Question Video: Finding the Measure of an Angle Subtended by an Arc in a Circle with Parallel Chords | Nagwa Question Video: Finding the Measure of an Angle Subtended by an Arc in a Circle with Parallel Chords | Nagwa

Question Video: Finding the Measure of an Angle Subtended by an Arc in a Circle with Parallel Chords Mathematics • Third Year of Preparatory School

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In the given circle with center π, chords π΄π΅β₯πΆπ· πβ π΅ππ· = 74Β°. Find πβ π΄πΈπΆ.

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Video Transcript

In the given circle with center π, chords π΄π΅ and πΆπ· are parallel and the measure of angle π΅ππ· is equal to 74 degrees. Find the measure of angle π΄πΈπΆ.

To answer this question, weβre going to use three theorems about circles.

The first tells us that the measures of the arcs between two parallel chords are equal. Applied to our circle, since chords π΄π΅ and πΆπ· are parallel, this means that the measures of arcs π΅π· and π΄πΆ are equal.

The second theorem we can use to find the measure of angle π΄πΈπΆ tells us that when two angles are subtended by the same arc, the measure of the angle at the center of the circle is twice the measure of the angle at the circumference. Applied to our circle, this means that any angle at the circumference subtended by the arc π΅π· must be half the measure of the angle subtended by π΅π· at the center. If we call a point on the circumference πΉ, then the measure of angle π΅πΉπ· is then one-half the measure of angle π΅ππ·. Thatβs one-half of 74 degrees, which is 37 degrees.

Our third theorem is that angles subtended by the same arc at the circumference have equal measure. Now how weβre going to apply this to our circle is by using the fact we noted earlier from the first theorem that arcs π΅π· and π΄πΆ are equal. This being the case, any angle subtended by arc π΅π· at the circumference will be equal in measure to any angle at the circumference subtended by arc π΄πΆ. So the measure of angle π΅πΉπ· we found earlier using the second theorem will be equal to the measure of angle π΄πΈπΆ, since angle π΄πΈπΆ is subtended by arc π΄πΆ. And thatβs 37 degrees.

Hence, if chords π΄π΅ and πΆπ· are parallel and the measure of angle π΅ππ· is 74 degrees, then the measure of angle π΄πΈπΆ is equal to 37 degrees.

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