Video: Finding the Measure of an Angle given Its Arc’s Measure Using Its Central Angle and Another Inscribed Angle

Given that line segment 𝐴𝐡 is a diameter in circle 𝑀, and π‘šβˆ π΅π‘€π· = 59Β°β€Ž, find π‘šβˆ π΄πΆπ· in degrees.

02:13

Video Transcript

Given that line segment 𝐴𝐡 is a diameter in circle 𝑀 and the measure of angle 𝐡𝑀𝐷 equals 59 degrees, find the measure of angle 𝐴𝐢𝐷 in degrees.

Let’s put what we know into the diagram. Angle 𝐡𝑀𝐷 measures 59 degrees, and we’re trying to find the measure of angle 𝐴𝐢𝐷. If we start with what we know about angle 𝐡𝑀𝐷, since 𝐡𝑀𝐷 has a vertex at the center of the circle, 𝐡𝑀𝐷 is a central angle. And because angle 𝐡𝑀𝐷 is a central angle, its subtended arc, arc 𝐡𝐷, also measures 59 degrees. We’re also interested in angle 𝐴𝐢𝐷. But angle 𝐴𝐢𝐷 is not a central angle. It’s an inscribed angle because its vertex is on the circumference of the circle, as are both endpoints.

The arc associated with angle 𝐴𝐢𝐷 would be arc 𝐴𝐷. We have a partial measurement for this arc, but we’re missing the distance from 𝐴 to 𝐡. But because we know that 𝐴𝐡 is a diameter, it cuts the circle in half. And that means the measure of arc 𝐴𝐡 is 180 degrees. If arc 𝐴𝐡 equals 180 and arc 𝐡𝐷 equals 59 degrees, we can say the measure of arc 𝐴𝐷 is equal to the measure of arc 𝐴𝐡 plus the measure of arc 𝐡𝐷.

If we plug in what we know, the measure of arc 𝐴𝐷 is 239 degrees. Because angle 𝐴𝐢𝐷 is an inscribed angle and it has a subtended arc measure of 239 degrees, we can find out the exact measure of angle 𝐴𝐢𝐷. The measure of the inscribed angle 𝐴𝐢𝐷 will be one-half its subtended arc, arc 𝐴𝐷. Since that arc is 239 degrees, we take half of that and we get 119.5 degrees for the measure of angle 𝐴𝐢𝐷.

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