### Video Transcript

In the following figure, a
rectangle π΄π΅πΆπ· is inscribed in a circle, where the measure of arc π΄π΅ equals 71
degrees. Find the measure of arc π΄π·.

Weβre going to use the fact that
π΄π΅πΆπ· is a rectangle. This means that the line segment or
chord π΄π΅ is parallel to chord π·πΆ. Similarly, line segment π·π΄ must
be parallel to line segment π΅πΆ. 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 π΄π΅ and π·πΆ
are parallel, the measure of arc π΄π· must be equal to the measure of arc π΅πΆ. Similarly, the measure of arc π΄π΅
must be equal to the measure of arc π·πΆ. But weβre actually told thatβs 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
π΄π΅ and the measure of arc π·πΆ is 71. So our equation is the measure of
arc π΄π· plus the measure of arc π΅πΆ plus 71 plus 71 equals 360. Since π΄π· and π΅πΆ are congruent
arcs, we can further simplify this. Two times the measure of arc π΄π·
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 π΄π· is 218
divided by two, which is equal to 109 or 109 degrees. The measure of arc π΄π· is 109
degrees.