AAS Congruence

Theorem: If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent.

Prerequisites:
ASA congruency (proof)
Angle sum property of triangle (proof)

Proof:

Let there be two triangles $\triangle ABC$ and $\triangle DEF$, such that the angles $\angle ABC = \angle DEF$, $\angle ACB = \angle DFE$ and the non-included sides AB $=$ DE.

Using angle sum property of a triangle,

$\qquad\quad\begin{align}\angle BAC &= 180^o - \angle ABC - \angle ACB\\ &= 180^o - \angle DEF - \angle DFE\qquad\;\;\!\qquad\text{(given)}\\ &= \angle EDF\qquad\qquad\qquad\qquad\qquad\qquad\text{(by angle sum property)}\end{align}$

Hence, $\triangle ABC\cong\triangle DEF$ by ASA rule.

Thus, the given triangles are congruent.


Recommended:
RHS congruence
SSS congruence
AAA similarity