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 CBD CBD 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 TR b. from part then, QS PR c. PQ reflexive property and PS QR given Therefore QPS PQR. By SSS congruence Postulate ANSWER