Classify each triangle by its sides. 1. 2 cm, 2 cm, 2 cm ANSWER equilateral 2. 7 ft, 11 ft, 7 ft ANSWER isosceles

Classify each triangle by its sides. 3. 9 m, 8 m, 10 m ANSWER scalene 4. In ∆ABC, if m A = 70º and m B = 50º, what is m C? ANSWER 60º

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 ANSWER 77º, 77º

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

GUIDED PRACTICE for Example 1 Copy and complete the statement. 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 its 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

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.

Use isosceles and equilateral triangles 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? 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 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 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 Find the values of x and y in the diagram. SOLUTION 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

Daily Homework Quiz Find the value of x. 1. ANSWER 8

Daily Homework Quiz Find the value of x. 2. ANSWER 3

Daily Homework Quiz If the measure of vertex angle of an isosceles triangle is 112°, what are the measures of the base angles? 3. ANSWER 34°, 34°

Daily Homework Quiz Find the perimeter of triangle. 4. ANSWER 66 cm