4.6 Isosceles Triangles

Objectives Use properties of isosceles triangles Use properties of equilateral triangles

Properties of Isosceles Triangles The  formed by the ≅ sides is called the vertex angle. The two ≅ sides are called legs. The third side is called the base. The two s formed by the base and the legs are called the base angles. vertex leg leg base

Isosceles Triangle Theorem Theorem 4.9 If two sides of a ∆ are ≅, then the s opposite those sides are ≅ (if AC ≅ AB, then B ≅ C). A B C

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

Example 1: Proof: Reasons Statements 1. Given 1. 2. Def. of Segments 3. Def. of Isosceles  3. ABC and BCD are isosceles triangles 5. 5. Given 4. 4. Isosceles  Theorem 6. 6. Substitution

Your Turn: Write a two-column proof. Given: . Prove:

Your Turn: 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

The Converse of Isosceles Triangle Theorem If two s of a ∆ are ≅, then the sides opposite those s are ≅ (if B ≅ C, then AC ≅ AB).

Example 2: Name two congruent angles. Answer:

Example 2: Name two congruent segments. By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So, Answer:

Your Turn: a. Name two congruent angles. Answer: b. Name two congruent segments. Answer:

Properties of Equilateral ∆s Corollary 4.3 A ∆ is equilateral iff it is equiangular. Corollary 4.4 Each  of an equilateral ∆ measures 60°.

Example 3a: EFG is equilateral, and bisects bisects Find and Since the angle was bisected, Each angle of an equilateral triangle measures 60°.

Example 3a: is an exterior angle of EGJ. Exterior Angle Theorem Substitution Add. Answer:

Example 3b: EFG is equilateral, and bisects bisects Find Linear pairs are supplementary. Substitution Subtract 75 from each side. Answer: 105

Your Turn: ABC is an equilateral triangle. bisects a. Find x. Answer: 30 b. Answer: 90