### Video Transcript

In the given figure, the diameter of circle π is line segment π΄π΅. If the measure of angle π΄π΅π· is 58 degrees, what is the measure of angle π·πΈπ΅?

Letβs start by adding the information we know. We know that π΄π΅ is a diameter. And the measure of angle π΄π΅π· is 58 degrees. We can add that 58 degrees to our diagram. And we want to know what the value of π·πΈπ΅ is. At first, it doesnβt seem like thereβs a direct way to answer that question. So, weβll need to fill in more of the values inside the circle.

Because we know that line segment π΄π΅ is a diameter, we know something about the inscribed angle whose endpoints fall on the diameter. That means the measure of angle π΅π·π΄ is 90 degrees. We know that the arc created between π΄ and π΅ is 180 degrees and that the subtended angle from this arc will be half that value, 90 degrees. Weβre still not quite there in finding the measure of angle π·πΈπ΅, but we are at a place where we could find the measure of angle π·π΄π΅.

The endpoints π·, π΅, and π΄ form a right triangle. And that means all three of these angles must add up to 180 degrees. If we plug in the values we do know, we can solve for that third angle in the triangle. 148 degrees plus the measure of angle π·π΄π΅ equals 180 degrees. So, we subtract 148 from both sides. And we get that the measure of angle π·π΄π΅ is 32 degrees.

And this is where we notice something about the angle π·π΄π΅. The arc associated with angle π·π΄π΅ is the arc π·π΅. Our missing angle, angle π·πΈπ΅, also has these two points as endpoints. We can write that as angle π·π΄π΅ and angle π·πΈπ΅ intercept the same arc, π·π΅. Therefore, the measure of angle π·πΈπ΅ will be equal to the measure of angle π·π΄π΅, which is 32 degrees. Our unknown angle, angle π·πΈπ΅, has a measure of 32 degrees.