The bisector of the vertex angle of an isosceles triangle separates the triangle into two congruent triangles

 Bisector of the vertex angle of an isosceles triangle

Given
Isosceles triangle ABC with AB = AC
AO bisects ,


Prove


Proof

1: AB=AC - Given
2: - Given

3: AO = AO - Identity

- From 1,2,3 - If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent (SAS)

Side by side (SAS) triangle congruence

Triangles are congruent if any pair of corresponding sides and their included angles are equal in both triangles.

In triangles ABC, DEF, AB=DE, BC=EF and   , it follows that: