Exercise 12 Page 402

We are given the following figure.

We want to find the measures of the two missing angles using properties of quadrilaterals and parallel lines. Let's start by marking these two angles.

First, we will find the measure of the $x^{\circ }$ angle using properties of parallel lines. Note that the given figure is a trapezoid because it has two bases that are parallel. Let's lengthen the bases and the right leg of this trapezoid.

Now we can see a line that intersects two parallel lines. 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. This means that we can use the $152^{\circ }$ angle 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 $152^{\circ }$ angle and $\angle 1.$

Notice that the $152^{\circ }$ angle and $\angle 1$ are the interior angles that lie on opposite sides of the transversal. This means that these angles are alternate interior angles. Since the two horizontal lines 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, these angles are supplementary and the sum of their measures is $180^{\circ }.$ Knowing that $m\angle 1=152^{\circ },$ we will find the measure of the $x^{\circ }$ angle. We got that $x=28.$

Now we will find the value of $y$ using properties of quadrilaterals. Let's start by recalling the rule for the sum of the measures of the interior angles of a polygon.

Interior Angle Sum of a Polygon

The sum of the measures of the interior angles of a polygon is $(n-2)180,$ where $n$ represents the number of sides.

To find the sum of the measures of the interior angles of the given quadrilateral, we will substitute $4$ for $n$ in this expression. We got that the sum of the interior angles of the trapezoid is $360^{\circ }.$ We can write an equation that represents this situation. Now we can solve this equation for $y.$ For simplicity, we will not write the degree symbol. We got that $y=130.$ Therefore, the measures of the two missing angles of the trapezoid are $28^{\circ }$ and $130^{\circ }.$