### Video Transcript

In the figure, π is the center and
the measure of angle ππ΄π΅ equals 59.5 degrees. What is the measure of angle
π΄ππ΅? What is the measure of angle
π΄πΆπ΅?

Weβre told that π is the center of
this circle and that the measure of angle ππ΄π΅ is 59.5 degrees. We want to find the measure of
angle π΄ππ΅ and the measure of π΄πΆπ΅. We see that the points π΄, π, and
π΅ form a triangle. Both line segment ππ΅ and line
segment ππ΄ 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 ππ΄ is equal to line segment ππ΅. And it will mean that triangle
π΄ππ΅ 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 ππ΄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΄ππ΅,
we subtract 119 degrees from both sides, and we find that angle π΄ππ΅ is equal to
61 degrees. Thatβs 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 theyβre both subtended by the arc π΄π΅. But we need to make a clarification
here. Angle π΄ππ΅ is a central angle
that is subtended by arc π΄π΅, while angle π΄πΆπ΅ 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΄πΆπ΅ will be equal to one-half the measure of angle π΄ππ΅. So, we plug in 61 degrees for angle
π΄ππ΅. Half of 61 degrees is 30.5
degrees. And so, the measure of angle π΄πΆπ΅
equals 30.5 degrees.