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LESSON

7.2

Name

Isosceles and

Equilateral Triangles

7.2

Class

Date

Isosceles and Equilateral

Triangles

Essential Question: What are the special relationships among angles and sides in isosceles

and equilateral triangles?

Common Core Math Standards

Resource

Locker

The student is expected to:

COMMON

CORE

G-CO.C.10

Explore

Prove theorems about triangles.

An isosceles triangle is a triangle with at least two congruent sides.

Mathematical Practices

COMMON

CORE

Investigating Isosceles Triangles

MP.3 Logic

The side opposite the vertex angle is the base.

Explain to a partner what you can deduce about a triangle if it has two

sides with the same length.

The angles that have the base as a side are the base angles.

ENGAGE

A

PREVIEW: LESSON

PERFORMANCE TASK

View the Engage section online. Discuss the photo,

explaining that the instrument is a sextant and that

long ago it was used to measure the elevation of the

sun and stars, allowing one¡¯s position on Earth¡¯s

surface to be calculated. Then preview the Lesson

Performance Task.

Base

Base angles

In this activity, you will construct isosceles triangles and investigate other potential

characteristics/properties of these special triangles.

Do your work in the space provided. Use a straightedge to draw an angle.

Label your angle ¡ÏA, as shown in the figure.

A

Check students¡¯ construtions.

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In an isosceles triangle, the angles opposite the

congruent sides are congruent. In an equilateral

triangle, all the sides and angles are congruent, and

the measure of each angle is 60¡ã.

B

Using a compass, place the point on the vertex and draw an arc that intersects the

sides of the angle. Label the points B and C.

A

C

B

Module 7

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Lesson 2

327

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GE_MNLESE385795_U2M07L2.indd 327

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Turn to these pages to

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HARDCOVER PAGES 283?292

Legs

Vertex angle

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Explore

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Lesson 2

327

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Lesson 7.2

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327

Legs

The angle formed by the legs is the vertex angle.

Language Objective

Essential Question: What are the

special relationships among angles and

sides in isosceles and equilateral

triangles?

Vertex angle

The congruent sides are called the legs of the triangle.

327

02/04/14

1:21 AM

02/04/14 1:20 AM

C

_

Use the straightedge to draw line segment BC.

EXPLORE

A

Investigating Isosceles Triangles

C

B

INTEGRATE TECHNOLOGY

D

Students have the option of completing the isosceles

triangle activity either in the book or online.

Use a protractor to measure each angle. Record the measures in the table under the column

for Triangle 1.

Triangle 1

Triangle 2

Triangle 3

Triangle 4

QUESTIONING STRATEGIES

m¡ÏA

What must be true about the triangles you

construct in order for them to be isosceles

triangles? They must have two congruent sides.

m¡ÏB

m¡ÏC

Possible answer for Triangle 1: m¡ÏA = 70¡ã; m¡ÏB = ¡Ï55¡ã; m¡ÏC = 55¡ã.

E

How could you draw isosceles triangles

without using a compass? Possible answer:

Draw ¡ÏA and plot point B on one side of ¡ÏA. Then

_

use a ruler to measure AB and plot point C on the

other side of ¡ÏA so that AC = AB.

Repeat steps A¨CD at least two more times and record the results in the table. Make sure ¡ÏA

is a different size each time.

Reflect

How do you know the triangles you constructed are isosceles triangles?

¨D ¨D

The compass marks equal lengths on both sides of ¡ÏA; therefore, AB ? AC.

2.

Make a Conjecture Looking at your results, what conjecture can be made about the base angles,

¡ÏB and ¡ÏC?

The base angles are congruent.

Explain 1

EXPLAIN 1

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1.

Proving the Isosceles Triangle Theorem

and Its Converse

In the Explore, you made a conjecture that the base angles of an isosceles triangle are congruent.

This conjecture can be proven so it can be stated as a theorem.

Isosceles Triangle Theorem

Proving the Isosceles Triangle

Theorem and Its Converse

CONNECT VOCABULARY

Ask a volunteer to define isosceles triangle and have

students give real-world examples of them. If

possible, show the class a baseball pennant or other

flag in the shape of an isosceles triangle. Tell students

they will be proving theorems about isosceles

triangles and investigating their properties in this

lesson.

If two sides of a triangle are congruent, then the two angles opposite the sides are

congruent.

This theorem is sometimes called the Base Angles Theorem and can also be stated as ¡°Base angles

of an isosceles triangle are congruent.¡±

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Lesson 2

PROFESSIONAL DEVELOPMENT

GE_MNLESE385795_U2M07L2.indd 328

Learning Progressions

20/03/14 1:32 PM

In this lesson, students add to their prior knowledge of isosceles and equilateral

triangles by investigating the Isosceles Triangle Theorem from both an inductive

and deductive perspective. The opening activity leads students to make a

conjecture about the measures of the base angles of an isosceles triangle. Students

prove their conjecture and its converse later in the lesson. They also prove the

Equilateral Triangle Theorem and its converse, and use the properties of both

types of triangles to find the unknown measure of angles and sides in a triangle.

All students should develop fluency with various types of triangles as they

continue their study of geometry.

Isosceles and Equilateral Triangles

328

Example 1

QUESTIONING STRATEGIES

Prove the Isosceles Triangle Theorem and its converse.

Step 1 Complete the proof of the Isosceles Triangle Theorem.

_ _

Given: AB ? AC

What can you say about an isosceles triangle,

?ABC, with base angles ¡ÏB and ¡ÏC, if you

know that m¡ÏA = 100¡ã? Explain. By the Isosceles

Triangle Theorem, ¡ÏB ? ¡ÏC, and m¡ÏB + m¡ÏC = 80¡ã

by the Triangle Sum Theorem, so m¡ÏB = m¡ÏC = 40¡ã.

Prove: ¡ÏB ? ¡ÏC

A

B

What can you say about the angles of an

isosceles right triangle? The angles of the

triangle measure 90¡ã, 45¡ã, and 45¡ã.

C

Statements

Reasons

_ _

1. BA ? CA

1. Given

2. ¡ÏA ? ¡ÏA

_ _

3. CA ? BA

2. Reflexive Property of Congruence

5. ¡ÏB ? ¡ÏC

5. CPCTC

3. Symmetric Property of Equality

4. ¡÷BAC ? ¡÷CAB

4. SAS Triangle Congruence Theorem

Step 2 Complete the statement of the Converse of the Isosceles Triangle Theorem.

If two

angles

those

angles

of a triangle are congruent, then the two

are congruent .

sides

opposite

Step 3 Complete the proof of the Converse of the Isosceles Triangle Theorem.

Given: ¡ÏB ? ¡ÏC

_ _

Prove: AB ? AC

A

? Houghton Mifflin Harcourt Publishing Company

B

C

Statements

1. ¡ÏABC ? ¡ÏACB

¨D

¨D

Reasons

1. Given

2. BC ? CB

2. Reflexive Property of Congruence

3. ¡ÏACB ? ¡ÏABC

3. Symmetric Property of Equality

4. ¡÷ABC ? ¡÷ACB

_ _

5. AB ? AC

4. ASA Triangle Congruence Theorem

5. CPCTC

Reflect

3.

Discussion In the proofs of the Isosceles Triangle Theorem and its converse, how

might it help to sketch a reflection of the given triangle next to the original triangle,

so that vertex B is on the right?

Possible answer: Sketching a copy of the triangle makes it easier to see the two pairs of

congruent corresponding sides and the two pairs of congruent corresponding angles.

Module 7

329

Lesson 2

COLLABORATIVE LEARNING

GE_MNLESE385795_U2M07L2 329

Small Group Activity

Geometry software allows students to explore the theorems in this lesson. For the

Isosceles Triangle Theorem (or the Equilateral Triangle Theorem), students

should construct an isosceles (or equilateral) triangle and measure the angles. As

students drag the vertices of the triangle to change its size or shape, the individual

base angle measures will change (for isosceles only), but the relationship between

the lengths of the sides and the measures of the angles will remain the same.

329

Lesson 7.2

5/22/14 4:40 PM

Explain 2

Proving the Equilateral Triangle Theorem

and Its Converse

EXPLAIN 2

An equilateral triangle is a triangle with three congruent sides.

Proving the Equilateral Triangle

Theorem and Its Converse

An equiangular triangle is a triangle with three congruent angles.

Equilateral Triangle Theorem

If a triangle is equilateral, then it is equiangular.

Example 2

COLLABORATIVE LEARNING

Prove the Equilateral Triangle Theorem and its converse.

Step 1 Complete the proof of the Equilateral Triangle Theorem.

_ _ _

Given: AB ? AC ? BC

Prove: ¡ÏA ? ¡ÏB ? ¡ÏC

_ _

Given that AB ? AC we know that ¡ÏB ? ¡Ï C by the

The converse of this theorem is proved interactively

using a paragraph proof. Have small groups of

students discuss the proof and highlight the

important statements (steps) and reasons for the

statements. Ask them how they would present the

same proof using the two-column method.

A

B

Isosceles Triangle Theorem .

It is also known that ¡ÏA ? ¡ÏB by the Isosceles Triangle Theorem, since

_ _

AC ? BC

C

.

Therefore, ¡ÏA ? ¡ÏC by substitution .

Finally, ¡ÏA ? ¡ÏB ? ¡ÏC by the

Transitive

QUESTIONING STRATEGIES

Property of Congruence.

The converse of the Equilateral Triangle Theorem is also true.

What is the connection between equilateral

triangles and equiangular triangles? If a

triangle is equilateral, then it is also equiangular. If a

triangle is equiangular, then it is also equilateral.

Converse of the Equilateral Triangle Theorem

If a triangle is equiangular, then it is equilateral.

Step 2 Complete the proof of the Converse of the Equilateral Triangle Theorem.

A

_

_

Because ¡ÏB ? ¡ÏC, AB ? AC by the

B

C

Converse of the Isosceles Triangle Theorem .

_ _

AC ? BC by the Converse of the Isosceles Triangle Theorem because

¡ÏA ? ¡ÏB.

AVOID COMMON ERRORS

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Given: ¡ÏA ? ¡ÏB ? ¡ÏC

_ _ _

Prove: AB ? AC ? BC

_

_

_ _ _

Thus, by the Transitive Property of Congruence, AB ? BC , and therefore, AB ? AC ? BC.

Reflect

Some students may confuse the theorems in this

lesson because they are so similar. Have students

draw and label diagrams to illustrate the theorems

and then add visual cues, if needed, to help them

remember how the theorems are applied.

To prove the Equilateral Triangle Theorem, you applied the theorems of isosceles

triangles. What can be concluded about the relationship between equilateral triangles

and isosceles triangles?

Possible answer: Equilateral/equiangular triangles are a special type of isosceles triangles.

4.

Module 7

Lesson 2

330

DIFFERENTIATE INSTRUCTION

GE_MNLESE385795_U2M07L2 330

5/22/14 4:44 PM

Visual Cues

Visually represent the Equilateral Triangle Theorem and its converse:

Equilateral Triangle Theorem

If

then

Converse

If

then

Isosceles and Equilateral Triangles

330

Using Properties of Isosceles and Equilateral

Triangles

Explain 3

EXPLAIN 3

You can use the properties of isosceles and equilateral triangles to solve problems involving these theorems.

Using Properties of Isosceles and

Equilateral Triangles

Example 3

?

INTEGRATE MATHEMATICAL

PRACTICES

Focus on Math Connections

MP.1 Encourage students to discuss how the

Find the indicated measure.

Katie is stitching the center inlay onto a banner that

she created to represent her new tutorial service. It is

an equilateral triangle with the following dimensions in

centimeters. What is the length of each side of the triangle?

A

6x - 5

Triangle Sum Theorem and the theorems in this

lesson help them solve for the unknown angles and

sides of an isosceles or equilateral triangle. Have them

share their ideas about the best method to use to

solve for the unknown quantities in each problem.

B

C

4x + 7

To find the length of each side of the triangle, first find the value of x.

_ _

AC ? BC

Converse of the Equilateral Triangle Theorem

AC = BC

Definition of congruence

6x ? 5 = 4x + 7

Substitution Property of Equality

x=6

Substitute 6 for x into either 6x ? 5 or 4x + 7.

? Houghton Mifflin Harcourt Publishing Company ? Image Credits: ?Nelvin C.

Cepeda/ZUMA Press/Corbis

QUESTIONING STRATEGIES

If the triangle is equiangular, how do you find

the measure of one of its angles? Divide the

sum of the interior angles by the number of interior

angles: 180¡ã ¡Â 3 = 60¡ã.

Solve for x.

6(6) ? 5 = 36 ? 5 = 31

4(6) + 7 = 24 + 7 = 31

or

So, the length of each side of the triangle is 31 cm.

?

m¡ÏT

T

3x¡ã

x¡ã

R

S

To find the measure of the vertex angle of the triangle, first find the value of x .

m¡ÏR = m¡ÏS = x¡ã

m¡ÏR + m¡ÏS + m¡ÏT = 180¡ã

Substitution Property of Equality

5x = 180

Addition Property of Equality

x = 36

( )=

So, m¡ÏT = 3x¡ã = 3 36

Theorem

Triangle Sum Theorem

x + x + 3x = 180

¡ã

Isosceles Triangle

Division

Property of Equality

¡ã

108 .

Module 7

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Lesson 2

LANGUAGE SUPPORT

GE_MNLESE385795_U2M07L2.indd 331

Connect Vocabulary

Help students understand the meanings of isosceles, equilateral, and equiangular

by having them make a poster showing each type of triangle along with its

definition. An isosceles triangle has two congruent sides, an equilateral triangle

has three congruent sides, and an equiangular triangle has three congruent

angles. Relate the prefix equi- to equal to help students make connections

between the terms.

331

Lesson 7.2

20/03/14 1:32 PM

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