Prove that two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of another triangle.

To prove that two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, we will use the ASA (Angle-Side-Angle) Congruence Criterion.

Given:
In triangle ABC, let ∠A = ∠D, ∠B = ∠E, and side AB = DE.
In triangle DEF, ∠D = ∠A, ∠E = ∠B, and side DE = AB.
We are required to prove that triangle ABC is congruent to triangle DEF.

Proof:
Start with two triangles: We have two triangles, triangle ABC and triangle DEF, where:

∠A = ∠D (Angle A in triangle ABC equals Angle D in triangle DEF).
∠B = ∠E (Angle B in triangle ABC equals Angle E in triangle DEF).
AB = DE (Side AB in triangle ABC equals side DE in triangle DEF).
Apply the ASA Criterion: According to the ASA Congruence Criterion, if two angles and the included side between them of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent. Here, the angle ∠A = ∠D, ∠B = ∠E, and the included side AB = DE.

Conclude that the triangles are congruent: Since we have two pairs of angles and the included side of the triangles equal, by the ASA criterion, we can conclude that triangle ABC is congruent to triangle DEF.

Thus, triangle ABC ≅ triangle DEF.

This completes the proof.