Question Video: Using Theories of Parallel Chords to Find the Measure of an Arc Mathematics

Video Transcript

In the given figure, if the measure of arc 𝐵𝐷 equals 65 degrees, find the measure of arc 𝐶𝐷.

In the diagram, we notice that we’ve been given a pair of parallel chords. That is, line segment 𝐴𝐵 is parallel to line segment 𝐶𝐷. And we recall that arcs formed by a pair of parallel chords are congruent. So arc 𝐴𝐶 and arc 𝐵𝐷 are congruent, which means the measures of these arcs must be equal. And so we see that the measure of arc 𝐴𝐶 is 65 degrees.

Next, we see that line segment 𝐴𝐵 in fact passes through the center of the circle. It must therefore be the diameter of the circle. And so it splits this circle exactly in half. And so we can say that the measure of both arcs 𝐴𝐵 are 180 degrees. Now, of course, we’re interested in the portion of the circle which passes through points 𝐶 and 𝐷. So we’ve called that the measure of arc 𝐴𝐶𝐷𝐵. The question wants us to find the measure of arc 𝐶𝐷. And we now know that the measure of all the individual arcs between 𝐴 and 𝐵 is 180 degrees. So we can say that the sum of the measure of arc 𝐴𝐶, the measure of arc 𝐶𝐷, and the measure of arc 𝐷𝐵 is 180. But remember, we said that arcs 𝐴𝐶 and 𝐷𝐵 are congruent and their measures are 65 degrees. So our equation becomes 65 degrees plus the measure of arc 𝐶𝐷 plus another 65 degrees equals 180. And then we simplify that left-hand side.

We can now solve this equation for the measure of arc 𝐶𝐷 by subtracting 130 from both sides. So it’s 180 minus 130, which is of course 50. So the measure of arc 𝐶𝐷 is 50 degrees.