Chapter 10.6, Problem 12PSA

To determine

Find the value for the given arc angle using diagram.

Answer to Problem 12PSA

The value of the given arc angle $m\text{arc}CD={122}^{°}$ .

Explanation of Solution

Given:

The given diagram is

  

$M$ is the midpoint of the arc $AB$ . Given arc angle is $marcCD$ .

Calculation:

In diagram, $M$ is the midpoint of the arc $AB$ .

  $m\angle MDA=m\angle MCB$

  $\begin{array}{l}3x-31=x+7\\ \Rightarrow 3x-x-31=x+7-x\\ \Rightarrow 2x-31=7\\ \Rightarrow 2x-31+31=7+31\\ \Rightarrow 2x=38\\ \Rightarrow \frac{2}{2}x=\frac{38}{2}\\ \Rightarrow x={19}^{°}\end{array}$

Now,

  $m\angle CMD=4x-15=4\cdot 19-15=76-15={61}^{°}$

Use the inscribed angle theorem $\to$ the measure of an inscribed angle is half the measure of its intercepted arc.

Hence the value of the given arc angle $m\text{arc}CD={122}^{°}$ .