The proof will be based on the given diagram, but it holds true for any pair of lines cut by a transversal. Consider only one pair of consecutive interior angles and one more angle.

It needs to be proven that ${\ell }_1$ and ${\ell }_2$ are parallel lines. It is already given that $\angle 1$ is supplementary to $\angle 3.$ The diagram shows that $\angle 3$ and $\angle \alpha$ form a linear pair, which means that these angles are supplementary. Notice that $\angle 3$ is common in both relationships. Subtracting the two equations gives a relation between $\angle 1$ and $\angle \alpha .$ The last equation tells that $\angle 1$ and $\angle \alpha$ have the same measure and therefore, they are congruent. The diagram also shows that $\angle 1$ and $\angle \alpha$ are corresponding angles. Given that relation, the Converse Corresponding Angles Theorem can be applied.

Converse Corresponding Angles Theorem

If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel.

Since $\angle 1$ and $\angle \alpha$ are corresponding congruent angles, then ${\ell }_1$ and ${\ell }_2$ are parallel lines. This completes the proof.