Chapter 6.4, Problem 31PPS

To determine

The measure of angle 8.

Answer to Problem 31PPS

  $m\angle 8=100$

Explanation of Solution

Given:

ABCD is a rectangle.

  $m\angle 2=40$

Calculation:

The diagonals in a rectangle are congruent and bisect each other.

Therefore, triangle AEB is an isosceles triangle.

According to Isosceles Triangle Theorem, $\angle 2\cong \angle CBA.$

  $\begin{array}{l}m\angle CBA=m\angle 2\\ m\angle CBA=40\end{array}$

The sum of the measures of all angles in triangle is 180 degrees.

So, consider triangle AEB;

  $\begin{array}{l}m\angle 2+m\angle CBA+m\angle AEB=180\\ 40+40+m\angle AEB=180\\ 80+m\angle AEB=180\\ m\angle AEB=100\end{array}$

From the given rectangle, it can be seen that $\angle 8\text{ and }\angle \text{AEB}$ are vertical angles, and hence are congruent.

  $\begin{array}{l}m\angle 8=m\angle AEB\\ m\angle 8=100\end{array}$