### Video Transcript

Given that line segment π΄π΅ is a diameter of the circle and line segment π·πΆ is parallel to line segment π΄π΅, find the measure of angle π΄πΈπ·.

The first thing when given a question such as this is to establish exactly which angle weβre trying to calculate. So, letβs mark on angle π΄πΈπ· here in pink in the circle. In this circle, we have a typical bowβtie type pattern, which usually indicates angles subtended by the same arc. So, if we use this arc π΄π·, then the angle π΄πΆπ· is going to be congruent to the angle π΄πΈπ· because we know that angles subtended by the same arc are congruent.

Now, we still canβt calculate angle π΄πΆπ· either, so letβs see if there are any other angles we could work out. Letβs use the fact that we are given that the line segment π΄π΅ 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 π΄πΆπ΅ must be 90 degrees. We might then notice that we have the triangle π΄πΆπ΅, and we know two of the angle measures. Therefore, we can work out the third angle measure of π΅π΄πΆ. By using the fact that the interior angle measures in a triangle sum to 180 degrees, we have that the measure of angle π΅π΄πΆ plus the measure of angle π΄πΆπ΅ plus the measure of angle π΄π΅πΆ 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 π΅π΄πΆ is 21.5 degrees. So, are we any closer to actually calculating the measure of angle π΄πΈπ· that we need to do in this question? Well, letβs observe that we have two parallel lines π·πΆ and π΄π΅. The line segment π΄πΆ 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 π΄πΆπ· must be equal to the measure of angle π΅π΄πΆ. 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 π΄πΈπ· must be equal to the measure of angle π΄πΆπ·.

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 π΄πΈπ· is 21.5 degrees.