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

In the figure, π is the center and πβ ππ΄π΅ = 59.5Β°β. What is the measure of angle π΄ππ΅? What is the measure of angle π΄πΆπ΅?

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### Video Transcript

In the figure, π is the center and the measure of angle ππ΄π΅ equals 59.5 degrees. What is the measure of angle π΄ππ΅? What is the measure of angle π΄πΆπ΅?

Weβre told that π is the center of this circle and that the measure of angle ππ΄π΅ is 59.5 degrees. We want to find the measure of angle π΄ππ΅ and the measure of π΄πΆπ΅. We see that the points π΄, π, and π΅ form a triangle. Both line segment ππ΅ and line segment ππ΄ 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 ππ΄ is equal to line segment ππ΅. And it will mean that triangle π΄ππ΅ 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 π΄π΅π 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 ππ΄π΅ and angle π΄π΅π. We add the two angles we know, and we get 119 degrees. And then to solve for angle π΄ππ΅, we subtract 119 degrees from both sides, and we find that angle π΄ππ΅ is equal to 61 degrees. Thatβs the answer to part one.

Part two is a little bit less straightforward. We notice that both of these angles share the endpoints π΄, π΅, which means theyβre both subtended by the arc π΄π΅. But we need to make a clarification here. Angle π΄ππ΅ is a central angle that is subtended by arc π΄π΅, while angle π΄πΆπ΅ is an inscribed angle subtended by arc π΄π΅. 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, the inscribed angle subtended by the same points will be π degrees.

Based on that, we can say that the measure of angle π΄πΆπ΅ will be equal to one-half the measure of angle π΄ππ΅. So, we plug in 61 degrees for angle π΄ππ΅. Half of 61 degrees is 30.5 degrees. And so, the measure of angle π΄πΆπ΅ equals 30.5 degrees.