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# 4.7 use isosceles and equilateral triangles

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4.7 4.7 Use Isosceles and Equilateral Triangles Bell Thinger Classify each triangle by its sides. 1. 2 cm, 2 cm, 2 cm ANSWER equilateral ANSWER isosceles 2. 7 ft, 11 ft, 7 ft 3. 9 m, 8 m, 10 m ANSWER scalene
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4.74.7 Use Isosceles and Equilateral TrianglesBell Thinger

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

4.7

4.7

4.7Example 1

SOLUTION

In DEF, DE ≅ DF . Name two congruent angles.

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

4.7Guided Practice

Copy and complete each statement.

1. If HG ≅ HK , then ? ≅ ? .

If KHJ ≅ KJH, then ? ≅ ? .2. 2.

4.7

4.7Example 2

Find the measures of P, Q, and R.

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

3(m P) = 180o

Triangle Sum Theorem

m P = 60o

Divide each side by 3.

The measures of ∠P, ∠Q, and ∠R are all 60°.ANSWER

4.7Guided Practice

3. Find ST in the triangle at the right.

4. 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.

4.7Example 3

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

SOLUTION

STEP 1 Find the value of y. Because KLN is equiangular, it is also equilateral and KN ≅ KL. Therefore, y = 4.

4.7Example 3

STEP 2 Find the value of x. Because LNM ≅ LMN, LN ≅ LM and LMN is isosceles. You also know that LN = 4 because KLN is equilateral.

LN = LM Definition of congruent segments

4 = x + 1 Substitute 4 for LN and x + 1 for LM.

3 = x Subtract 1 from each side.

4.7Example 4

Lifeguard Tower

In the lifeguard tower, PS ≅ QR and QPS ≅ PQR.

QPS ≅ PQR?

a. What congruence postulate can you use to prove that

SOLUTION

Draw and label QPS and PQR so that they do not overlap. You can see that PQ ≅ QP, PS ≅ QR, and ∠QPS ≅ ∠PQR. So, by the SAS ≅ Postulate, QPS ≅ PQR.

a.

4.7Example 4

Lifeguard Tower

In the lifeguard tower, PS ≅ QR and QPS ≅ PQR.

b. Explain why PQT is isosceles.

SOLUTION

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

PQT is isosceles.

4.7Example 4

Lifeguard Tower

In the lifeguard tower, PS ≅ QR and QPS ≅ PQR.

c. Show that PTS ≅ QTR.

SOLUTION

c. You know that PS ≅ QR , and 3 ≅ 4 because corresp. parts of ≅ are ≅. Also, PTS ≅ QTR by the Vertical Angles Congruence Theorem. So, PTS ≅ QTR by the AAS Congruence Theorem.

4.7Exit Slip

Find the value of x.

1.

4.7

Find the value of x.

2.

Exit Slip

4.7

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

3.

Exit Slip

4.7

Find the perimeter of triangle.4.

Exit Slip

4.7HomeworkPg 279-282#8, 11, 15, 20, 40Classwork (if you finish HW)

pg 279#12, 13, 16, 21, 38

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