Question

what is the measure of angle oac if major arc ab measures 220 degrees? 55° 70° 110° 140°

Explanation:

Step1: Find measure of minor arc AB

The sum of major and minor arcs of a circle is 360°. So, minor arc $AB=360 - 220=140^{\circ}$.

Step2: Find central - angle AOB

The measure of a central - angle is equal to the measure of the arc it subtends. So, $\angle AOB = 140^{\circ}$.

Step3: Consider triangle AOC

Since $OA = OC$ (radii of the same circle), $\triangle AOC$ is isosceles. Let $\angle OAC=\angle OCA = x$.

Step4: Use angle - sum property of a triangle

In $\triangle AOC$, we know that the sum of interior angles of a triangle is 180°. Also, $\angle AOC$ is half of the measure of minor arc $AB$ (the central - angle is twice the inscribed - angle subtended by the same arc when the inscribed - angle is on the circle and the central - angle is at the center). So, $\angle AOC = 70^{\circ}$. Then, in $\triangle AOC$, $x + x+\angle AOC=180^{\circ}$, $2x+70^{\circ}=180^{\circ}$, $2x = 110^{\circ}$, $x = 55^{\circ}$.

Answer:

$55^{\circ}$