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 equilateral isosceles scalene 4. In ∆ABC, if m A = 70º and m B = 50º, what is m C? ANSWER 60º 5. In ∆DEF, if m D = m E and m F = 26º, What are the measure of D and E ANSWER 77º, 77º

Proving triangles congruent. Target Proving triangles congruent. GOAL: 4.8 Use theorems about isosceles and equilateral triangles.

Vocabulary vertex angle legs isosceles triangle – base base angles vertex angle is always opposite the base base angles are always opposite the legs

Vocabulary Base Angles Theorem 4.7 – If two sides of a triangle are congruent, then the angles opposite them are congruent. Corollary – If a triangle is equilateral, then it is equiangular. Base Angles Converse Theorem 4.8 – If two angles of a triangle are congruent, then the sides opposite them are congruent. Corollary – If a triangle is equiangular, then it is equilateral.

EXAMPLE 1 Apply the Base Angles Theorem In DEF, DE DF . Name two congruent angles. SOLUTION DE DF , so by the Base Angles Theorem, E F.

GUIDED PRACTICE for Example 1 Copy and complete the statement. If HG HK , then ? ? . SOLUTION HGK HKG If KHJ KJH, then ? ? . SOLUTION If KHJ KJH, then , KH KJ

Find measures in a triangle 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

GUIDED PRACTICE for Example 2 Find ST in the triangle at the right. SOLUTION STU is equilateral, then it is equiangular Thus ST = 5 ( Base angle theorem ) ANSWER

GUIDED PRACTICE for Example 2 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.

Use isosceles and equilateral triangles EXAMPLE 3 Use isosceles and equilateral triangles 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. 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.

EXAMPLE 4 Solve a multi-step problem Lifeguard Tower In the lifeguard tower, PS QR and QPS PQR. QPS PQR? What congruence postulate can you use to prove that Explain why PQT is isosceles. 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, QPS PQR. From part (a), you know that 1 2 because corresponding parts of congruent triangles are congruent. By the Converse of the Base Angles Theorem, PT QT , and PQT is isosceles.

EXAMPLE 4 Solve a multi-step problem You know that PS QR , and 3 4 because corresponding parts of congruent triangles are congruent. Also, PTS QTR by the Vertical Angles Congruence Theorem. So, PTS QTR by the AAS Congruence Theorem.

GUIDED PRACTICE for Examples 3 and 4 Find the values of x and y in the diagram. ANSWER y° = 120° x° = 60°

GUIDED PRACTICE for Examples 3 and 4 Use parts (b) and (c) in Example 4 and the SSS Congruence Postulate to give a different proof that PTS QTR SOLUTION QPS PQR. Can be shown by segment addition postulate i.e a. QT + TS = QS and PT + TR = PR

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