# Question Video: Using Circle Theorems to Calculate Unknown Angles Mathematics

In the figure, 𝑂 is the center and 𝑚∠𝑂𝐴𝐵 = 59.5°‎. What is the measure of angle 𝐴𝑂𝐵? What is the measure of angle 𝐴𝐶𝐵?

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### Video Transcript

In the figure, 𝑂 is the center and the measure of angle 𝑂𝐴𝐵 equals 59.5 degrees. What is the measure of angle 𝐴𝑂𝐵? What is the measure of angle 𝐴𝐶𝐵?

We’re told that 𝑂 is the center of this circle and that the measure of angle 𝑂𝐴𝐵 is 59.5 degrees. We want to find the measure of angle 𝐴𝑂𝐵 and the measure of 𝐴𝐶𝐵. We see that the points 𝐴, 𝑂, and 𝐵 form a triangle. Both line segment 𝑂𝐵 and line segment 𝑂𝐴 are radii of this circle because any line drawn from the center of the circle to the circumference of the circle will be a radius. This means we can say that line segment 𝑂𝐴 is equal to line segment 𝑂𝐵. And it will mean that triangle 𝐴𝑂𝐵 is an isosceles triangle.

In an isosceles triangle, the two angles opposite the radii are equal to each other. And that means we could say that angle 𝐴𝐵𝑂 is also equal to 59.5 degrees. Since these three angles form a triangle, they must sum to 180 degrees. And so, we substitute the values we do know for angle 𝑂𝐴𝐵 and angle 𝐴𝐵𝑂. We add the two angles we know, and we get 119 degrees. And then to solve for angle 𝐴𝑂𝐵, we subtract 119 degrees from both sides, and we find that angle 𝐴𝑂𝐵 is equal to 61 degrees. That’s the answer to part one.

Part two is a little bit less straightforward. We notice that both of these angles share the endpoints 𝐴, 𝐵, which means they’re both subtended by the arc 𝐴𝐵. But we need to make a clarification here. Angle 𝐴𝑂𝐵 is a central angle that is subtended by arc 𝐴𝐵, while angle 𝐴𝐶𝐵 is an inscribed angle subtended by arc 𝐴𝐵. And we remember that the central angle subtended by two points on a circle is twice the inscribed angle subtended by those two points. We might see it represented something like this: if the central angle measures two 𝑎, the inscribed angle subtended by the same points will be 𝑎 degrees.

Based on that, we can say that the measure of angle 𝐴𝐶𝐵 will be equal to one-half the measure of angle 𝐴𝑂𝐵. So, we plug in 61 degrees for angle 𝐴𝑂𝐵. Half of 61 degrees is 30.5 degrees. And so, the measure of angle 𝐴𝐶𝐵 equals 30.5 degrees.