4.6 Isosceles Triangles Theorem 4.9 Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Theorem 4.10 If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Corollaries 4.3 A triangle is equilateral if and only if it is equiangular 4.4 Each angle of an equilateral triangle measures 60°

Write a two-column proof. Given: Prove: Example 6-1a

3. ABC and BCD are isosceles 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 Example 6-1b

Write a two-column proof. Given: . Prove: Example 6-1c

2. Def. of isosceles triangles 1. 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 Example 6-1d

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 Example 6-2a

Step 1 The base angles of CDE are congruent. Let 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. Example 6-2b

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. Example 6-2c

Answer: D Example 6-2d

Multiple-Choice Test Item If and what is the measure of A. 25 B. 35 C. 50 D. 130 Answer: A Example 6-2e

Name two congruent angles. Answer: Example 6-3a

Name two congruent segments. By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So, Answer: Example 6-3b

a. Name two congruent angles. Answer: b. Name two congruent segments. Answer: Example 6-3c

EFG is equilateral, and bisects bisects Find and Since the angle was bisected, Each angle of an equilateral triangle measures 60°. Example 6-4a

is an exterior angle of EGJ. Exterior Angle Theorem Substitution Add. Answer: Example 6-4b

EFG is equilateral, and bisects bisects Find Linear pairs are supplementary. Substitution Subtract 75 from each side. Answer: 105 Example 6-4c

ABC is an equilateral triangle. bisects a. Find x. Answer: 30 b. Answer: 90 Example 6-4d