Chapter 10, Problem 27RP

To determine

To prove: The expression for the angle $\mathrm{m}\angle \mathrm{P}=\frac{\mathrm{m}\stackrel{⌢}{\mathrm{W}\mathrm{X}}+\mathrm{m}\stackrel{⌢}{\mathrm{Y}\mathrm{Z}}}{2}$

Explanation of Solution

Given:

The line WZ is the diameter of the circle.

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

Consider the angle $\mathrm{m}\angle \mathrm{P}$ is calculated as,

  $\begin{array}{c}\mathrm{m}\angle \mathrm{P}=\frac{\stackrel{⌢}{\mathrm{W}\mathrm{Z}}-\stackrel{⌢}{\mathrm{X}\mathrm{Y}}}{2}\\ =\frac{\stackrel{⌢}{\mathrm{W}\mathrm{X}}+\stackrel{⌢}{\mathrm{X}\mathrm{Y}}+\stackrel{⌢}{\mathrm{Y}\mathrm{Z}}-\stackrel{⌢}{\mathrm{X}\mathrm{Y}}}{2}\\ =\frac{\stackrel{⌢}{\mathrm{W}\mathrm{X}}+\stackrel{⌢}{\mathrm{Y}\mathrm{Z}}}{2}\end{array}$

Hence, proved.