Triangle Angle-Side-Angle and Side-Angle-Angle Congruence/Proof

Theorem

In the words of Euclid:

If two triangles have two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining sides equal to the remaining sides and the remaining angle equal to the remaining angle.

(The Elements: Book $\text{I}$: Proposition $26$)

Proof

Both parts of this proposition follow trivially from the other part and Sum of Angles of Triangle equals Two Right Angles.

However, it is important to note that both of these are provable without the parallel postulate, which the proof of that theorem requires.