Question

in the figure below, o is the center of the circle. find each measurement below. a. m∠aco b. m\\(widehat{ac}\\)

Explanation:

Step1: Recall the central - inscribed angle relationship

The measure of an inscribed angle is half the measure of the central angle that subtends the same arc.

Step2: Identify the central angle corresponding to arc $\overset{\frown}{AC}$

The central angle corresponding to arc $\overset{\frown}{AC}$ is $\angle AOC$. In $\triangle AOC$, $OA = OC$ (radii of the same circle), so $\angle OAC=\angle ACO = 32^{\circ}$. Then, using the angle - sum property of a triangle ($\angle AOC+ \angle OAC+\angle ACO = 180^{\circ}$), we find $\angle AOC=180^{\circ}-32^{\circ}-32^{\circ}=116^{\circ}$.

Step3: Find the measure of arc $\overset{\frown}{AC}$

The measure of an arc is equal to the measure of its central angle. So $m\overset{\frown}{AC}=116^{\circ}$.

Answer:

b. $116$