### If two angles and any side of a triangle are equal to the corresponding angles and side of another triangle, then prove that the two triangles are congruent.

1. Let $\triangle ABC$ and $\triangle DEF$ be the two triangles with $BC = EF, \angle A = \angle D, \text{ and } \angle B = \angle E$.
2. We have to prove that $$\triangle ABC \cong \triangle DEF$$
3. We know that the sum of the angles of a triangle is $180^ \circ$. \begin{aligned} \implies \angle A + \angle B + \angle C = \angle D + \angle E + \angle F = 180^ \circ && \ldots (1) \end{aligned} It is given that $\angle A = \angle D \text{ and } \angle B = \angle E.$
Thus, from equation (1), we conclude that $\angle C = \angle F.$
4. In $\triangle ABC \text{ and } \triangle DEF$, we have \begin{aligned} &BC = EF && \text{[By step 1]}\\ &\angle C = \angle F && \text{[By step 3]} \\ &\angle B = \angle E && \text{[By step 1]} \\ &\therefore \triangle ABC \cong \triangle DEF && \text{[By ASA criterion]} \end{aligned}