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

Video Transcript

In the given circle with center 𝑀, chords 𝐴𝐵 and 𝐶𝐷 are parallel and the measure of angle 𝐵𝑀𝐷 is equal to 74 degrees. Find the measure of angle 𝐴𝐸𝐶.

To answer this question, we’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 𝐴𝐵 and 𝐶𝐷 are parallel, this means that the measures of arcs 𝐵𝐷 and 𝐴𝐶 are equal.

The second theorem we can use to find the measure of angle 𝐴𝐸𝐶 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 𝐵𝐷 must be half the measure of the angle subtended by 𝐵𝐷 at the center. If we call a point on the circumference 𝐹, then the measure of angle 𝐵𝐹𝐷 is then one-half the measure of angle 𝐵𝑀𝐷. That’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 we’re going to apply this to our circle is by using the fact we noted earlier from the first theorem that arcs 𝐵𝐷 and 𝐴𝐶 are equal. This being the case, any angle subtended by arc 𝐵𝐷 at the circumference will be equal in measure to any angle at the circumference subtended by arc 𝐴𝐶. So the measure of angle 𝐵𝐹𝐷 we found earlier using the second theorem will be equal to the measure of angle 𝐴𝐸𝐶, since angle 𝐴𝐸𝐶 is subtended by arc 𝐴𝐶. And that’s 37 degrees.

Hence, if chords 𝐴𝐵 and 𝐶𝐷 are parallel and the measure of angle 𝐵𝑀𝐷 is 74 degrees, then the measure of angle 𝐴𝐸𝐶 is equal to 37 degrees.