5.4 Equilateral and Isosceles Triangles - Big Ideas Learning

5.4

TEXAS ESSENTIAL

KNOWLEDGE AND SKILLS

Equilateral and Isosceles Triangles

Essential Question

What conjectures can you make about the side

lengths and angle measures of an isosceles triangle?

Writing a Conjecture about Isosceles Triangles

G.5.C

G.6.D

Work with a partner. Use dynamic geometry software.

a. Construct a circle with a radius of 3 units centered at the origin.

b. Construct ¡÷ABC so that B and C are on the circle and A is at the origin.

Sample

Points

A(0, 0)

B(2.64, 1.42)

C(?1.42, 2.64)

Segments

AB = 3

AC = 3

BC = 4.24

Angles

m¡ÏA = 90¡ã

m¡ÏB = 45¡ã

m¡ÏC = 45¡ã

3

C

2

B

1

0

?4

?3

?2

?1

A

0

1

2

3

4

?1

MAKING

MATHEMATICAL

ARGUMENTS

To be proficient in math,

you need to make

conjectures and build a

logical progression of

statements to explore the

truth of your conjectures.

?2

?3

c. Recall that a triangle is isosceles if it has at least two congruent sides. Explain why

¡÷ABC is an isosceles triangle.

d. What do you observe about the angles of ¡÷ABC?

e. Repeat parts (a)¨C(d) with several other isosceles triangles using circles of different

radii. Keep track of your observations by copying and completing the table below.

Then write a conjecture about the angle measures of an isosceles triangle.

A

Sample

1.

(0, 0)

2.

(0, 0)

3.

(0, 0)

4.

(0, 0)

5.

(0, 0)

B

C

(2.64, 1.42) (?1.42, 2.64)

AB

AC

BC

3

3

4.24

m¡ÏA m¡ÏB m¡ÏC

90¡ã

45¡ã

45¡ã

f. Write the converse of the conjecture you wrote in part (e). Is the converse true?

Communicate Your Answer

2. What conjectures can you make about the side lengths and angle measures of an

isosceles triangle?

3. How would you prove your conclusion in Exploration 1(e)? in Exploration 1(f)?

Section 5.4

Equilateral and Isosceles Triangles

255

5.4 Lesson

What You Will Learn

Use the Base Angles Theorem.

Core Vocabul

Vocabulary

larry

legs, p. 256

vertex angle, p. 256

base, p. 256

base angles, p. 256

Use isosceles and equilateral triangles.

Using the Base Angles Theorem

vertex angle

A triangle is isosceles when it has at least two congruent

sides. When an isosceles triangle has exactly two congruent

sides, these two sides are the legs. The angle formed by the

legs is the vertex angle. The third side is the base of the

isosceles triangle. The two angles adjacent to the base are

called base angles.

leg

leg

base

angles

base

Theorems

Theorem 5.6

Base Angles Theorem

A

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

opposite them are congruent.

¡ª ? AC

¡ª, then ¡ÏB ? ¡ÏC.

If AB

Proof p. 256; Ex. 33, p. 272

B

C

Theorem 5.7 Converse of the Base Angles Theorem

A

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

opposite them are congruent.

¡ª ? AC

¡ª.

If ¡ÏB ? ¡ÏC, then AB

Proof Ex. 27, p. 279

B

C

Base Angles Theorem

B

¡ª ? AC

¡ª

Given AB

A

Prove ¡ÏB ? ¡ÏC

¡ª so that it bisects ¡ÏCAB.

Plan a. Draw AD

for

Proof b. Use the SAS Congruence Theorem to show that ¡÷ADB ? ¡÷ADC.

D

C

c. Use properties of congruent triangles to show that ¡ÏB ? ¡ÏC.

Plan STATEMENTS

in

¡ª

Action a. 1. Draw AD , the angle

REASONS

1. Construction of angle bisector

bisector of ¡ÏCAB.

2. ¡ÏCAD ? ¡ÏBAD

¡ª ¡ª

3. AB ? AC

¡ª ? DA

¡ª

4. DA

2. Definition of angle bisector

3. Given

4. Reflexive Property of Congruence (Thm. 2.1)

b. 5. ¡÷ADB ? ¡÷ADC

5. SAS Congruence Theorem (Thm. 5.5)

c. 6. ¡ÏB ? ¡ÏC

6. Corresponding parts of congruent triangles

are congruent.

256

Chapter 5

Congruent Triangles

Using the Base Angles Theorem

¡ª ? DF

¡ª. Name two congruent angles.

In ¡÷DEF, DE

F

E

D

SOLUTION

¡ª ? DF

¡ª, so by the Base Angles Theorem, ¡ÏE ? ¡ÏF.

DE

Monitoring Progress

Help in English and Spanish at

Copy and complete the statement.

1.

¡ª ? HK

¡ª, then ¡Ï

If HG

H

?¡Ï

2. If ¡ÏKHJ ? ¡ÏKJH, then

.

?

.

G

K

J

Recall that an equilateral triangle has three congruent sides.

Corollaries

Corollary 5.2 Corollary to the Base Angles Theorem

READING

If a triangle is equilateral, then it is equiangular.

Proof Ex. 37, p. 262; Ex. 10, p. 357

The corollaries state that a

triangle is equilateral if and

only if it is equiangular.

A

Corollary 5.3 Corollary to the Converse

of the Base Angles Theorem

If a triangle is equiangular, then it is equilateral.

Proof Ex. 39, p. 262

B

C

Finding Measures in a Triangle

Find the measures of ¡ÏP, ¡ÏQ, and ¡ÏR.

P

SOLUTION

R

The diagram shows that ¡÷PQR is equilateral. So, by

the Corollary to the Base Angles Theorem, ¡÷PQR is

equiangular. So, m¡ÏP = m¡ÏQ = m¡ÏR.

3(m¡ÏP) = 180¡ã

m¡ÏP = 60¡ã

Triangle Sum Theorem (Theorem 5.1)

Q

Divide each side by 3.

The measures of ¡ÏP, ¡ÏQ, and ¡ÏR are all 60¡ã.

S

T

5

U

Monitoring Progress

3.

Help in English and Spanish at

¡ª for the triangle at the left.

Find the length of ST

Section 5.4

Equilateral and Isosceles Triangles

257

Using Isosceles and Equilateral Triangles

Constructing an Equilateral Triangle

¡ª. Use a

Construct an equilateral triangle that has side lengths congruent to AB

compass and straightedge.

A

B

SOLUTION

Step 1

Step 2

Step 3

Step 4

C

A

B

A

¡ª.

Copy a segment Copy AB

B

C

A

Draw an arc Draw an

arc with center A and

radius AB.

B

A

Draw an arc Draw an arc

with center B and radius

AB. Label the intersection

of the arcs from Steps 2

and 3 as C.

B

Draw a triangle Draw

¡ª and

¡÷ABC. Because AB

¡ª

AC are radii of the same

¡ª ? AC

¡ª. Because

circle, AB

¡ª

¡ª

AB and BC are radii of the

¡ª ? BC

¡ª. By

same circle, AB

the Transitive Property of

Congruence (Theorem 2.1),

¡ª ? BC

¡ª. So, ¡÷ABC is

AC

equilateral.

Using Isosceles and Equilateral Triangles

Find the values of x and y in the diagram.

K

4

y

N

COMMON ERROR

You cannot use N to refer

to ¡ÏLNM because three

angles have N as

their vertex.

Chapter 5

x+1

M

SOLUTION

Step 1 Find the value of y. Because ¡÷KLN is equiangular, it is also equilateral and

¡ª ? KL

¡ª. So, y = 4.

KN

¡ª ? LM

¡ª, and ¡÷LMN is

Step 2 Find the value of x. Because ¡ÏLNM ? ¡ÏLMN, LN

isosceles. You also know that LN = 4 because ¡÷KLN is equilateral.

LN = LM

258

L

Congruent Triangles

Definition of congruent segments

4=x+1

Substitute 4 for LN and x + 1 for LM.

3=x

Subtract 1 from each side.

Solving a Multi-Step Problem

¡ª ? QR

¡ª and ¡ÏQPS ? ¡ÏPQR.

In the lifeguard tower, PS

P

2

1

Q

T

4

3

S

R

a. Explain how to prove that ¡÷QPS ? ¡÷PQR.

b. Explain why ¡÷PQT is isosceles.

COMMON ERROR

When you redraw the

triangles so that they do

not overlap, be careful to

copy all given information

and labels correctly.

SOLUTION

a. Draw and label ¡÷QPS and ¡÷PQR so that they do not overlap. You can see that

¡ª ? QP

¡ª, PS

¡ª ? QR

¡ª, and ¡ÏQPS ? ¡ÏPQR. So, by the SAS Congruence Theorem

PQ

(Theorem 5.5), ¡÷QPS ? ¡÷PQR.

P

Q

3

Q

P

2

1

T

T

S

4

R

b. From part (a), you know that ¡Ï1 ? ¡Ï2 because corresponding parts of congruent

¡ª ? QT

¡ª,

triangles are congruent. By the Converse of the Base Angles Theorem, PT

and ¡÷PQT is isosceles.

Monitoring Progress

Help in English and Spanish at

4. Find the values of x and y in the diagram.

y¡ã x¡ã

5. In Example 4, show that ¡÷PTS ? ¡÷QTR.

Section 5.4

Equilateral and Isosceles Triangles

259

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