Work Step by Step
Let the angle whose measure is labeled as $x^o$ be $\angle X$. Let the angle whose measure is labeled as $y^o$ be $\angle Y$. The unlabeled angle is complementary to $\angle Y$. Call this angle $\angle U$. The problem and diagram tell us that $m\angle U$ is 40, since the angles adjacent to the perpendicular line are the same. $m\angle U+m\angle Y=90\longrightarrow$ definition of complementary angles $40+m\angle Y=90\longrightarrow$ substitute the given value for $m\angle U$ $m\angle Y=50$ Theorem 2.3 tells that angles that are complementary to congruent angles are also congruent, so $m\angle X$ is also 50.