### Video Transcript

What is the measure of angle ππ΄π΅?

In this figure, we can notice that the angle ππ΄π΅ appears here in the triangle. It might not immediately be obvious how weβre going to calculate this angle measure because we do have quite a few unknown angle measures. Letβs begin by seeing if we can calculate the acute angle of π΄ππ΅. To do this, weβll use the fact that the angle measures about a point sum to 360 degrees. So that means that this acute angle measure of π΄ππ΅ must be equal to 360 degrees subtract 306 degrees, which leaves us with 54 degrees.

Next, we can use the fact that we have this triangle π΄ππ΅ and the fact that the interior angles in a triangle add up to 180 degrees. However, we do have a bit of a problem because we donβt actually know the angle measure of angle π΄π΅π. We can observe though that the line segments π΄π and π΅π are both radii of the circle. And that means that they will be the same length. We can therefore say that triangle π΄π΅π must be an isosceles triangle, which has two equal sides and two equal angle measures.

The two angles which will have equal angle measures are angle ππ΄π΅ and angle ππ΅π΄. Letβs define these both as π₯ degrees. We know that these three angle measures will add to give 180 degrees. We can therefore say that 54 degrees plus two π₯ degrees is equal to 180 degrees. Subtracting 54 degrees from both sides, we have two π₯ degrees is equal to 126 degrees. Dividing both sides of the equation, we have that π₯ degrees is equal to 63 degrees. Since we defined the measure of angle ππ΄π΅ as π₯ degrees, then we can give the answer that this angle measure is 63 degrees.