Chapter 2.4, Problem 34WE

To determine

To prove: If the measure of a complement of an angle ever equal exactly halt the measure of a supplement of the angle

Answer to Problem 34WE

The measure of a complement of an angle can be equal to exactly half the measure of supplement of the angle only for the angle zero

Explanation of Solution

Given information:

The measure of a complement of an angle is exactly halt the measure of a supplement of the angle

Suppose that the measure of the angle be $x$

Then the measure of its supplement is $180-x$

Then the measure of its complement is $90-x$

The measure of a complement of the angle is exactly half the measure of the supplement of the angle implies that

  $90-x=\frac{1}{2}(180-x)$

Simplifying this gives

  $90-x=\frac{1}{2}(180-x)$

  $90-x=90-\frac{x}{2}$

  $\begin{array}{cc}-x& =-\frac{x}{2}\\ x& =0\end{array}$

This is possible only if the measure of the angle is $0.$

Thus, the measure of a complement of an angle can be equal to exactly half the measure of supplement of the angle only for the angle zero