Question Video: Using Circle Theorems to Calculate Unknown Angles Mathematics

In the figure, 𝑂 is the center and π‘šβˆ π‘‚π΄π΅ = 59.5Β°β€Ž. What is the measure of angle 𝐴𝑂𝐡? What is the measure of angle 𝐴𝐢𝐡?

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Video Transcript

In the figure, 𝑂 is the center and the measure of angle 𝑂𝐴𝐡 equals 59.5 degrees. What is the measure of angle 𝐴𝑂𝐡? What is the measure of angle 𝐴𝐢𝐡?

We’re told that 𝑂 is the center of this circle and that the measure of angle 𝑂𝐴𝐡 is 59.5 degrees. We want to find the measure of angle 𝐴𝑂𝐡 and the measure of 𝐴𝐢𝐡. We see that the points 𝐴, 𝑂, and 𝐡 form a triangle. Both line segment 𝑂𝐡 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 𝑂𝐡. And it will mean that triangle 𝐴𝑂𝐡 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 𝐴𝐡𝑂 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 𝑂𝐴𝐡 and angle 𝐴𝐡𝑂. We add the two angles we know, and we get 119 degrees. And then to solve for angle 𝐴𝑂𝐡, we subtract 119 degrees from both sides, and we find that angle 𝐴𝑂𝐡 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, 𝐡, which means they’re both subtended by the arc 𝐴𝐡. But we need to make a clarification here. Angle 𝐴𝑂𝐡 is a central angle that is subtended by arc 𝐴𝐡, while angle 𝐴𝐢𝐡 is an inscribed angle subtended by arc 𝐴𝐡. 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 π‘Ž, the inscribed angle subtended by the same points will be π‘Ž degrees.

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

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