Given that line segment 𝐴𝐵 is a diameter of the circle and line segment 𝐷𝐶 is parallel to line segment 𝐴𝐵, find the measure of angle 𝐴𝐸𝐷.
The first thing when given a question such as this is to establish exactly which angle we’re trying to calculate. So, let’s mark on angle 𝐴𝐸𝐷 here in pink in the circle. In this circle, we have a typical bow–tie type pattern, which usually indicates angles subtended by the same arc. So, if we use this arc 𝐴𝐷, then the angle 𝐴𝐶𝐷 is going to be congruent to the angle 𝐴𝐸𝐷 because we know that angles subtended by the same arc are congruent.
Now, we still can’t calculate angle 𝐴𝐶𝐷 either, so let’s see if there are any other angles we could work out. Let’s use the fact that we are given that the line segment 𝐴𝐵 is a diameter of the circle. So, by using the property that the angle inscribed in a semicircle is 90 degrees, we can work out that the angle 𝐴𝐶𝐵 must be 90 degrees. We might then notice that we have the triangle 𝐴𝐶𝐵, and we know two of the angle measures. Therefore, we can work out the third angle measure of 𝐵𝐴𝐶. By using the fact that the interior angle measures in a triangle sum to 180 degrees, we have that the measure of angle 𝐵𝐴𝐶 plus the measure of angle 𝐴𝐶𝐵 plus the measure of angle 𝐴𝐵𝐶 equals 180 degrees.
The two known angle measures of 90 degrees and 68.5 degrees will add to 158.5 degrees. So, by subtracting this value from 180 degrees, we have that the measure of angle 𝐵𝐴𝐶 is 21.5 degrees. So, are we any closer to actually calculating the measure of angle 𝐴𝐸𝐷 that we need to do in this question? Well, let’s observe that we have two parallel lines 𝐷𝐶 and 𝐴𝐵. The line segment 𝐴𝐶 is a transversal of these two parallel lines. This means that we can find a pair of alternate angles. So, in fact, the measure of angle 𝐴𝐶𝐷 must be equal to the measure of angle 𝐵𝐴𝐶. Then, as previously mentioned, we use the fact that angles subtended by the same arc are equal to work out that the measure of angle 𝐴𝐸𝐷 must be equal to the measure of angle 𝐴𝐶𝐷.
We now know that both of these angle measures will be 21.5 degrees. Therefore, we can give the answer that the measure of angle 𝐴𝐸𝐷 is 21.5 degrees.