# Question Video: Finding the Measure of the Sum of Two Arcs given the Rest of the Arcsβ Measure in the Same Circle Mathematics • 11th Grade

Given that the line segment π΄π΅ is a diameter of the circle and line segment π·πΆ β«½ line segment π΄π΅, find πβ π΄πΈπ·.

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### Video Transcript

Given that the line segment π΄π΅ is a diameter of the circle and line segment π·πΆ is parallel to line segment π΄π΅, find the measure of angle π΄πΈπ·.

Weβre interested in the measure of angle π΄πΈπ·; thatβs this measure. And weβve been given a few other pieces of information. We know line segment π·πΆ is parallel to line segment π΄π΅. We know line segment π΄π΅ is the diameter. And on the figure, angle πΆπ΅π΄ has been labeled as 68.5 degrees.

At first, it might not seem like thereβs a very clear direction for where to go here. But if we start with the measure of angle πΆπ΄π΅, using that information, we could find the measure of arc πΆπ΄. Since angle πΆπ΄π΅ is an inscribed angle, its arc will be two times the measure of that inscribed angle. Arc π΄πΆ will then be equal to two times 68.5, which is 137 degrees. And because we know that line segment π΄π΅ is a diameter, arc π΄π΅ must be equal to 180 degrees. We can also say that arc π΄π΅ will be equal to arc π΅πΆ plus arc πΆπ΄.

We know π΄π΅ needs to be 180 degrees and arc πΆπ΄ is 137 degrees. To solve for the measure of arc π΅πΆ, we can subtract 137 from both sides of the equation. And we get the measure of arc π΅πΆ is 43 degrees. And hereβs where our parallel chords come into play. When you have parallel chords, their intercepted arcs are going to be congruent. And that means because arc πΆπ΅ equals 43 degrees, arc π·π΄ also equals 43 degrees.

And at this point, we began to see that arc π·π΄ is subtended by the angle π΄πΈπ·. Since angle π΄πΈπ· is an inscribed angle, its angle measure, the measure of angle π΄πΈπ·, is going to be equal to one-half the measure of arc π΄π·. We know that the measure of arc π΄π· is 43 degrees, and one-half of 43 is 21.5. And so, we can say that the measure of angle π΄πΈπ· is 21.5 degrees.