Exercise 31 Page 427

Let's analyze the given quadrilateral so that we can find the measure of $\angle 8.$

Firstly, notice that $\angle 2$ and $\angle 3$ form alternate interior angles. Because our quadrilateral is a rectangle, both pairs of opposite sides are parallel. Recall the theorem.

Alternate Interior Angles Theorem

If parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

Therefore, $\angle 2$ and $\angle 3$ are congruent and their measures are equal. Since the measure of $\angle 2$ is given, we can use it. Because our quadrilateral is a rectangle, its diagonals are congruent and bisect each other. Therefore, the triangle formed by $\angle 3,$ $\angle 7,$ and $\angle 8$ is an isosceles triangle, and because of the definition of an isosceles triangle, $m\angle 7$ and $m\angle 3$ are congruent. Recall the Interior Angles Theorem.

Interior Angles Theorem

The sum of the interior angles of a triangle is $180^{\circ }.$

Using that, we can write an equation for the angles of the triangle. Finally, let's solve it to find $m\angle 8.$