Exercise 4 Page 481

We are asked to use the given diagram and the given angle measure to find the other three angles measures. Looking at the figure, we notice three things we need to figure out the remaining angles.

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

By the Vertical Angles Congruence Theorem, $\angle 1$ and $\angle 3$ are congruent. Therefore, we get the following equation. Since we are given that $m\angle 1=117^{\circ },$ it must follow that $m\angle 3=117^{\circ }.$ Recall that supplementary angles have measures that sum to $180^{\circ }.$ Additionally, we also know that $\angle 2$ and $\angle 4$ are vertical angles. This means that we can write the following three equations. Since the measure of $\angle 2$ is equal to the measure of $\angle 4,$ we can only solve the first equation for $m\angle 2$ to get all the measures that we need. Let's summarize what we have found!