# Question Video: Finding the Measure of an Arc Using the Relationship Between the Sides of an Inscribed Rectangle Mathematics

In the following figure, a rectangle π΄π΅πΆπ· is inscribed in a circle, where the measure of arc π΄π΅ = 71Β°. Find the measure of arc π΄π·.

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### Video Transcript

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

Weβre going to use the fact that π΄π΅πΆπ· is a rectangle. This means that the line segment or chord π΄π΅ is parallel to chord π·πΆ. Similarly, line segment π·π΄ must be parallel to line segment π΅πΆ. 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·πΆ are parallel, the measure of arc π΄π· must be equal to the measure of arc π΅πΆ. Similarly, the measure of arc π΄π΅ must be equal to the measure of arc π·πΆ. But weβre actually told thatβ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 π΄π΅ and the measure of arc π·πΆ is 71. So our equation is the measure of arc π΄π· plus the measure of arc π΅πΆ plus 71 plus 71 equals 360. Since π΄π· and π΅πΆ 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.