Question Video: Using the Relationship Between Parallel Chords to Determine the Measure of an Arc Expressed Algebraically Mathematics

Video Transcript

In the following figure, 𝐴𝐵 and 𝐸𝐹 are two equal chords. 𝐵𝐶 and 𝐹𝐸 are two parallel chords. If the measure of arc 𝐴𝐶 is 120 degrees, find the measure of arc 𝐶𝐸.

Let’s begin by using the fact that these two line segments 𝐴𝐵 and 𝐸𝐹 are two equal chords. Since they’re equal in length, we can deduce that the measure of their arcs must also be equal. So the measure of arc 𝐴𝐵 must be equal to the measure of arc 𝐸𝐹. In fact, we’re told that this is equal to 𝑥 degrees. Then, we use the information about 𝐵𝐶 and 𝐹𝐸; they’re parallel chords. This means that the measures of the arcs between those two chords is equal. That is, the measure of arc 𝐶𝐸 must be equal to the measure of arc 𝐵𝐹. And this time we’re also told that that is equal to 𝑥 plus 30 degrees.

Using this information alongside the measure of arc 𝐴𝐶, we know that the sum of all the arc measures is 360 degrees. So we can form and solve an equation. The sum of the arcs is 𝑥 plus 𝑥 plus 30 plus 𝑥 plus 𝑥 plus 30 plus 120. And that must be equal to 360. And so that left-hand side simplifies to four 𝑥 plus 180. So four 𝑥 plus 180 degrees equals 360. We can therefore say that four 𝑥 must be equal to 180. And we can then solve for 𝑥 by dividing through by four. So 𝑥 degrees equals 45 degrees. We want to find the measure of arc 𝐶𝐸, and we said that that was equal to 𝑥 plus 30. So the measure of arc 𝐶𝐸 is 45 plus 30, which is equal to 75 degrees. The measure of arc 𝐶𝐸 then is 75 degrees.