It is given that ${\ell }_1$ and ${\ell }_2$ are parallel lines and the transversal $t$ is perpendicular to the line ${\ell }_1.$ This means that the lines ${\ell }_1$ and $t$ intersect at a right angle.

Angles $\angle 1$ and $\angle 2$ are corresponding angles. By the Corresponding Angles Theorem, $\angle 1$ and $\angle 2$ are congruent. This means that $\angle 2$ is also a right angle.

Therefore, $t$ is perpendicular to ${\ell }_2.$ This proves the theorem.