Using Properties of Parallel Lines

We should find the measure of $\alpha$ and we know that one angle is $114^{\circ }.$

These angles are formed by the parallel lines and a transversal. They lie on opposite sides of the transversal and on different sides of the parallel lines. According to the definition, these are alternate exterior angles. To find the measure of $\alpha ,$ we can use Alternate Exterior Angles Theorem. $$\begin{array}{c}\text{If two parallel lines are cut by a transversal,}\\ \text{then each pair of alternate exterior angles}\\ \text{is congruent.}\end{array}$$ With the theorem, we conclude that $\alpha$ and $114^{\circ }$ are congruent and their measures are the same. Therefore, the measure of $\alpha$ is also $114^{\circ }.$