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

We name the triangle as ABC and CBDCBD is equilateral triangle which guarantees

the angle measure 60° therefore x° = 60°

y° = 180° – x°y° = 180° – 60°

y° = 120°x° = 180° – y° = 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 and

TS TRb. from part then,

QS PRc.

PQ PQ reflexive property and

PS QR given

Therefore QPS PQR . By SSS congruence Postulate

ANSWER