Exercise 31 Page 396

We are given the following diagram.

In this diagram, there is a new maintenance garage at the northeast corner of Park and Main. We know that the street maintenance vehicles for the city cannot make turns less than $70^{\circ }$ safely. We want to determine whether these vehicles can safely make turns at the given crossroads. Let's start by marking all the remaining angles formed by this crossroads!

Now we will find the measure of each angle one at a time. Note that $\angle 3$ and the $108^{\circ }$ angle form a straight line. Therefore, these angles are supplementary and the sum of their measures is $180^{\circ }.$ Let's find the measure of $\angle 3.$ For simplicity, we will not write the degree symbol. We found that the measure of $\angle 3$ is $72^{\circ }.$ Notice that $\angle 2$ and the $108^{\circ }$ angle also form a straight line. Therefore, these angles are also supplementary and $m\angle 2=72^{\circ }.$ Finally, let's find the measure of $\angle 1.$ To find $m\angle 1,$ we will use the fact that $\angle 1$ and $\angle 2$ form a straight line. It follows that these angles are also supplementary and the sum of their measures is $180^{\circ }.$ Since we know that $m\angle 2=72^{\circ },$ we can find the measure of $\angle 1.$ Finally, we know the measures of the angles formed by the crossroads.

Notice that the measure of each angle formed by the crossroads is greater than $70^{\circ }.$ This means that the street maintenance vehicles can safely make turns at this crossroads. Therefore, the proposed site of the new maintenance garage should be approved.