Exercise 8 Page 483

Looking at the figure, we notice three things we need to figure out the remaining angles.

1. $\angle 3$ and $\angle 1$ are vertical angles.
2. $\angle 3$ is supplementary to both $\angle 2$ and $\angle 4.$
3. $\angle 2$ and $\angle 4$ are vertical angles.

By the Vertical Angles Congruence Theorem, $\angle 3$ and $\angle 1$ are congruent. As $m\angle 3=159^{\circ }$ it must follow that $m\angle 1=159^{\circ }.$ Supplementary angles have measures that sum to $180^{\circ }.$ Additionally, since we also know that $\angle 2$ and $\angle 4$ are vertical angles, we can write the following three equations. By solving the first equation for $m\angle 4,$ we also figure out the measure of $\angle 2.$ Let's summarize what we have found.