
FH
If FG
, F
J , and mGJH 40, find mH.
GHJ is isosceles with base GH
, so JGHH by the Isosceles Triangle
Theorem. Thus, mJGH mH.
2(mH) 140 Subtract 40 from each side.
mH 70
Exercises
D(6m, 0) x
22.
23.
24.
25.
y
J(0, b)
L(a, 0)
See pages
222–226.
Example
Answers (p. 231)
10. D P, E Q, F R,
DE PQ
, EF QR
, DF PR
11. F H, M N,
G J, FM
H,
N MG
NJ,
FG
HJ
12. X Z, Y Y, Z X,
XY
ZY
, YZ YX
, XZ ZX
K
J
L
N
230
For Exercises 22–25, refer to the figure at the right.
If P
Q
U
Q
and mP 32, find mPUQ. 32
If P
Q
U
Q
, P
R
R
T
, and mPQU 40, find mR. 40
If R
Q
R
S
and mRQS 75, find mR. 30
If R
RP
Q
R
S
, R
T
, and mRQS 80, find mP. 80
Chapter 4 Congruent Triangles
H
R
Q
P
S
U
T
Concept Summary
• Coordinate proofs use algebra to prove geometric concepts.
• The Distance Formula, Slope Formula, and Midpoint Formula
are often used in coordinate proof.
Position and label isosceles right triangle ABC with legs of length a units
on the coordinate plane.
• Use the origin as the vertex of ABC that has the right angle.
A(0, 0)
O
• Place each base along an axis.
• Since B is on the x-axis, its y-coordinate is 0. Its x-coordinate
C (0, – a )
is a because the leg A
B
of the triangle is a units long.
• Since ABC is isosceles, C should also be a distance of a units
from the origin. Its coordinates are (0, a).
Exercises
Position and label each triangle on the coordinate plane.
See Example 1 on page 222.
26– 28. See margin.
26. isosceles TRI with base T
I 4a units long
27. equilateral BCD with side length 6m units long
28. right JKL with leg lengths of a units and b units
230 Chapter 4 Congruent Triangles
Proof: JKM JNM
Given
JK JN
KJL NJL
M
J
4-7 Triangles and Coordinate Proof
x
13. JK 10
, KL 17
, JL 5,
MN 80, NP 53,
221. Corresponding sides
MP are not congruent, so JKL is not
congruent to MNP.
14. Given: JKM JNM
Prove: JKL JNL
Divide each side by 2.
See Example 2 on page 217.
28. Sample answer:
K(0, 0)
G
40 2(mH) 180 Substitution
C(3m, n)
B(0, 0)
F
mGJH mJGH mH 180 Angle Sum Theorem
27. Sample answer:
y
Concept Summary
• Two sides of a triangle are congruent if and only if the angles opposite
those sides are congruent.
• A triangle is equilateral if and only if it is equiangular.
CPCTC
JKL JNL
JL JL
SAS
Reflexive Prop.
B (a , 0)
Practice Test
Vocabulary and Concepts
Assessment Options
Choose the letter of the type of triangle that best matches each phrase.
1. triangle with no sides congruent b
a. isosceles
2. triangle with at least two sides congruent a
b. scalene
3. triangle with all sides congruent c
c. equilateral
Vocabulary Test A vocabulary
test/review for Chapter 4 can be
found on p. 238 of the Chapter 4
Resource Masters.
Chapter Tests There are six
Chapter 4 Tests and an OpenEnded Assessment task available
in the Chapter 4 Resource Masters.
Skills and Applications
Identify the indicated triangles in the figure
AD
PA
PC
if PB
and . 6. PBA, PBC, PBD
4. obtuse PCD 5. isosceles PAC 6. right
P
1
3
Find the measure of each angle in the figure.
7. m1 80
8. m2 105
9. m3 25
A
B
C
100˚
2 75˚
D
Questions 4–6
Form
Questions 7–9
1
2A
2B
2C
2D
3
Name the corresponding angles and sides for each pair of congruent triangles. 10–12. See margin.
10. DEF PQR
11. FMG HNJ
12. XYZ ZYX
13. Determine whether JKL MNP given J(1, 2), K(2, 3), L(3, 1),
M(6, 7), N(2, 1), and P(5, 3). Explain. See margin.
14. Write a flow proof. See margin.
Given: JKMJNM
Prove: JKL JNL
K
F
J
In the figure, FJ FH
GF
GH
and .
15. If mJFH 34, find mJ. 73
16. If mGHJ 152 and mG 32, find mJFH. 24
G
J
N
H
Question 14
Questions 15–16
17. LANDSCAPING A landscaper designed a garden shaped as shown
in the figure. The landscaper has decided to place point B 22 feet
east of point A, point C 44 feet east of point A, point E 36 feet south of
point A, and point D 36 feet south of point C. The angles at points A
and C are right angles. Prove that ABECBD. See p. 233J.
B
A
B
62
C
56
D
D
H
J
28
F
www.geometryonline.com/chapter_test
Pages
225–226
227–228
229–230
231–232
233–234
235–236
MC = multiple-choice questions
FR = free-response questions
ExamView® Pro
18. STANDARDIZED TEST PRACTICE In the figure, FGH is a right triangle
with hypotenuse F
H
and GJ GH. What is the measure of JGH? C
104
basic
average
average
average
average
advanced
C
E
A
MC
MC
MC
FR
FR
FR
Open-Ended Assessment
Performance tasks for Chapter 4
can be found on p. 237 of the
Chapter 4 Resource Masters. A
sample scoring rubric for these
tasks appears on p. A28.
L
M
Chapter 4 Tests
Type
Level
28˚
G
Chapter 4 Practice Test 231
Portfolio Suggestion
Introduction As students progress with writing geometric proofs, it can help for
them to have an example of their own work to refer to for self-encouragement
or to review for method or style.
Ask Students Go back through your notes and problems that you worked and
find a well-organized, clear and concise proof that you composed. Place an
example of this proof in your portfolio, and write a paragraph explaining why you
selected this particular example. Highlight important concepts and methods that
you used to form your proof.
Use the networkable
ExamView® Pro to:
• Create multiple versions of
tests.
• Create modified tests for
Inclusion students.
• Edit existing questions and
add your own questions.
• Use built-in state curriculum
correlations to create tests
aligned with state standards.
• Apply art to your tests from a
program bank of artwork.
Chapter 4 Practice Test 231
Standardized
Test Practice
These two pages contain practice
questions in the various formats
that can be found on the most
frequently given standardized
tests.
Part 1 Multiple Choice
Record your answers on the answer sheet
provided by your teacher or on a sheet of
paper.
(Lesson 3-3)
1. In 2002, Capitol City had a population of 2010,
and Shelbyville had a population of 1040. If
Capitol City grows at a rate of 150 people a
year and Shelbyville grows at a rate of 340
people a year, when will the population of
Shelbyville be greater than that of Capitol
City? (Prerequisite Skill) B
A practice answer sheet for these
two pages can be found on p. A1
of the Chapter 4 Resource Masters.
O
Practice
4Standardized
Standardized Test
Test Practice
Student Record
Sheet (Use with Sheet,
pages 232–233 of
Student
Recording
p.the Student
A1 Edition.)
Part 1 Multiple Choice
A
2004
B
2008
C
2009
D
2012
Select the best answer from the choices given and fill in the corresponding oval.
1
A
B
C
D
4
A
B
C
D
7
A
B
C
D
2
A
B
C
D
5
A
B
C
D
8
A
B
C
D
3
A
B
C
D
6
A
B
C
D
For Questions 12 and 14, also enter your answer by writing each number or
symbol in a box. Then fill in the corresponding oval for that number or symbol.
10
11
12
(grid in)
13
14
(grid in)
B
x 2y1
8
C
x2y 1
4
D
2x y 1
2
.
/
.
/
.
.
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
.
/
.
/
.
.
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
A
35
B
70
A
grams
B
feet
C
90
C
liters
D
meters
D
110
Part 3 Extended Response
3. A 9-foot tree casts a shadow on the ground.
The distance from the top of the tree to the end
of the shadow is 12 feet. To the nearest foot,
how long is the shadow? (Lesson 1-3) B
Record your answers for Questions 15–16 on the back of this paper.
9 ft
12 ft
10
6
0 1
14
Answers
12
12
2x y 1
2 3 4 5 6
6. What is mEFG?
2. Which unit is most appropriate for measuring
liquid in a bottle? (Lesson 1-2) C
Solve the problem and write your answer in the blank.
D
A
(Lesson 4-2)
Part 2 Short Response/Grid In
9
5. Students in a math classroom simulated stock
trading. Kris drew the graph below to model
the value of his shares at closing. The graph
that modeled the value of Mitzi’s shares was
parallel to the one Kris drew. Which equation
might represent the line for Mitzi’s graph?
A
7 ft
B
8 ft
C
10 ft
D
13 ft
G
B
F
(9x 7)˚
5x˚
5x˚
D
E
7. In the figure, ABDCBD. If A has the
coordinates (2, 4), what are the coordinates
of C? (Lesson 4-3) C
A
(4, 2)
B
(4, 2)
C
(2, 4)
D
(2, 4)
y
A
B
D
x
O
C
? ft
Additional Practice
See pp. 243–244 in the Chapter 4
Resource Masters for additional
standardized test practice.
4. Which of the following is the inverse of the
statement If it is raining, then Kamika carries an
umbrella? (Lesson 2-2) D
A
If Kamika carries an umbrella, then it is
raining.
B
If Kamika does not carry an umbrella,
then it is not raining.
C
If it is not raining, then Kamika carries an
umbrella.
D
If it is not raining, then Kamika does not
carry an umbrella.
232 Chapter 4 Standardized Test Practice
ExamView® Pro
Special banks of standardized test
questions similar to those on the SAT,
ACT, TIMSS 8, NAEP 8, and state
proficiency tests can be found on this
CD-ROM.
232 Chapter 4 Congruent Triangles
8. The wings of some butterflies can be modeled
by triangles as shown. If AC
D
C
and
ACB ECD, which additional statements
are needed to prove that ACB ECD?
(Lesson 4-4)
A
A
B
C C
E
B
A
B E
D
C
BAC CED
D
ABCCDE
D
A
C
E
B
Preparing for Standardized Tests
For test-taking strategies and more
practice, see pages 795– 810.
Evaluating Extended
Response Questions
Part 2 Short Response/Grid In
Record your answers on the answer sheet
provided by your teacher or on a sheet of
paper.
9. Find the product
(Prerequisite Skill)
3s2(2s3 6s5 21s2
Test-Taking Tip
• If you are not permitted to write in your test booklet,
make a sketch of the figure on scrap paper.
7).
• Mark the figure with all of the information you know
so that you can determine the congruent triangles
more easily.
10. After a long workout, Brian noted, “If I
do not drink enough water, then I will
become dehydrated.” He then made another
statement, “If I become dehydrated, then I
did not drink enough water.” How is the
second statement related to the original
statement? (Lesson 2-2) converse
• Make a list of postulates or theorems that you might
use for this case.
Part 3 Extended Response
Record your answers on a sheet of paper.
Show your work.
11. On a coordinate map, the towns of Creston
and Milford are located at (1, 1) and
(1, 3), respectively. A third town, Dixville, is
located at (x, 1) so that Creston and Dixville
are endpoints of the base of the isosceles
triangle formed by the three locations. What
is the value of x? (Lesson 4-1) 3
15. Train tracks a and b are parallel lines
although they appear to come together to
give the illusion of distance in a drawing. All
of the railroad ties are parallel to each other.
x˚
12. A watchtower, built to help
prevent forest fires, was
designed as an isosceles
triangle. If the side of the
tower meets the ground
at a 105° angle, what is
the measure of the angle
at the top of the tower?
(Lesson 4-2)
1
105˚
2
b. What is the relationship between the
tracks and the ties that run across the
tracks? (Lesson 1-5) perpendicular lines
1
c. What is the relationship between 1
and 2? Explain. (Lesson 3-2) See margin.
0
16. The measures of the angles of ABC are 5x,
4x 1, and 3x13.
a. Draw a figure to illustrate ABC.
D
E
b
a. What is the value of x? (Lesson 3-1) 90
ASA
C
A correct solution that is
supported by well-developed,
accurate explanations
A generally correct solution,
but may contain minor flaws
in reasoning or computation
A partially correct interpretation
and/or solution to the problem
A correct solution with no
supporting evidence or
explanation
An incorrect solution indicating
no mathematical understanding
of the concept or task, or no
solution is given
90˚
(Lesson 4-1)
B
Criteria
4
3
a
30
A
Score
2
13. During a synchronized flying show, airplane
A and airplane D are equidistant from the
ground. They descend at the same angle to
land at points B and E, respectively. Which
postulate would prove that ABC DEF?
(Lesson 4-4)
Extended Response questions
are graded by using a multilevel
rubric that guides you in
assessing a student’s knowledge
of a particular concept.
Goal: Students find angle
measures, describe angle
relationships, and prove that a
triangle is isosceles.
Sample Scoring Rubric: The
following rubric is a sample
scoring device. You may wish to
add more detail to this sample to
meet your individual scoring
needs.
Question 8
See margin.
b. Find the measure of each angle of ABC.
Explain. (Lesson 4-2) 70, 55, and 55
F
14. ABC is an isosceles triangle with AB
B
C
,
and the measure of vertex angle B is three
times mA. What is mC? (Lesson 4-6) 36
See margin for explanation.
c. Prove that ABC is an isosceles triangle.
(Lesson 4-6)
www.geometryonline.com/standardized_test
See margin.
Chapter 4 Standardized Test Practice 233
Answers
15c. They are congruent.
Sample answers: Both
are right angles; they
are supplementary
angles; they are
corresponding angles.
16a. Sample answer:
A
5x B
(4x 1) (3x 13)
C
16b. From the Angle Sum Theorem,
we know that
mA mB mC 180.
Substituting the given measures,
5x 4x 1 3x 13 180.
Solve for x to find that x 14.
Substitute 14 for x to find the
measures: 5x 5(14) or 70,
4x 1 4(14) 1 or 55, and
3x 13 3(14) 13 or 55.
16c. If two angles of a are , then
the sides opposite those angles
are (Converse of the
Isosceles Theorem). Since two
sides of this are , it is an
isosceles (Definition of
Isosceles ).
Chapter 4 Standardized Test Practice 233
Page 180–184 Lesson 4-1
38. Given: EUI is equiangular.
QL
|| U
I
Prove: EQL is equiangular.
Pages 188–191, Lesson 4-2
39. Given: FGI IGH
GI ⊥ FH
Prove: F H
Proof:
E
Q
L
G
F
GI ⊥ FH
U
Proof:
Statements (Reasons)
1. EUI is equiangular. QL
|| U
I (Given)
2. E EUI EIU (Def. of equiangular )
3. EUI EQL; EIU ELQ (Corr. Post.)
4. E E (Reflexive Prop.)
5. E EQL ELQ (Trans. Prop.)
6. EQL is equiangular. (Def. of equiangular )
39. Given: mNPM 33
Prove: RPM is obtuse.
M
I
N
33
Additional Answers for Chapter 4
P
R
Proof: NPM and RPM form a linear pair. NPM
and RPM are supplementary because if two angles
form a linear pair, then they are supplementary. So,
mNPM mRPM 180. It is given that
mNPM 33. By substitution, 33 mRPM 180.
Subtract to find that mRPM 147. RPM is obtuse
by definition. RPM is obtuse by definition.
40. TS (7 (4))2
(8
14
)2 9 36
or 45
SR (10
(7
))2 (
2 8
)2 9 36
or 45
S is the midpoint of R
T
.
UT (0 (
4))2 (8 14)2 16 36
or 52
VU (4 0
)2 (2
8)2 16 36
or 52
U is the midpoint of TV
.
41. AD CD 0 a2 (0 b)2 a2 (b)2
2
2
a
b
4
2
2
a a2 (0 b)2 a2 (b)2
2
2
a
b
4
2
2
AD CD, so A
D
C
D
. ADC is isosceles by
definition.
233A
Chapter 4 Additional Answers
I
H
Given
GIF and GIH
are right angles.
⊥ lines form rt. .
GIF GIH
All rt. are .
FGI IGH
Given
F H
Third Angle Theorem
40. Given: ABCD is a quadrilateral.
A
B
1 2
Prove: mDAB mB mBCD mD 360
3
4
C
Proof:
D
Statements (Reasons)
1. ABCD is a quadrilateral. (Given)
2. m2 m3 mB 180;
m1 m4 mD 180 ( Sum Theorem)
3. m2 m3 mB m1 m4 mD 360 (Addition Prop.)
4. mDAB m1 m2; mBCD m3 m4
( Addition)
5. mDAB mB mBCD mD 360
(Substitution)
41. Given: ABC
C
Prove: mCBD mA mC
D
A
Proof:
B
Statements (Reasons)
1. ABC (Given)
2. CBD and ABC form a linear pair. (Def. of linear
pair)
3. CBD and ABC are supplementary. (If 2 form
a linear pair, they are suppl.)
4. mCBD mABC 180 (Def. of suppl.)
5. mA mABC mC 180 ( Sum Theorem)
6. mA mABC mC mCBD mABC
(Substitution)
7. mA mC mCBD (Subtraction Prop.)
42. Given: RST
R is a right angle.
Prove: S and T are
complementary.
Proof:
S
R
T
Pages 195–198, Lesson 4-3
33. Given: RST XYZ
Prove: XYZ RST
R
S X
R is a rt. .
T
Given
Proof:
mR 90
Def. of rt. mR mS mT 180
Y
Z
RST XYZ
Given
Angle Sum Theorem
R X, S Y, T Z,
RS XY, ST YZ, RT XZ
90 mS mT 180
Substitution
CPCTC
X R, Y S, Z T,
XY RS, YZ ST, XZ RT
Congruence of and
mS mT 90
Subtraction Prop.
segments is symmetric.
S and T are complementary
Def. of complementary XYZ RST
Def. of Pages 203–206, Lesson 4-4
and BC
bisect each other.
5. Given: DE
Prove: DGB EGC
Proof:
B
D
G
E
DG and GE bisect each other.
Given
C
DE BC, BG GC
Def. of bisector
of segments
DGB EGC
SAS
DGB EGC
Vertical are .
6. Given: KM
|| JL
, KM
JL
Prove: JKM MLJ
K
J
Proof:
Statements (Reasons)
1.
2.
3.
4.
M
L
KM
|| JL
, KM
JL
(Given)
KMJ LJM (Alt. Int. Theorem)
JM
JM
(Reflexive Prop.)
JKM MLJ (SAS)
Chapter 4 Additional Answers 233B
Additional Answers for Chapter 4
43. Given: MNO
N
M is a right angle.
Prove: There can be at most one
M
O
right angle in a triangle.
Proof: In MNO, M is a right angle.
mM mN mO 180. mM 90, so
mN mO 90. If N were a right angle, then
mO 0. But that is impossible, so there cannot be
two right angles in a triangle.
Given: PQR
Q
P is obtuse.
Prove: There can be at most one
R
obtuse angle in a triangle.
P
Proof: In PQR, P is obtuse. So mP 90.
mP mQ mR 180. It must be that
mQ m R 90. So, Q and R must be acute.
44. Given: A D
F
B E
C
D
E
Prove: C F
A
B
Proof:
Statements (Reasons)
1. A D, B E (Given)
2. mA mD, mB mE (Def. of )
3. mA mB mC 180,
mD mE mF 180 (Angle Sum Theorem)
4. mA mB mC mD mE mF
(Transitive Prop.)
5. mD mE mC mD mE mF
(Substitution Prop.)
6. mC mF (Subtraction Prop.)
7. C F (Def. of )
14. Given: A
E
F
C
, AB
B
C
, BE
B
F
Prove: AFB CEB
GC
18. Given: AC
C
E
bisects AG
.
Prove: GEC AEC
B
F
E
Given
AE FC
Def. of seg.
AE EF EF FC
EF EF
Addition Prop.
AE EF AF
EF FC EC
Substitution
Seg. Addition Post.
Additional Answers for Chapter 4
AF EC
Def. of seg.
AFB CEB
AB BC
BE BF
SAS
Given
15. Given: RQ
TQ
YQ
WQ
RQY WQT
Prove: QWT QYR
R
Y
Q
W
Proof:
Given
Proof:
Statements (Reasons)
T
RQY WQT
QWT QYR
SAS
233C
Chapter 4 Additional Answers
J
R
1. R
S
PN
, RT
MP
(Given)
2. S N, and T M (Given)
3. R P (Third Angle Theorem)
4. RST MNP (SAS)
21. Given: EF
H
F
G is the midpoint of EH
.
F
Prove: EFG HFG
Given
16. Given: CDE is an isosceles
D
triangle.
G is the midpoint of CE
.
Prove: CDG EDG
Proof:
C
G
Statements (Reasons)
1. CDE is an isosceles triangle,
G is the midpoint of CE
. (Given)
2. C
D
DE
(Def. of isos. )
3. CG
GE
(Midpoint Th.)
4. DG
DG
(Reflexive Prop.)
5. CDG EDG (SSS)
17. Given: MRN QRP
M
MNP QPN
R
Prove: MNP QPN
Proof:
N
P
Statements (Reasons)
1. MRN QRP, MNP QPN (Given)
2. MN
QP
(CPCTC)
3. N
P
NP
(Reflexive Prop.)
4. MNP QPN (SAS)
K
Proof:
G
L
Statements (Reasons)
1. GHJ LKJ (Given)
2. HJ KJ, GJ L
J, GH
LK
(CPCTC)
3. HJ KJ, GJ LJ (Def. of segments)
4. HJ LJ KJ JG (Addition Prop.)
5. KJ GJ KG; HJ LJ HL (Segment Addition)
6. KG
HL
(Substitution)
7. KG
HL
(Def. of segments)
8. GL
GL
(Reflexive Prop.)
9. GHL LKG (SSS)
N
S
20. Given: RS
, RT
PN
MP
S N, T M
M
T
Prove: RST PNM
P
Reflexive Prop.
AE EC
C
G
1. A
C
GC
, EC
bisects A
G
. (Given)
2. AE
EG
(Def. of segment bisector)
3. E
C
EC
(Reflexive Prop.)
4. GEC AEC (SSS)
19. Given: GHJ LKJ
H
Prove: GHL LKG
Proof: AE FC
RQ TQ YQ WQ
E
Proof:
Statements (Reasons)
C
A
A
E
Q
Proof:
Statements (Reasons)
1. E
F
H
F
; G is the midpoint of EH
. (Given)
2. E
G
GH
(Def. of midpoint)
3. FG
FG
(Reflexive Prop.)
4. EFG HFG (SSS)
26. Given: T
S
S
F
F
H
H
T
S
TSF, SFH, FHT, and
HTS are right angles.
P
T
Prove: HS
TF
H
G
E
F
Proof:
H
Statements (Reasons)
1. T
S
S
F
F
H
H
T
(Given)
2. TSF, SFH, FHT, and HTS are right angles.
(Given)
3. STH THF (All right are .)
4. STH THF (SAS)
5. HS
TF
(CPCTC)
27. Given: TS
S
F
F
H
H
T
TSF, SFH, FHT, and
HTS are right angles.
Prove: SHT SHF
S
T
Proof:
Statements (Reasons)
P
F
GK || HJ
Given
G
H
K
KGJ HJG
Alt. int. are .
J
XWY ZYW
Alt. int. are .
GJ GJ
Reflexive Property
W
Z
WY WY
Reflexive
Property
WXY YZW
AAS
6. Given: QS
bisects RST;
R
R T.
Prove: QRS QTS
S
Q
Proof: We are given that
R T and QS
bisects
T
RST, so by definition of angle
bisector, RSQ TSQ. By the Reflexive Property,
QS
QS
. QRS QTS by AAS.
7. Given: E K,
D
DGH DHG,
EG
KH
Prove: EGD KHD
E
G
H
K
Proof: Since EGD and
DGH are linear pairs, the angles are supplementary.
Likewise, KHD and DHG are supplementary. We are
given that DGH DHG. Angles supplementary to
congruent angles are congruent so EGD KHD.
Since we are given that E K and EG
KH
,
EGD KHD by ASA.
9. Given: EF
|| G
H
, EF
GH
F
E
Prove: EK
KH
Proof:
K
EF || GH
Given
E H
Alt. int. are .
H
EF GH
G
Given
EKF HKG
AAS
EK KH
CPCTC
10. Given: D
E
|| JK
K
D
bisects JE
.
Prove: EGD JGK
Proof:
WXY YZW
ASA
Y
Given
GKH EKF
Vert. are .
Given
HGJ KJG
Alt. int. are .
X Z
X
J
D
G
DK bisects JE.
DE || JK
Given
E J
Alt. int. are .
DGE KGJ
Vert. are .
K
E
Given
EG GJ
Def. of seg. bisector
EGD JGK
ASA
Chapter 4 Additional Answers 233D
Additional Answers for Chapter 4
1. TS
S
F
F
H
H
T
(Given)
2. TSF, SFH, FHT, and HTS are right angles.
(Given)
3. STH SFH (All right are .)
4. STH SFH (SAS)
5. SHT SHF (CPCTC)
28. Given: DE
FB
, AE
FC
, A
B
AE
⊥
DB
, CF
⊥
DB
F
Prove: ABD CDB
Plan: First use SAS to
E
show that ADE CBF. D
C
Next use CPCTC and
Reflexive Property for
segments to show ABD CDB.
Proof:
Statements (Reasons)
1. D
E
, AE
(Given)
FB
FC
2. A
E
⊥
DB
, CF
⊥
DB
(Given)
3. AED is a right angle. CFB is a right angle.
(⊥ lines form right .)
4. AED CFB (All right angles are .)
5. ADE CBF (SAS)
6. AD
BC
(CPCTC)
7. D
B
DB
(Reflexive Prop. for Segments)
8. CBD ADB (CPCTC)
9. ABD CDB (SAS)
GH || KS
XW || YZ
Given
H
Pages 210–213, Lesson 4-5
4. Given: GH
K
J, GK
H
J
Prove: GJK JGH
Proof:
5. Given: XW
|| Y
Z
, X Z
Prove: WXY YZW
Proof:
11. Given: V S, TV
Q
S
Prove: VR
SR
Proof:
V S
TV QS
1
2
R
Q
V
1 2
Vert. are .
Given
S
T
14. Given: Z is the midpoint of CT
.
CY
|| T
E
Prove: YZ
E
Z
Proof:
Given
TZ CZ
Midpt. Th.
CPCTC
12. Given: E
J || F
K
, JG
|| K
H
, E
F
GH
Prove: EJG FKH
Proof:
J
Additional Answers for Chapter 4
Substitution
AAS
E
F
G
H
FG FG
Reflexive Prop.
EF FG EG
FG GH FH
Seg. Addition Post.
EG FH
Def. of seg.
EJ || FK, JG || KH
EJG FKH
JEG KFH
JGE HFK
Corr. are .
SAS
Given
13. Given: MN
PQ
, M Q
2 3
Prove: MLP QLN
Proof:
MN PQ
L
1
M
MN PQ
Def. of seg.
Addition Prop.
MP NQ
Substitution
NP NP
Reflexive Prop.
MN NP MP
NP PQ NQ
Seg. Addition Post.
MP NQ
Def. of seg.
MLP QLN
ASA
M Q
2 3
Given
233E
N
2
3
4
P
Q
19. Given: MYT NYT
MTY NTY
Prove: RYM RYN
M
R
T
Y
Proof:
N
Statements (Reasons)
1. MYT NYT, MTY NTY (Given)
2. YT
Y
T
, RY
(Reflexive Property)
RY
3. MYT NYT (ASA)
4. MY
(CPCTC)
NY
5. RYM and MYT are a linear pair; RYN and
NYT are a linear pair (Def. of linear pair)
6. RYM and MYT are supplementary and RYN
and NYT are supplementary. (Suppl.Th.)
7. RYM RYN ( suppl. to are .)
8. RYM RYN (SAS)
20. Given: BMI KMT
I
B
P
IP
PT
Prove: IPK TPB
M
T
Given
MN NP NP PQ
Given
EZT YZC
K
EF GH
Def. of seg.
EG FH
Y
CPCTC
Given
Addition Prop.
T
YZ EZ
EF GH
EF FG FG GH
Z
ETC YCT
TEY CYE
Alt. int. are .
AAS
VR SR
C
CY ⊥ TE
Z is the
midpoint of CT.
TRV QRS
E
Chapter 4 Additional Answers
Proof:
K
Statements (Reasons)
1. BMI KMT (Given)
2. B K (CPCTC)
3. IP
PT
(Given)
4. P P (Reflexive Prop.)
5. IPK TPB (AAS)
21. CD
, because the segments have the same
GH
measure. CFD HFG because vertical angles are
congruent. Since F is the midpoint of DG
, DF
FG
. It
cannot be determined whether CFD HFG. The
information given does not lead to a unique triangle.
22. Since F is the midpoint of DG
, DF
FG
. F is also the
midpoint of CH
, so C
F
FH
. Since D
G
C
H
,
DF
CF
and F
G
FH
. CFD HFG because
vertical angles are congruent. CFD HFG by SAS.
23. Since N is the midpoint of J
L, J
N
N
L. JNK LNK because perpendicular lines form right angles
and right angles are congruent. By the Reflexive
Property, KN
. JKN LKN by SAS.
KN
24. It is given that JM
L
M
and NJM NLM. By the
Reflexive Property, NM
N
M
. It cannot be determined
whether JNM LNM. The information given does
not lead to a unique triangle.
33. Given: BA
DE
, DA
BE
B
D
C
Prove: BEA DAE
Proof:
A
E
DA BE
11. Given: JK
⊥
KM
, JM
K
L
,
ML
|| JK
Prove: ML
JK
Proof:
Statements (Reasons)
1.
2.
3.
4.
5.
6.
7.
M
J
L
K
JK
⊥
KM
, JM
K
L
, ML
|| JK
(Given)
JKM is a rt. (⊥ lines form rt. .)
KM
⊥
ML
(Perpendicular Transversal Th.)
LMK is a rt. (⊥ lines form rt. .)
MK
MK
(Reflexive Property)
JMK LMK (HL)
ML
JK
(CPCTC)
Given
BA DE
BEA DAE
Given
ASA
AE AE
Reflexive Prop.
W
Z
Y
X
XZ ⊥ WY
Given
WZX and
YZX are rt .
Def. of ⊥ lines.
WZX YZX
All rt. are .
XZ XZ
XZ bisects WY.
Given
WZ ZY
Def. of seg. bisector
WZX YZX
SAS
Reflexive Prop.
Pages 214–215, Geometry Activity
10. Given: ML
⊥
MK
, JK
⊥
KM
,
J L
Prove: JM
KL
Proof:
Statements (Reasons)
1.
2.
3.
4.
5.
6.
M
J
L
1
K
ML
⊥
MK
, JK
⊥
KM
, J L (Given)
LMK and JKM are rt. (⊥ lines form 4 rt. .)
LMK and JKM are rt. s (Def. of rt. )
MK
(Reflexive Property)
MK
LMK JKM (LA)
JM
KL
(CPCTC)
Proof:
K
J
Statements (Reasons)
1. XKF is equilateral. (Given)
2. 1 2 (Equilateral s are equiangular.)
3. KX
FX
(Definition of equilateral )
4. XJ bisects X (Given)
5. KXJ FXJ (Def. of bisector)
6. KXJ FXJ (ASA)
7. KJ JF
(CPCTC)
8. J is the midpoint of K
F
. (Def. of midpoint)
2
F
Chapter 4 Additional Answers 233F
Additional Answers for Chapter 4
34. Given: XZ
⊥
WY
,
X
Z
bisects W
Y
.
Prove: WZX YZX
Proof:
Pages 219–221, Lesson 4-6
7. Given: CTE is isosceles
T
with vertex C.
60
mT 60
C
Prove: CTE is equilateral.
E
Proof:
Statements (Reasons)
1. CTE is isosceles with vertex C. (Given)
2. C
T
(Def. of isosceles )
CE
3. E T (Isosceles Th.)
4. mE mT (Def. of )
5. mT 60 (Given)
6. mE 60 (Substitution)
7. mC mE mT 180 (Angle Sum Theorem)
8. mC 60 60 180 (Substitution)
9. mC 60 (Subtraction)
10. CTE is equiangular. (Def. of equiangular )
11. CTE is equilateral. (Equiangular s are
equilateral.)
29. Given: XKF is equilateral.
X
XJ bisects X.
Prove: J is the midpoint of KF
.
Additional Answers for Chapter 4
30. Given: MLP is isosceles.
N is the midpoint of MP
.
Prove: LN
⊥
MP
L
Proof:
M
N
P
Statements (Reasons)
1. MLP is isosceles. (Given)
2. M
L
L
P
(Definition of isosceles )
3. M P (Isosceles Theorem)
4. N is the midpoint of MP
. (Given)
5. MN
NP
(Midpoint Theorem)
6. MNL PNL (SAS)
7. LNM LNP (CPCTC)
8. mLNM mLNP ( have equal measures.)
9. LNM and LNP are a linear pair (Def. of linear pair)
10. mLNM mLNP 180 (Sum of measures of
linear pair of 180)
11. 2mLNM 180 (Substitution)
12. mLNM 90 (Division)
13. LNM is a right angle. (Definition of right )
14. LN
⊥
MP
(Definition of ⊥)
31. Case I:
Given: ABC is an equilateral triangle.
B
Prove: ABC is an equiangular
triangle.
A
C
Proof:
Statements (Reasons)
1. ABC is an equilateral triangle. (Given)
2. AB
AC
B
C
(Def. of equilateral )
3. A B C (Isosceles Th.)
4. ABC is an equiangular triangle. (Def. of
equiangular )
Case II:
Given: ABC is an equiangular
B
triangle.
Prove: ABC is an equilateral triangle.
A
C
Proof:
Statements (Reasons)
1. ABC is an equiangular triangle. (Given)
2. A B C (Def. of equiangular )
3. A
B
AC
B
C
(If 2 of a are , then the sides
opp. those are .)
4. ABC is an equilateral triangle. (Def. of
equiangular )
233G
Chapter 4 Additional Answers
32. Given: MNO is an equilateral triangle.
Prove: mM mN mO 60
O
Proof:
M
N
Statements (Reasons)
1. MNO is an equilateral triangle. (Given)
2. MN
MO
NO
(Def. of equilateral )
3. M N O (Isosceles Th.)
4. mM mN mO (Def. of )
5. mM mN mO 180 ( Sum Theorem)
6. 3mM 180 (Substitution)
7. mM 60 (Division prop.)
8. mM mN mO 60 (Substitution)
33. Given: ABC
B
A C
Prove: AB
CB
A
C
D
Proof:
Statements (Reasons)
bisect ABC. (Protractor Post.)
1. Let BD
2. ABD CBD (Def. bisector)
3. A C (Given)
(Reflexive Prop.)
4. BD
BD
5. ABD CBD (AAS)
6. AB
(CPCTC)
CB
38. There are two sets of 12 isosceles triangles.
One black set forms a circle with their bases
on the outside of the circle. Another black
set encircles a circle in the middle.
39. The triangles in each set appear to be acute.
44. Given: N D, G I, AN
S
D
Prove: ANG SDI
G
N
S
A
D
I
Proof: We are given N D and G I. By the
Third Angle Theorem, A S. We are also given
AN
SD
. ANG SDI by ASA.
45. Given: VR
⊥
RS
, UT
⊥S
U
,
R
V
RS
US
S
Prove: VRS TUS
Proof: We are given that
T
U
VR
⊥
RS
, UT
⊥
SU
, and R
S
U
S
.
Perpendicular lines form four right angles, so R and
U are right angles. R U because all right angles
are congruent. RSV UST since vertical angles
are congruent. Therefore, VRS TUS by ASA.
48.
50.
49.
p
q p q p or q
T
T
T
a
b
a and b
T
T
F
F
F
T
F
F
T
F
F
T
T
F
T
F
F
T
T
F
T
F
F
F
F
F
T
T
T
k
m m k and m
T
T
F
T
F
F
F
51.
y y or z
y
z
F
T
T
F
T
T
T
T
F
F
F
T
F
F
F
T
T
T
F
T
F
F
F
T
T
9. Given: ABC
Prove: ABC is isosceles.
Proof: Use the Distance
Formula to find AB and BC.
B(2, 8)
AB (2 0
)2 (8
0)2
4 64
or 68
BC (4 2
)2 (0
8)2
A(0, 0)
C (4, 0)
x
4 64
or 68
Since AB BC, AB
B
C
. Since the legs are
congruent, ABC is isosceles.
10. Sample answer:
11. Sample answer:
y
y
T (b–2, c)
12. Sample answer:
N (2a, 0) x
M (0, 0)
R(b, 0) x
Q (0, 0)
P (a, b)
13. Sample answer:
y
y
X(1–4b, c)
J(0, c)
E
A
J
Proof:
Statements (Reasons)
1. A H, AEJ HJE (Given)
2. EJ E
J (Reflexive Prop.)
3. AEJ HJE (AAS)
4. AJ E
H
(CPCTC)
Pages 224–226, Lesson 4-7
y
8. Given: ABC is a right
B(0, 2b)
triangle with
hypotenuse B
C
.
M
M is the midpoint
of BC
.
A(0, 0)
C(2c, 0) x
Prove: M is equidistant from
the vertices.
Proof: The coordinates of M, the midpoint of BC
, will
22c
2b
2
be , (c, b). The distance from M to each of
the vertices can be found using the Distance Formula.
2 b2
MB (c 0
)2 (b
2b
)2 c
MC (c 2
c)2 (b 0
)2
(c 0
)2 (b
0)2
2 b2
c
2 b2
MA c
Thus, MB MC MA, and M is equidistant from the
vertices.
14. Sample answer:
y
15. Sample answer:
ab
2 ,c
Y(
Z (1–2b, 0) x
W(0, 0)
M (c, 0) x
L(0, 0)
H
y
)
X(0, b)
P(0, 0)
W(a b, 0) x
25. Given: isosceles ABC
with A
C
BC
R and S are midpoints
of legs A
C
and BC
.
Prove: AS
BR
Proof: The coordinates of R
2a 2 0
2b 0
2
Z(2b, 0) x
Y (0, 0)
y
C(2a, 2b)
S
R
B(4a, 0) x
A(0, 0)
are , or (a, b).
2a 2 4a
2b 0
2
The coordinates of S are , or (3a, b).
BR (4a a)2 (0 b)2
(3a)2 (b)2 or 9a 2 b2
AS (3a 0)2 (b 0)2
(3a)2 (b)2 or 9a 2 b2
Since BR AS, AS
B
R
.
Chapter 4 Additional Answers 233H
Additional Answers for Chapter 4
Page 221, Practice Quiz 2
1. JM 5
, ML 26
, JL 5, BD 5
, DG 26
,
and BG 5. Each pair of corresponding sides have
the same measure so they are congruent.
JML BDG by SSS.
2. Given: A H, AEJ HJE
Prove: AJ E
H
y
y
26. Given: isosceles triangle ABC
AB
BC
A
C
R, S, and T are
R
midpoints of their
respective sides.
A(0, 0)
Prove: RST is isosceles.
Proof:
a0 b0
a b
Midpoint R is , or , .
C(a, b)
S
Additional Answers for Chapter 4
2
B(a, 0) x
ST
and AB
have the same slope so S
T || A
B.
y
28. Given: ABC
C(b, c)
S is the midpoint of AC
.
T is the midpoint of B
C
.
T
S
1
Prove: ST AB
2
c
c 2
ab
b 2
2
2
2
2
a 2
02
2
a 2
a
or 2
2
ST AB (a 0
)2 (0
0)2
a2 02 or a
1
2
ST AB
233I
Chapter 4 Additional Answers
x
B(800, 800)
A(1600, 0)
P(0, 0)
R(800, 800)
x
Proof:
Since PR and BR have the same measure, PR
.
BR
00
800 0
800 0
The slope of BR , which is undefined.
800 800
0
a
Proof:
y
The slope of PR or 0.
Slope of A
B
or 0.
A(0, 0)
O
BF (6 3
)2 (1
4)2 9 9 or 32
Since AB BF, AB
BF
.
ABD FBD by SSS.
30. Given: BPR
PR 800, BR 800
Prove: BPR is an isosceles right triangle.
Slope of S
T
or 0.
ab
b
00
a0
F(6, 1)
AB (3 0
)2 (4
1)2 9 9 or 32
2
2 2
2 ab 0c
ab c
Midpoint T is , or , .
2
2
2
2
2
2
D(3, 1)
Proof: BD
B
D
by the
Reflexive Property.
DF (6 3
)2 (4
1)2 9 0 or 3
Since AD DF, AD
DF
.
Proof:
b0 c0
b c
Midpoint S is , or , .
0
a
B(3, 4)
AD (3 0
)2 (1
1)2 9 0 or 3
RT ST and RT
S
T
and RST is isosceles.
y
27. Given: ABC
C(b, c)
S is the midpoint of AC
.
T is the midpoint of B
C
.
T
S
Prove: ST
|| A
B
c c
2
2
y
A(0, 1)
T B(2a, 0) x
2
2 2
2 a 2a b 0
3a b
Midpoint S is , or , .
2
2
2 2
2a 0 0 0
Midpoint T is , or (a, 0).
2
2
2
2
2
2
a
b
RT 2 a 2 0 a2 2b
2
2
a2 2b
2
2
ST 32a a b2 0
2
2
a2 b2
A(0, 0)
29. Given: ABD, FBD
AF 6, BD 3
Prove: ABD FBD
B(a, 0) x
PR
⊥
BR
, so PRB is a right angle. BPR is an
isosceles right triangle.
31. Given: BPR, BAR
PR 800, BR 800, RA 800
Prove: P
B
BA
y
B(800, 800)
A(1600, 0)
P(0, 0)
R(800, 800)
x
Proof:
PB (800 0)2 (800 0)2 or 1,280
,000
BA (800 1600
)2 (8
00 0)2 or 1,280
,000
PB BA, so P
B
BA
.
32. Given: JCT
Prove: JCT is a right triangle.
y
43. Given: A
D
CE
,
A
D
|| C
E
Prove: ABD EBC
T(300, 500)
J
A
Proof:
Statements (Reasons)
(–500, 300)
C(0, 0)
x
Proof:
300 0
3
The slope of JC
or .
500 0
5
500 0
5
The slope of T
C or .
300 0
3
The slope of T
C
is the opposite reciprocal of the slope
of JC
. JC
⊥
TC
, so TCJ is a right angle. So JCT is
a right triangle.
41. Given: 3 4
Q
Prove: QR
QS
3
1
2
4
1
3
L
Proof:
J
Statements (Reasons)
1. isosceles triangle JKN with vertex N (Given)
2. NJ NK
(Def. of isosceles triangle)
3. 2 1 (Isosceles Triangle Theorem)
4. JK
|| L
M
(Given)
5. 1 3, 4 2 (Corr. Post.)
6. 2 3, 4 1 (Congruence of is transitive.
Statements 3 and 5)
7. 4 3 (Congruence of is transitive.
Statements 3 and 6)
8. LN
M
N
(If 2 of a are , then the sides opp.
those are .)
9. NML is an isosceles triangle. (Def. of isosceles )
D
E
1. A
D
|| C
E
(Given)
2. A E, D C (Alt. Int. Theorem)
3. AD
CE
(Given)
4. ABD EBC (ASA)
44. Given: W
X
XY
, V Z
V
Prove: WV
Y
Z
W
X
Z
Proof:
Statements (Reasons)
1.
2.
3.
4.
Y
WX
X
Y
, V Z (Given)
WXV YXZ (Vertical )
WXV YXZ (AAS)
WV
Y
Z
(CPCTC)
Page 231, Chapter 4 Practice Test
17. Given: ABE, BCE
AB 22, AC 44,
AE 36, CD 36
A and C are right
angles.
Prove: ABE CBD
A
B
E
C
D
Proof: We are given that AB 22 and AC 44.
By the Segment Addition Postulate,
AB BC AC
22 BC 44
BC 22
Substitution
Subtract 22 from each side.
Since AB BC, then by the definition of congruent
segments, AB
BC
.
We are given that AE 36 and CD 36. Then also
by the definition of congruent segments, AE
CD
.
We are additionally given that both A and C are
right angles. Since all right angles are congruent,
A C.
Since AB
B
C
, A C, and AE
C
D
, then by
SAS, ABE CBD.
Chapter 4 Additional Answers 233J
Additional Answers for Chapter 4
Proof:
S
R
Statements (Reasons)
1. 3 4 (Given)
2. 2 and 4 form a linear pair. 1 and 3 form a
linear pair. (Def. of linear pair)
3. 2 and 4 are supplementary. 1 and 3 are
supplementary. (If 2 form a linear pair, then they
are suppl.)
4. 2 1 (Angles that are suppl. to are .)
5. Q
R
QS
(If 2 of a are , then the sides
opposite those are .)
42. Given: isosceles JKN
K
M
2
with vertex N, JK
|| L
M
4
Prove: NML is isosceles
N
C
B