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Use Isosceles and Equilateral Triangles. Warm Up. Lesson Presentation. Lesson Quiz. equilateral. scalene. ANSWER. ANSWER. isosceles. ANSWER. Warm-Up. Classify each triangle by its sides. 1. 2 cm, 2 cm, 2 cm. 2. 7 ft, 11 ft, 7 ft. 3. 9 m, 8 m, 10 m.

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Warm Up

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Use Isosceles and Equilateral Triangles

Warm Up

Lesson Presentation

Lesson Quiz

equilateral

scalene

isosceles

Warm-Up

Classify each triangle by its sides.

1.2 cm, 2 cm, 2 cm

2. 7 ft, 11 ft, 7 ft

3.9 m, 8 m, 10 m

5. In ∆DEF, ifm D = m E andm F = 26º, What are the measure of D and E

77º, 77º

4. In ∆ABC, ifm A = 70º andm B = 50º, what ism C?

60º

Warm-Up

DE DF, so by the Base Angles Theorem, E F.

Example 1

In DEF, DEDF. Name two congruent angles.

SOLUTION

If KHJKJH, then ?? .

If KHJKJH, then ?? .

If HG HK, then ?? .

HGK, HKG

KH, KJ

Guided Practice

Copy and complete each statement.

Find the measures of P, Q, and R.

The diagram shows that PQRis equilateral. Therefore, by the Corollary to the Base Angles Theorem, PQRis equiangular. So, m P = m Q = m R.

o

3(m P)

= 180

Triangle Sum Theorem

o

m P

= 60

Divide each side by 3.

The measures of P, Q, and Rare all 60°.

Example 2

Find STin the triangle at the right.

Is it possible for an equilateral triangle to have an angle measure other than 60°? Explain.

No; The Triangle Sum Theorem and the fact that the triangle is equilateral guarantees the angles measure 60° because all pairs of angles could be considered base angles of an isosceles triangle.

5

Guided Practice

Find the values of x and yin the diagram.

ALGEBRA

STEP 1

Find the value of y. Because KLNis equiangular, it is also equilateral and KN KL. Therefore, y = 4.

Example 3

SOLUTION

STEP 2

Find the value of x. Because LNM LMN, LN LMand LMNis isosceles. You also know that LN = 4 because KLNis equilateral.

Example 3

LN = LM

Definition of congruent segments

4 = x + 1

Substitute 4 for LNand x + 1 for LM.

3 = x

Subtract 1 from each side.

In the lifeguard tower, PS QRand QPS PQR.

What congruence postulate can you use to prove that

QPS PQR?

Draw and label QPSand PQRso that they do not overlap. You can see that PQ QP, PS  QR, and QPS PQR. So, by the SASPostulate, QPS PQR.

Example 4

Lifeguard Tower

SOLUTION

In the lifeguard tower, PS QRand QPS PQR.

Explain why PQTis isosceles.

From part (a), you know that 1 2 because corresp. parts of are . By the Converse of the Base Angles Theorem, PT QT, and

PQTis isosceles.

Example 4

Lifeguard Tower

SOLUTION

You know that PS QR, and 3 4 because corresp. parts of are . Also, PTS QTRby the Vertical Angles Congruence Theorem. So,

In the lifeguard tower, PS QRand QPS PQR.

PTS QTRby the AAS Congruence Theorem.

Show thatPTS QTR.

Example 4

Lifeguard Tower

SOLUTION

Find the values of x and yin the diagram.

x = 60

y = 120

Guided Practice

Use parts (b) and (c) in Example 4 and the SSS Congruence Postulate to give a different proof that

PTS QTR

By the Segment Addition PostulateQT + TS=QSandPT + TR=PR. SincePTQTfrom part (b) andTSTRfrom part (c), thenQSPR. PQPQby the Reflexive Property and it is given thatPSQR, thereforeQPSPQRby the SSS Congruence Postulate.

Guided Practice

1.

8

Lesson Quiz

Find the value of x.

2.

3

Lesson Quiz

Find the value of x.

If the measure of vertex angle of an isosceles triangle is 112°, what are the measures of the base angles?

3.

34°, 34°

Lesson Quiz

Find the perimeter of triangle.

4.