# Question Video: Finding the Measure of an Inscribed Angle given That the Diameter and a Chord Are Parallel Mathematics

Given that line segment π΄π΅ is a diameter of the circle and line segment π·πΆ β₯ line segment π΄π΅, find πβ π΄πΈπ·.

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### Video Transcript

Given that line segment π΄π΅ is a diameter of the circle and line segment π·πΆ is parallel to line segment π΄π΅, find the measure of angle π΄πΈπ·.

The first thing when given a question such as this is to establish exactly which angle weβre trying to calculate. So, letβs mark on angle π΄πΈπ· here in pink in the circle. In this circle, we have a typical bowβtie type pattern, which usually indicates angles subtended by the same arc. So, if we use this arc π΄π·, then the angle π΄πΆπ· is going to be congruent to the angle π΄πΈπ· because we know that angles subtended by the same arc are congruent.

Now, we still canβt calculate angle π΄πΆπ· either, so letβs see if there are any other angles we could work out. Letβs 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΄πΆπ΅ must be 90 degrees. We might then notice that we have the triangle π΄πΆπ΅, and we know two of the angle measures. Therefore, we can work out the third angle measure of π΅π΄πΆ. 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΄πΆ plus the measure of angle π΄πΆπ΅ plus the measure of angle π΄π΅πΆ 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΄πΆ is 21.5 degrees. So, are we any closer to actually calculating the measure of angle π΄πΈπ· that we need to do in this question? Well, letβs observe that we have two parallel lines π·πΆ and π΄π΅. The line segment π΄πΆ 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΄πΆπ· must be equal to the measure of angle π΅π΄πΆ. 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΄πΈπ· must be equal to the measure of angle π΄πΆπ·.

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΄πΈπ· is 21.5 degrees.