Chapter 6.CT, Problem 1CT

a) If $m\stackrel{⌢}{AB}=88°$, then $m\stackrel{⌢}{ACB}=$____________.

b) If $m\stackrel{⌢}{AB}=92°$ and C is the midpoint of major are ACB, then $m\stackrel{⌢}{AC}=$_____________.

To determine

(a)

To find:

The measure of $m\stackrel{⌢}{ACB}=$

Answer to Problem 1CT

Solution:

$m\stackrel{⌢}{ACB}=272°$.

Explanation of Solution

Given:

$m\stackrel{⌢}{AB}=88°$

Approach:

The degree measured around a circle is $360°$.

Calculation:

From the above circle we have,

$\begin{array}{l}m\stackrel{⌢}{ACB}+m\stackrel{⌢}{AB}=360°\\ m\stackrel{⌢}{ACB}+88°=360°\\ m\stackrel{⌢}{ACB}=360°-88°\\ m\stackrel{⌢}{ACB}=272°\end{array}$

To determine

(b)

To find:

The measure of $m\stackrel{⌢}{AC}.$

Answer to Problem 1CT

Solution:

$m\stackrel{⌢}{AC}=134°$.

Explanation of Solution

Given:

$m\stackrel{⌢}{AB}=92°$.

C is the midpoint of major are ACB.

Approach:

The degree measured around a circle is $360°$.

Calculation:

From the above circle we have,

$\begin{array}{l}m\stackrel{⌢}{ACB}+m\stackrel{⌢}{AB}=360°\\ m\stackrel{⌢}{ACB}+92°=360°\\ m\stackrel{⌢}{ACB}=360°-92°\\ m\stackrel{⌢}{ACB}=268°\end{array}$

Now to find the measure of $m\stackrel{⌢}{AC}.$

It is given that C is the midpoint of major are ACB.

C is the midpoint of the arc ACB, then it cuts the arc into two congruent parts.

Thus, measure of the arc is $m\stackrel{⌢}{AC}=\frac{m\stackrel{⌢}{ACB}}{2}$.

$\begin{array}{l}m\stackrel{⌢}{AC}=\frac{268}{2}\\ m\stackrel{⌢}{AC}=134°\end{array}$