Chapter 9.5, Problem 2CE

3

To determine

To explain: the fact that $arc\left(MXN\right)$ is a semicircle leads to a conclusion that $\angle X$ is a right angle.

Explanation of Solution

Given information:

Given figure,

Calculation:

The corollary is;

In a circle, any angle inscribed in a semi circle is a right angle.

Consider the theorem:

The measure of an inscribed angle is equal to half the measure of its intercepted arc.

Proof:

The intercepted arc for an angle inscribed in a semi circle is 180 degrees.

Therefore, the measure of the angle must be half of 180, or 90 degrees. In other words, the angle is a right angle.