# Question Video: Finding the Measure of an Inscribed Angle given a Diameter of the Circle and the Measure of Another Inscribed Angle Mathematics • 11th Grade

Given that πβ πΆπ΄π΅ = 64Β°, find πβ πΆπ΅π·.

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### Video Transcript

Given that the measure of angle πΆπ΄π΅ is 64 degrees, find the measure of angle πΆπ΅π·.

And then weβve been given a circle with center π and four points π΄, π΅, πΆ, π· on the circumference of that circle. Weβre told that the measure of angle πΆπ΄π΅ is 64 degrees. So we add that to our circle. Similarly, weβre trying to find the measure of angle πΆπ΅π·. Thatβs this angle here.

Now, we might notice that angle πΆπ΅π· is the sum of two individual angles πΆπ΅π΄ and π΄π΅π·. So if we can find the measure of angles πΆπ΅π΄ and π΄π΅π· individually, their sum will tell us the angle πΆπ΅π·. Well, the key to finding these two individual angles is to spot that the line segment π΅π΄ passes through point π, the center of the circle. So π΅π΄ is a diameter of this circle.

Then, we can use a special version of the inscribed angle theorem to calculate the measure of angle π΄πΆπ΅ and π΅π·π΄. This tells us that angles subtended by the diameter are equal to 90 degrees. Since angles π΅πΆπ΄ and π΅π·π΄ are subtended by the diameter, they are both equal to 90 degrees. Then, we might notice that π΄π΅πΆ and π΄π΅π· are triangles. And angles in a triangle sum to 180 degrees. So letβs define angle πΆπ΅π΄ to be π₯ degrees and π΄π΅π· to be π¦ degrees. Thinking purely about triangle π΄π΅πΆ, we know that π₯ plus 90 plus 64 is the interior angle sum. And then we know that this is equal to 180. This sum simplifies to π₯ plus 154. So we can say that π₯ plus 154 equals 180. Then, we subtract 154 from both sides. So π₯ is equal to 26 degrees.

With that in mind, now letβs perform a similar process to find the value of π¦. To find the value of π¦, we spot that arc π΅π· is congruent to arc π·π΄. This means line segments π΅π· and π·π΄ are also congruent to one another. The triangle therefore is isosceles, and so angle π΅π΄π· is also equal to π¦.

We can therefore say that 90 plus two π¦ equals 180. Subtracting 90 degrees from both sides, we get two π¦ equals 90. And then we divide through by two. So π¦ is equal to 45 degrees. We now know the measure of angles πΆπ΅π΄ and π΄π΅π·. So the measure of angle πΆπ΅π·, which is the sum of these, is 26 plus 45. And thatβs 71 degrees. So the measure of angle πΆπ΅π· is 71 degrees.