Chapter 6.5, Problem 56SPR

To determine

The measure of angle 5.

Answer to Problem 56SPR

  $m\angle 5=104$

Explanation of Solution

Given:

ABDC is a rectangle.

  $m\angle 1=38$

Calculation:

The diagonals of a rectangle are congruent and also bisect each other.

So, triangle AEC is an isosceles triangle.

According to Isosceles Triangle Theorem, $\angle 1\cong \angle ACE$ i.e.;

  $\begin{array}{l}m\angle ACE=m\angle 1\\ m\angle ACE=38\end{array}$

The sum of all the angles of a triangle is 180.

In triangle AEC:

  $\begin{array}{l}m\angle ACE+m\angle 1+m\angle AEC=180\\ 38+38+m\angle AEC=180\\ m\angle AEC=180-76\\ m\angle AEC=104\end{array}$

Also,

  $\angle 5\cong \angle AEC$ as vertical angles are congruent.

So,

  $\begin{array}{l}m\angle 5=m\angle AEC\\ m\angle 5=104\end{array}$