If parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Consider the following diagram.

To prove that alternate interior angles are congruent, it will be shown that $\angle 3$ and $\angle 5$ are congruent.
Notice that, by definition, $\angle 1$ and $\angle 3$ are vertical angles. Thus, Notice also that, by definition, $\angle 1$ and $\angle 5$ are corresponding angles. Thus, By transitivity, $m\angle 3=m\angle 5.$ Thus, it follows that $\angle 3\cong \angle 5.$ The same reasoning applies to the other pair of alternate interior angles. Thererefore, when a pair of parallel lines is cut by a transversal, the pairs of alternate interior angles are congruent.
This can be summarized in the following two-column proof.