Video Transcript
In the following figure, a
rectangle 𝐴𝐵𝐶𝐷 is inscribed in a circle, where the measure of arc 𝐴𝐵 equals 71
degrees. Find the measure of arc 𝐴𝐷.
We’re going to use the fact that
𝐴𝐵𝐶𝐷 is a rectangle. This means that the line segment or
chord 𝐴𝐵 is parallel to chord 𝐷𝐶. Similarly, line segment 𝐷𝐴 must
be parallel to line segment 𝐵𝐶. This means we can use the theorem
that tells us that the measure of arcs between parallel chords of a circle are
equal. Since line segments 𝐴𝐵 and 𝐷𝐶
are parallel, the measure of arc 𝐴𝐷 must be equal to the measure of arc 𝐵𝐶. Similarly, the measure of arc 𝐴𝐵
must be equal to the measure of arc 𝐷𝐶. But we’re actually told that’s 71
degrees.
Now, since the sum of all the arc
measures that make up the circle is 360 degrees, we can form and solve an
equation. We know that the measure of arc
𝐴𝐵 and the measure of arc 𝐷𝐶 is 71. So our equation is the measure of
arc 𝐴𝐷 plus the measure of arc 𝐵𝐶 plus 71 plus 71 equals 360. Since 𝐴𝐷 and 𝐵𝐶 are congruent
arcs, we can further simplify this. Two times the measure of arc 𝐴𝐷
plus 142 degrees equals 360 degrees. Then, we subtract 142 degrees from
both sides, and our final stage is to divide through by two. So the measure of arc 𝐴𝐷 is 218
divided by two, which is equal to 109 or 109 degrees. The measure of arc 𝐴𝐷 is 109
degrees.
