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Lesson 4-6 Isosceles Triangles 5-Minute Check on Lesson 4-5 Transparency 4-6 Refer to the figure. Complete each congruence statement and the postulate or theorem that applies. 1. WXY _____ by _____. 2. WYZ _____ by _____. 3. VWZ _____ by _____. 4. What additional congruence statement is necessary to prove RST UVW by the ASA Postulate? Standardized Test Practice: A T W B R U C ST UW D RT VW 5-Minute Check on Lesson 4-5 Transparency 4-6 Refer to the figure. Complete each congruence statement and the postulate or theorem that applies. 1. WXY VZY by ASA . or AAS 2. WYZ VYX by AAS . 3. VWZ WVX by ASA . or SAS 4. AAS, SSS What additional congruence statement is necessary to prove RST UVW by the ASA Postulate? Standardized Test Practice: A T W B R U C ST UW D RT VW Objectives • Use properties of isosceles triangles • Use properties of equilateral triangles Vocabulary • Vertex angle – the angle formed by the two congruent sides • Base angle – the angle formed by the base and one of the congruent sides Theorems • Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. • Converse of Isosceles Triangle Theorem: If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Corollaries • A triangle is equilateral if, and only if, it is equiangular • Each angle of an equilateral triangle measures 60° Isosceles Triangle B Vertex angle leg A leg base C Base Angles A C A + B + C = 180° Write a two-column proof. Given: Prove: Proof: Statements Reasons 1. 1. Given 2. 2. Def. of 3. ABC and BCD are isosceles 4. 3. Def. of isosceles 5. 6. 5. Given 6. Substitution segments 4. Isosceles Theorem Write a two-column proof. Given: . Prove: Proof: Statements 1. Reasons 1. Given 2. ADB is isosceles. 2. Def. of isosceles triangles 3. 4. 3. Isosceles Theorem 4. Given 5. 6. ABC ADC 7. 5. Def. of midpoint 6. SAS 7. CPCTC Multiple-Choice Test Item If and measure of A. 45.5 B. 57.5 C. 68.5 D. 75 Read the Test Item CDE is isosceles with base isosceles with what is the Likewise, CBA is Solve the Test Item Step 1 The base angles of CDE are congruent. Let Angle Sum Theorem Substitution Add. Subtract 120 from each side. Divide each side by 2. Step 2 are vertical angles so they have equal measures. Def. of vertical angles Substitution Step 3 The base angles of CBA are congruent. Angle Sum Theorem Substitution Add. Subtract 30 from each side. Divide each side by 2. Answer: D Multiple-Choice Test Item If and measure of A. 25 Answer: A B. 35 what is the C. 50 D. 130 Name two congruent angles. Answer: Name two congruent segments. By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So, Answer: a. Name two congruent angles. Answer: b. Name two congruent segments. Answer: ABC is an equilateral triangle. a. Find x. Answer: 30 b. Answer: 90 bisects Summary & Homework • Summary: – Two sides of a triangle are congruent if, and only if, the angles opposite those sides are congruent. – A triangle is equilateral if, and only if, it is equiangular. • Homework: – pg 219 - 20: 9, 10, 13-18, 27