1. 4.6 Isosceles Triangles

2. Objectives Use properties of isosceles triangles
Use properties of equilateral triangles

3. 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

4. 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

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

6. 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

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

8. 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

9. 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).

10. Example 2:
Name two congruent angles.
Answer:

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

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

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

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

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

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

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