Chapter 14.6, Problem 12CE

To determine

To explain the theorem 14-8.

Answer to Problem 12CE

The theorem 14-8 explained.

Explanation of Solution

Given information: Given transformation $S$ to another transformation $T$ can be written as $SoT.$

Calculation:

Calculation:

Theorem 14-8 states that;

A composite of reflections in two intersect lines is a rotation about the point of intersection of the two lines.

And the measure of the angle of rotation is twice the Measure of the angle from the first line of reflection to the second.

Now consider if a line $j$ intersects line $k$ then it forms an angle of measure $y$ at o, which is the point of intersection.

Therefore, $R_{k}o{R}_j=R_{0,2y}$

And if the line of reflection is perpendicular then the composite of the reflection will give; $R_{k}o{R}_j=R_{0,180}$

It means that a composite of the reflections in perpendicular lines is a half-turn about the point

Where the lines intersects, which is the given Corollary.

Thus, the theorem 14-8 explained.