Chapter 9.6, Problem 12WE

To determine

To find: find the measure of numbered angle

Answer to Problem 12WE

The measure of angle $\angle$ 1 is 65

Explanation of Solution

Given information: we have given $m\stackrel{⌢}{RU}=130$ , $m\stackrel{⌢}{TS}=100$ and the figure of circle

  

We can see from the figure

  $\begin{array}{c}m\stackrel{⌢}{RT}=180-m\stackrel{⌢}{RU}\\ m\stackrel{⌢}{RT}=180-130\\ m\stackrel{⌢}{RT}=50\end{array}$

Also

  $\begin{array}{c}m\stackrel{⌢}{US}=180-m\stackrel{⌢}{TS}\\ m\stackrel{⌢}{US}=180-100\\ m\stackrel{⌢}{US}=80\end{array}$

As we know

The measure of an angle formed by the two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs.

Therefore from given figure we must have

  $\begin{array}{c}m\angle 1=\frac{1}{2}\left(m\stackrel{⌢}{RT}+m\stackrel{⌢}{US}\right)\\ m\angle 1=\frac{1}{2}\left(50+80\right)\\ m\angle 1=\frac{130}{2}\\ m\angle 1=65\end{array}$

Therefore, the measure of the angle $\angle$ 1 is 65