AAS Congruence (Angle-Angle-Side Congruence)

AAS Congruence (Angle-Angle-Side Congruence)

How to prove the congruence of triangles by using the AAS congruence theorem: theorem, example, and its solution.

Theorem

For two triangles, if two angles and a non-included side of each triangle are congruent, then those two triangles are congruent. This is the AAS congruence theorem.

For two triangles,

if two angles and a non-included side of each triangle
are congruent,
then those two triangles are congruent.

This is the AAS congruence theorem.

Example

Given: [line segment AB] is congruent to [line segment CD], [angle PAB] is congruent to [angle PCD]. Prove: [triangle PAB] is congruent to [triangle PCD].

Start from the given statement.

ABCD

Two-column proof

Use the other given statement.

PAB ≅ ∠PCD

APB and ∠CPD are vertical angles.

Vertical angles are congruent.

So ∠APB ≅ ∠CPD.

Vertical angles (Opposite angles)

For △PAB and △PCD,
two angles and a non-included side of each triangle
are congruent.

Then, by the AAS congruence theorem,
PAB and △PCD are congruent.

So this is the two-column proof of the example.