Exercise 41 Page 39

Let's begin by reviewing the definition of vertical and supplementary angles. Then we can decide when vertical angles are also supplementary.

Vertical angles

Vertical angles are two angles whose sides are opposite rays.

Notice that the pairs: $\angle 1\&\angle 4,\angle 1\&\angle 3,$ $\angle 3\&\angle 2$ and $\angle 2\&\angle 4$ are linear pairs. Thus, we can conclude the following. We can tell that $\angle 1$ and $\angle 2$ are vertical and equal, and $\angle 3$ and $\angle 4$ are also vertical and equal.

Supplementary angles

Supplementary angles are two angles whose measures have a sum of $180.$

Both

Let's take $\angle 1$ and $\angle 2$ as angles that are supplementary. Let's write out what this means. If we want them to be vertical, then they must be equal: $m\angle 1=m\angle 2.$ Therefore, we can write the following equation Let's solve it! Thus, the measure of both $\angle 1$ and $\angle 2$ is $90^{\circ }.$ Since they are vertical angles, we still need the measures of $\angle 3$ and $\angle 4$ to graph them. From the equations that we wrote in the definition of vertical angles, we get the following. We can tell that the measure of $\angle 3$ and $\angle 4$ is also $90^{\circ }.$ Therefore, there is only one way to represent these vertical angles.

As we can see, all four vertical angles are right angles.