In which of these figures will line segment 𝐴𝐵 be parallel to line segment 𝐶𝐷?
In all of these images, segment 𝐴𝐵 and segment 𝐶𝐷 are chords inside a circle. And we know that the measures of the arcs between parallel chords of a circle are equal. To find which of these figures has parallel chords 𝐴𝐵 and 𝐶𝐷, we’ll need to find the place where the arcs between the two chords are equal in measure. Notice that in all five of these images we’re given three of the four arc measures. So we need to identify the measure of these missing arcs and then compare that measure to the other arc between those two chords.
We know that all the arcs in a circle sum to 360 degrees. And that means our missing arc 𝐵𝐷 in option (A) will be equal to 360 degrees minus 55 degrees plus 65 degrees plus 185 degrees, which equals 55 degrees. This means arc 𝐵𝐷 is not equal in measure to arc 𝐴𝐶. And therefore these chords are not parallel. In figure (B), we have 360 degrees minus 55 degrees plus 65 degrees plus 175 degrees, which equals 65 degrees. Here, arc 𝐵𝐷 is equal in measure to arc 𝐴𝐶. And this makes the chords 𝐴𝐵 and 𝐶𝐷 parallel. So we’ll select option (B). But let’s go ahead and double-check our remaining three missing arcs.
In option (C), arc 𝐵𝐷 is 64 degrees, which means chords 𝐴𝐵 and 𝐶𝐷 are not parallel. In option (D), the arc measure of 𝐵𝐷 is 70 degrees, which is not equal to the measure of arc 𝐴𝐶. And in our final example, arc 𝐵𝐷 measures 55 degrees, which again means that the chords cannot be parallel. By showing that the measure of arc 𝐴𝐶 was equal to the measure of arc 𝐵𝐷 in option (D), we were able to prove that the line segments 𝐴𝐵 and 𝐶𝐷 were parallel by the measure of arcs between parallel chords theorem.