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.
