Exercise 23 Page 525

From the figure, we can identify a number of vertical angles. According to the Vertical Angles Congruence Theorem, vertical angles are congruent. Therefore, we know that $\angle 1\cong \angle 5,$ $m\angle 2=40,$ and $m\angle 3=30^{\circ }.$ Let's show this in our diagram. Note that $\angle 1$ is a right angle and therefore measures $90^{\circ }.$

To find the remaining $m\angle 4,$ notice that $30^{\circ },$ $40^{\circ },$ and $\angle 4$ in the figure below make up a right angle.

By adding these three angle measures and setting their sum equal to $90^{\circ },$ we can solve for $m\angle 4.$ The last angle has a measure of $20^{\circ }.$