Isosceles Triangle Theorem

Consider a triangle ABC with two congruent sides, or an isosceles triangle.

In this triangle, let P be the point of intersection of BC and the angle bisector of ∠ A. From the diagram, the following facts about △ BAP and △ CAP can be observed.

Statement	Reason
∠ BAP ≅ ∠ CAP	Definition of an angle bisector
BA ≅ CA	Given
AP ≅ AP	Reflexive Property of Congruence

Therefore, △ BAP and △ CAP have two pairs of corresponding congruent sides and one pair of congruent included angles. By the Side-Angle-Side Congruence Theorem, △ BAP and △ CAP are congruent triangles. △ BAP ≅ △ CAP Corresponding parts of congruent figures are congruent. Therefore, ∠ B and ∠ C are congruent. ∠ B ≅ ∠ C It has been proven that if two sides of a triangle are congruent, then the angles opposite them are congruent.

Consider an isosceles triangle △ ABC.

A line passing through A and the midpoint of BC will be drawn. Let P be the midpoint.

Since BP and PC are congruent, the distance between B and P is equal to the distance between C and P. Therefore, B is the image of C after a reflection across AP. Also, because A lies on AP, a reflection across AP maps A onto itself. The same is true for P.

Reflection Across AP
Preimage	Image
C	B
A	A
P	P

The table shows that the images of the vertices of △ CAP are the vertices of △ BAP. It can be concluded that △ BAP is the image of △ CAP after a reflection across AP. Since a reflection is a rigid motion, this proves that the triangles are congruent. Corresponding parts of congruent figures are congruent, so ∠ B and ∠ C are congruent. ∠ B≅∠ C