Lesson 4-6 Isosceles Triangles

Standardized Test Practice: Transparency 4-6 5-Minute Check on Lesson 4-5 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

Standardized Test Practice: Transparency 4-6 5-Minute Check on Lesson 4-5 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 . AAS, SSS or SAS 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

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 A + B + C = 180° B Vertex angle leg leg A base C Base Angles A  C A + B + C = 180°

Write a two-column proof. Given: Prove: Proof: Reasons Statements 1. Given 1. 2. Def. of segments 2. 3. Def. of isosceles  3. ABC and BCD are isosceles 5. 5. Given 4. 4. Isosceles  Theorem 6. 6. Substitution

Write a two-column proof. Given: . Prove: Proof: Reasons Statements 1. Given 3. Isosceles  Theorem 2. Def. of isosceles triangles 1. 2. ADB is isosceles. 3. 4. 5. 4. Given 5. Def. of midpoint 6. SAS 7. 7. CPCTC 6. ABC ADC

Multiple-Choice Test Item If and what is the measure of A. 45.5 B. 57.5 C. 68.5 D. 75 Read the Test Item CDE is isosceles with base Likewise, CBA is isosceles with 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 what is the measure of A. 25 B. 35 C. 50 D. 130 Answer: A

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. bisects a. Find x. Answer: 30 b. Answer: 90

Summary & Homework Summary: Homework: 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