Video Transcript
In the diagram, the measure of arc
𝐴𝐵 is equal to 62 degrees, the measure of arc 𝐵𝐶 equals 110 degrees, and the
measure of arc 𝐴𝐷 equals 126 degrees. What can we conclude about the line
segments 𝐴𝐷 and 𝐵𝐶? (A) They are parallel. (B) They are neither parallel nor
perpendicular. (C) They are perpendicular. (D) They’re the same length. (E) They are parallel and of the
same length.
Let’s begin by adding the measure
of each of our arcs to the diagram. We’re told the measure of arc 𝐴𝐵
equals 62 degrees, the measure of arc 𝐵𝐶 equals 110, and the measure of arc 𝐴𝐷
equals 126. Since the measure of each arc is
the angle that arc makes at the center of the circle, it follows that the sum of all
arc measures that make up that circle is 360 degrees. And this is really useful because
it will allow us to calculate the measure of arc 𝐶𝐷.
We say that the sum of the arc
measures is 360 degrees. And then we can replace the various
arc measures with their values. So the measure of arc 𝐴𝐵 is 62,
the measure of arc 𝐵𝐶 is 110, and so on. This left-hand side simplifies to
298 degrees plus the measure of arc 𝐶𝐷. And then we can find that measure
of arc 𝐶𝐷 by subtracting 298 from both sides. It’s 360 minus 298, which is of
course 62 degrees.
So why is this useful? Well, we know that the measure of
arcs between parallel chords of a circle are equal, and the opposite is also
true. That is, if the measure of two arcs
between two distinct chords is equal, those chords must in fact be parallel. We in fact have that the measure of
arc 𝐴𝐵 is equal to the measure of arc 𝐶𝐷. And so that must mean that line
segments 𝐴𝐷 and 𝐵𝐶 are in fact parallel. We’re therefore able to disregard
options (B), (C), and (D). And so we need to choose between
option (A) and option (E), where option (A) is that they are parallel and option (E)
is that they’re not only parallel, but they’re of the same length.
Well, if the chords are parallel
and equal in length, then the arcs between the endpoints of the chords will be equal
in measure. But we can see that the measure of
arc 𝐵𝐶 is not equal to the measure of arc 𝐴𝐷. They are in fact 110 and 126
degrees, respectively. So 𝐵𝐶 and 𝐴𝐷 cannot be of equal
length. And so the answer is (A). They’re parallel.
