Start by noticing that $\angle 1$ and $\angle 4$ form a linear pair as well as $\angle 2$ and $\angle 3.$

Therefore, these pair of angles are supplementary. On the other hand, since the lines ${\ell }_1$ and ${\ell }_2$ are parallel, by the Alternate Interior Angles Theorem, the alternate interior angles are congruent. This allows to write the following relations. Next, in Equation I, substitute $m\angle 3$ for $m\angle 4.$ The last equation implies that $\angle 1$ and $\angle 3$ are supplementary. Similarly, in Equation II, substitute $m\angle 4$ for $m\angle 3.$ As before, the last equation implies that $\angle 2$ and $\angle 4$ are supplementary, which completes the proof.