1. Isosceles and Equilateral Triangles

2. Isosceles Triangle – Triangle with two congruent sides.
The congruent sides are the legs.
The third side is the base.
The two legs form the vertex angle.
The other two angles are the base angles.
Legs of an isosceles triangle are congruent.
Base angles of an isosceles triangle are congruent.

3. Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

4. Proof of Isosceles Triangle Theorem
Given:
Prove:
This proof requires an auxiliary line.
Statements
Reasons 1.
1. Given 2.
2. Given 3.
3. A bisector divides an angle into two congruent angles. (2) 4.
4. Reflexive Property 5.
5. SAS (1, 3, 4) 6.
6. CPCTC (5)

5. Converse of Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

6. Theorem
If a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular bisector of the base.

7. Equilateral Triangle – Triangle with three congruent sides.

8. Corollary – A theorem that can be proved using another theorem.

9. Corollary
If a triangle is equilateral, then the triangle is equiangular.

10. Corollary
If a triangle is equiangular, then the triangle is equilateral.