Exercise 21 Page 377

We are given the following diagram.

In this diagram, a line intersects the parallel lines $a$ and $b.$ When two parallel lines are cut by a transversal, special angle relationships exist. If we know the measure of one of the angles, we can find the measures of all of the angles. We can use these relationships to find the measure of the $x^{\circ }$ angle. Let's start by marking an extra angle in the diagram.

To find the the measure of $\angle 1,$ we can analyze the relationship between the $70^{\circ }$ angle and $\angle 1.$

The $70^{\circ }$ angle and $\angle 1$ are in the same position on the two lines in relation to the transversal. This means that the $70^{\circ }$ angle and $\angle 1$ are corresponding angles. Since $a$ and $b$ are parallel, the measures of these angles are equal. Now let's take a closer look at $\angle 1$ and the $x^{\circ }$ angle. We can see that $\angle 1$ and the $x^{\circ }$ angle form a straight line. Therefore, $\angle 1$ and the $x^{\circ }$ angle are supplementary and the sum of their measures is $180^{\circ }.$ Since we know that $m\angle 1=70^{\circ },$ we can find the value of $x.$ We found that $x=110.$ This means that the correct option is D.