Given that the measure of angle 𝐴𝑀𝐷 equals 41.5 degrees, find the measure of angle 𝐵 in degrees.
And then we have a diagram showing a circle, center 𝑀, with four points on its circumference 𝐴, 𝐵, 𝐶, and 𝐷. We’re given that the measure of angle 𝐴𝑀𝐷 is 41.5 degrees. So, let’s add that to our diagram. Angle 𝐴𝑀𝐷 lies at the center of the circle as shown. This angle is said to subtend the arc 𝐷𝐴. Now, in fact, there’s an arc that’s equal in length to arc 𝐷𝐴. It’s arc 𝐶𝐷 as represented by these dashed lines. And since the arc lengths are equal, we can say that the measure of arc 𝐷𝐴 must be equal to the measure of arc 𝐶𝐷. But of course, the measure of an arc is simply the angle that that arc makes at the center of the circle. So, the measure of arc 𝐷𝐴 is 41.5 degrees, meaning that the measure of arc 𝐶𝐷 must also be 41.5 degrees.
We can now add that angle to our diagram too. And in fact, this is really useful because we can now calculate the measure of angle 𝐶𝑀𝐴. It’s 41.5 plus 41.5, which is 83 degrees. Now, we’re trying to find the measure of angle 𝐵 in degrees, and we now have everything we need to do so. We can use the inscribed angle theorem that links the angle at the center of the circle with the inscribed angle at its circumference. This theorem tells us that an inscribed angle is going to be equal to half of the central angle that subtends the same arc on that circle. The inscribed angle at 𝐵 subtends arc 𝐶𝐴, as does the angle at the center of the circle 𝐶𝑀𝐴. So, the measure of angle 𝐵 must be a half of the measure of angle 𝐶𝑀𝐴. In other words, it’s a half of 83 degrees, which is 41.5 degrees.
The measure of angle 𝐵 then is 41.5 degrees.