Using Properties of Parallel Lines

To find the measure of angle $\alpha$ we should determine the relation between $108^{\circ }$ and $\alpha .$

These angles are formed by the parallel lines and a transversal. They both lie on the same side of the transversal and on the same left side of the parallel lines. According to the definition, these are called corresponding angles. Thus, to find the measure of $\alpha ,$ we can use Corresponding Angles Theorem. $$\begin{array}{c}\text{If two parallel lines are cut by a transversal,}\\ \text{then each pair of corresponding angles is congruent.}\end{array}$$ By the theorem, we conclude that $108^{\circ }$ and $\alpha$ are congruent, so their measures are the same. Therefore, the measure of $\alpha$ is also $108^{\circ }.$