In the given figure, the diameter of circle 𝑀 is line segment 𝐴𝐵. If the measure of angle 𝐴𝐵𝐷 is 58 degrees, what is the measure of angle 𝐷𝐸𝐵?
Let’s start by adding the information we know. We know that 𝐴𝐵 is a diameter. And the measure of angle 𝐴𝐵𝐷 is 58 degrees. We can add that 58 degrees to our diagram. And we want to know what the value of 𝐷𝐸𝐵 is. At first, it doesn’t seem like there’s a direct way to answer that question. So, we’ll need to fill in more of the values inside the circle.
Because we know that line segment 𝐴𝐵 is a diameter, we know something about the inscribed angle whose endpoints fall on the diameter. That means the measure of angle 𝐵𝐷𝐴 is 90 degrees. We know that the arc created between 𝐴 and 𝐵 is 180 degrees and that the subtended angle from this arc will be half that value, 90 degrees. We’re still not quite there in finding the measure of angle 𝐷𝐸𝐵, but we are at a place where we could find the measure of angle 𝐷𝐴𝐵.
The endpoints 𝐷, 𝐵, and 𝐴 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 𝐷𝐴𝐵 equals 180 degrees. So, we subtract 148 from both sides. And we get that the measure of angle 𝐷𝐴𝐵 is 32 degrees.
And this is where we notice something about the angle 𝐷𝐴𝐵. The arc associated with angle 𝐷𝐴𝐵 is the arc 𝐷𝐵. Our missing angle, angle 𝐷𝐸𝐵, also has these two points as endpoints. We can write that as angle 𝐷𝐴𝐵 and angle 𝐷𝐸𝐵 intercept the same arc, 𝐷𝐵. Therefore, the measure of angle 𝐷𝐸𝐵 will be equal to the measure of angle 𝐷𝐴𝐵, which is 32 degrees. Our unknown angle, angle 𝐷𝐸𝐵, has a measure of 32 degrees.