Warm-Up Exercises Classify each triangle by its sides. Lesson 4.7, For use with pages cm, 2 cm, 2 cm ANSWER equilateral ANSWER isosceles 2. 7 ft, 11 ft, 7 ft

Warm-Up Exercises 3.9 m, 8 m, 10 m ANSWER scalene ANSWER 60º Lesson 4.7, For use with pages Classify each triangle by its sides. 4. In ∆ABC, if m A = 70º and m B = 50º, what is m C ?

Warm-Up Exercises ANSWER 77º, 77º Lesson 4.7, For use with pages Classify each triangle by its sides. 5. In ∆DEF, if m D = m E and m F = 26º, What are the measure of D and E

Warm-Up Exercises EXAMPLE 1 The Isosceles Triangle

Warm-Up Exercises EXAMPLE 1 The Base Angles Theorem

Warm-Up Exercises 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.

Warm-Up Exercises GUIDED PRACTICE for Example 1 SOLUTION Copy and complete the statement. 1. If HG HK, then ? ?. HGK HKG

Warm-Up Exercises GUIDED PRACTICE for Example 1 Copy and complete the statement. If KHJ KJH, then ? ?. 2. SOLUTION If KHJ KJH, then, KH KJ

Warm-Up Exercises EXAMPLE 2 Corollaries to the Base Angle Theorem

Warm-Up Exercises 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) = 180 o Triangle Sum Theorem m P = 60 o Divide each side by 3. The measures of P, Q, and R are all 60 °. ANSWER

Warm-Up Exercises 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

Warm-Up Exercises 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

Warm-Up Exercises 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.

Warm-Up Exercises 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.

Warm-Up Exercises 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.

Warm-Up Exercises 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.

Warm-Up Exercises 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.

Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 5. Find the values of x and y in the diagram. SOLUTION y ° = 120 ° x ° = 60 °

Warm-Up Exercises 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

Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 Since PT QT from part (b) and TS TR from part (c) then, QS PR PQ Reflexive Property and PS QR Given Therefore QPS PQR. By SSS Congruence Postulate ANSWER