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7.2 L E S S O N

Isosceles and Equilateral Triangles

Common Core Math Standards

The student is expected to:

COMMON CORE

G-CO.C.10

Prove theorems about triangles.

Mathematical Practices

COMMON CORE

MP.3 Logic

Language Objective

Explain to a partner what you can deduce about a triangle if it has two sides with the same length.

ENGAGE

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

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?.

Name

Class

Date

7.2 Isosceles and Equilateral Triangles

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

Resource Locker

Explore Investigating Isosceles Triangles

An isosceles triangle is a triangle with at least two congruent sides. The congruent sides are called the legs of the triangle.

Vertex angle

Legs

The angle formed by the legs is the vertex angle. The side opposite the vertex angle is the base. The angles that have the base as a side are the base angles.

Base Base angles

In this activity, you will construct isosceles triangles and investigate other potential characteristics/properties of these special triangles.

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

Label your angle A, as shown in the figure.

A

Check students' construtions.

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

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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.

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B

C

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327

Date

Class

7.2 Isosceles and Equilateral Name Triangles Essential Question: Wanhdateaqrueiltahteersapletcriiaanl grelelast?ionships among angles and sides in isosceles

ReLsoocukrecre

Investigating Isosceles Triangles COACMTOTMnRThOETheiNIhceshneEhcoeaotassGxnhanrci-dganieCpgslceOlgreeta.lulDoeCsLeocfer.postao1tniiprrrsb0tvtmityhoeieaiPstocaslenyiritsuddo,ytg/ehreyvoplbsaeeoturwyhvouatrioeeshprtwaehreetavnkheroicelttgerarlirlieelntleicbelemagseoaxntsdonshsgaiefsaeAtslnthebtart,sgheoshuwplaeueecalassietttecivsghsstieiehsrssdtirapoopahoettesnwreefclcaogxeteinrbvlaahleeaaesileisndststnt.hetgrettr.eiwtrdliaehia.bona.enUnagcgfsglsoelieleegnse.au.sganrraAseugnt.lerdenasiit.ngshvidetesetdsi.ggaetetoodCthrhearewcpakontseantVnutigedallreet.enxtsa'ncgolBenasstBerauastenigoLlenegsss.

Usisdinesgoafctohme apnagssle, .pLlaacbeelththeeppoionint tosnBthane dveCrt.Aex and draw an arc that intersects the

C B

Lesson 2

HARDCOVER PAGES 283292

Turn to these pages to find this lesson in the hardcover student edition.

02/04/14 1:20 AM

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Lesson 2 02/04/14 1:21 AM

327 Lesson 7.2

C _ Use the straightedge to draw line segment BC. A

B

C

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

for Triangle 1.

mA mB mC

Triangle 1 Triangle 2 Triangle 3 Triangle 4

Possible answer for Triangle 1: mA = 70?; mB = 55?; mC = 55?.

E Repeat steps A?D at least two more times and record the results in the table. Make sure A

is a different size each time.

Reflect

1.

How The

do you know the triangles you constructed are isosceles triangles? compass marks equal lengths on both sides of A; therefore,

AB

AC.

2. Make a Conjecture Looking at your results, what conjecture can be made about the base angles, B and C? The base angles are congruent.

Explain 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

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

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Learning Progressions

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.

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EXPLORE

Investigating Isosceles Triangles

INTEGRATE TECHNOLOGY

Students have the option of completing the isosceles triangle activity either in the book or online.

QUESTIONING STRATEGIES

What must be true about the triangles you construct in order for them to be isosceles triangles? They must have two congruent sides. How could you draw isosceles triangles without using a compass? Possible answer: Draw A and plot point_B on one side of A. Then use a ruler to measure AB and plot point C on the other side of A so that AC = AB.

EXPLAIN 1

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.

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Isosceles and Equilateral Triangles 328

QUESTIONING STRATEGIES

What can you say about an isosceles triangle, ABC, with base angles B and C, if you know that mA = 100?? Explain. By the Isosceles Triangle Theorem, B C, and mB + mC = 80? by the Triangle Sum Theorem, so mB = mC = 40?.

What can you say about the angles of an isosceles right triangle? The angles of the triangle measure 90?, 45?, and 45?.

Example 1 Prove the Isosceles Triangle Theorem and its converse.

Step 1 Complete the proof of the Isosceles Triangle Theorem. _ _

Given: AB AC

Prove: B C

A

B

C

Statements _ _ 1. BA CA 2. A A _ _ 3. CA BA 4. BAC CAB 5. B C

Reasons 1. Given 2. Reflexive Property of Congruence 3. Symmetric Property of Equality 4. SAS Triangle Congruence Theorem 5. CPCTC

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

If two angles of a triangle are congruent, then the two those angles are congruent.

sides

opposite

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

Given: B C

_ _

Prove: AB AC

A

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B

Statements 1. ABC ACB

2. BC CB

3. ACB ABC 4. ABC ACB

_ _ 5. AB AC

C

Reasons 1. Given 2. Reflexive Property of Congruence 3. Symmetric Property of Equality 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.

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Lesson 2

COLLABORATIVE LEARNING

Small Group Activity GE_MNLESE385795_U2M07L2 329

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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

Explain 2 Proving the Equilateral Triangle Theorem and Its Converse

An equilateral triangle is a triangle with three congruent sides. An equiangular triangle is a triangle with three congruent angles.

Equilateral Triangle Theorem

If a triangle is equilateral, then it is equiangular.

Example 2 Prove the Equilateral Triangle Theorem and its converse.

Step 1 Complete the proof of the Equilateral Triangle Theorem. _ _ _

A

Given: AB AC BC

Prove: A B C

_ _ Given that AB AC we know that B

C

by the

Isosceles Triangle Theorem .

B

C

_ _

It is also known that A B by the Isosceles Triangle Theorem, since AC BC .

Therefore, A C by substitution .

Finally, A B C by the Transitive Property of Congruence.

The converse of the Equilateral Triangle Theorem is also true.

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

Given: A B C

_ _ _ Prove: AB AC BC

_ _ Because B C, AB AC by the

B

C

Converse of the Isosceles Triangle Theorem . _ _ AC BC by the Converse of the Isosceles Triangle Theorem because

A B. Thus, by the

Transitive

Property

of

Congruence,

_ AB

_ BC

,

and

therefore,

_ AB

_ AC

_ BC.

Reflect

4. 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.

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DIFFERENTIATE INSTRUCTION Visual Cues GE_MNLESE385795_U2M07L2 330

Visually represent the Equilateral Triangle Theorem and its converse:

Equilateral Triangle Theorem

Converse

Lesson 2

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EXPLAIN 2

Proving the Equilateral Triangle Theorem and Its Converse

COLLABORATIVE LEARNING

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.

QUESTIONING STRATEGIES

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.

AVOID COMMON ERRORS

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.

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If

then

If

then

Isosceles and Equilateral Triangles 330

EXPLAIN 3

Using Properties of Isosceles and Equilateral Triangles

INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Encourage students to discuss how the

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.

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? ? 3 = 60?.

Explain 3 Using Properties of Isosceles and Equilateral Triangles

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

Example 3 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

B

4x + 7

C

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

Solve for x.

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

6(6) - 5 = 36 - 5 = 31 or 4(6) + 7 = 24 + 7 = 31

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

mT

T

3x?

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x?

R

S

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

mR = mS = x?

Isosceles Triangle Theorem

mR + mS + mT = 180? x + x + 3x = 180

Triangle Sum Theorem Substitution Property of Equality

5x = 180

Addition Property of Equality

x = 36

( ) So, mT = 3x? = 3 36 ? = 108 ?.

Division Property of Equality

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Lesson 2

LANGUAGE SUPPORT

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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.

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331 Lesson 7.2

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