4.7 Isosceles and Equilateral TrianglesObjective: You will use theorems about isosceles and

equilateral triangles.

Base Angle TheoremIf two sides of a triangle are congruent,

then the angles opposite them are congruent.

Converse of Base Angles Theorem

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

Corollary

• Corollary to the Base Angle Theorem- if a triangle is equilateral, then it is equiangular.

• Corollary to the Converse of Base Angle Theorem- If a triangle is equiangular, then it is equilateral.

EXAMPLE 1 Apply the Base Angles Theorem

SOLUTION

In DEF, DE DF . Name two congruent angles.

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

GUIDED PRACTICE for Example 1

SOLUTION

Copy and complete the statement.

1. If HG HK , then ? ? .

HGK HKG

GUIDED PRACTICE for Example 1

Copy and complete the statement.

If KHJ KJH, then ? ? .If KHJ KJH, then ? ? .2. 2.

SOLUTION

If KHJ KJH, then , KH KJ

EXAMPLE 2 Find measures in a triangle

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

GUIDED PRACTICE for Example 2

3. Find ST in the triangle at the right.

SOLUTION

STU is equilateral, then its is equiangular

Thus ST = 5 ( Base angle theorem )

ANSWER

GUIDED PRACTICE for Example 2

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

SOLUTION

No; it is not possible for an equilateral triangle to have angle measure other then 60°. Because the triangle sum theorem and the fact that the triangle is equilateral guarantees the angle measure 60° because all pairs of angles could be considered base of an isosceles triangle

EXAMPLE 3 Use isosceles and equilateral triangles

ALGEBRA

Find the values of x and y in the diagram.

SOLUTION

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.

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

EXAMPLE 3 Use isosceles and equilateral triangles

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.

EXAMPLE 4 Solve a multi-step problem

Lifeguard Tower

In the lifeguard tower, PS QR and QPS PQR.

QPS PQR?

a. What congruence postulate can you use to prove that

b. Explain why PQT is isosceles.

c. Show that PTS QTR.

EXAMPLE 4 Solve a multi-step problem

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 Congruence Postulate,

a.

QPS PQR.

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.

EXAMPLE 4 Solve a multi-step problem

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.

GUIDED PRACTICE for Examples 3 and 4

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

SOLUTION

y° = 120°

x° = 60°

GUIDED PRACTICE for Examples 3 and 4

SOLUTION

QPS PQR. Can be shown by segment addition postulate i.e

a. QT + TS = QS and PT + TR = PR

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

GUIDED PRACTICE for Examples 3 and 4

Since PT QT from part (b) and

TS TR from part (c) then,

QS PR

PQ PQ Reflexive Property and

PS QR Given

Therefore QPS PQR . By SSS Congruence Postulate

ANSWER