Question Video: Finding the Measure of an Inscribed Angle given the Measure of Another Inscribed Angle and the Measure of an Arc Mathematics

Video Transcript

Find the measure of angle 𝐷𝐶𝐵.

Angle 𝐷𝐶𝐵 is this angle in the triangle. And we notice that we’ve been given the measure of angle 𝐵𝐷𝐴 and a second bit of information. There’s a 60 degree angle here. Now in fact, this angle is the measure of the arc 𝐴𝐷. And we know that the measure of the arc is the angle it makes at the center of the circle. So defining the center of our circle to be equal to 𝑀, angle 𝐴𝑀𝐷 must be equal to 60 degrees.

And in fact, there’s a number of extra angles that subtend arc 𝐴𝐷. The first is angle 𝐴𝐶𝐷. We can use the fact that these angles subtend the same arc to find the measure of 𝐴𝐶𝐷. The inscribed angle theorem tells us that the inscribed angle is half the measure of the central angle that subtends the same arc. So the measure of angle 𝐴𝐶𝐷 must be equal to a half the measure of angle 𝐴𝑀𝐷. That’s a half of 60 degrees, which is equal to 30 degrees.

So we found a portion of angle 𝐷𝐶𝐵. But we’re not quite finished. We still need to work out the other bit. To do so, we note that we have a pair of inscribed angles that subtend a second arc. That’s arc 𝐵𝐴. If the inscribed angle is half the measure of the central angle that subtends the same arc, then if we have two inscribed angles that subtend the same arc, they must both be half the measure of the central angle. And therefore they must themselves be equal to one another.

In other words, the angle that we’re trying to find, the measure of angle 𝐵𝐶𝐴, must be equal to the measure of 𝐵𝐷𝐴. And that’s 52 degrees. Then the measure of angle 𝐷𝐶𝐵 is the sum of the measure of angle 𝐴𝐶𝐷 and 𝐵𝐶𝐴. It’s 52 plus 30, which is equal to 82 degrees. And so, the measure of angle 𝐷𝐶𝐵 is 82 degrees.