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

Given that the line segment 𝐴𝐡 is a diameter of the circle and line segment 𝐷𝐢 β«½ line segment 𝐴𝐡, find π‘šβˆ π΄πΈπ·.


Video Transcript

Given that the line segment 𝐴𝐡 is a diameter of the circle and line segment 𝐷𝐢 is parallel to line segment 𝐴𝐡, find the measure of angle 𝐴𝐸𝐷.

We’re interested in the measure of angle 𝐴𝐸𝐷; that’s this measure. And we’ve been given a few other pieces of information. We know line segment 𝐷𝐢 is parallel to line segment 𝐴𝐡. We know line segment 𝐴𝐡 is the diameter. And on the figure, angle 𝐢𝐡𝐴 has been labeled as 68.5 degrees.

At first, it might not seem like there’s a very clear direction for where to go here. But if we start with the measure of angle 𝐢𝐴𝐡, using that information, we could find the measure of arc 𝐢𝐴. Since angle 𝐢𝐴𝐡 is an inscribed angle, its arc will be two times the measure of that inscribed angle. Arc 𝐴𝐢 will then be equal to two times 68.5, which is 137 degrees. And because we know that line segment 𝐴𝐡 is a diameter, arc 𝐴𝐡 must be equal to 180 degrees. We can also say that arc 𝐴𝐡 will be equal to arc 𝐡𝐢 plus arc 𝐢𝐴.

We know 𝐴𝐡 needs to be 180 degrees and arc 𝐢𝐴 is 137 degrees. To solve for the measure of arc 𝐡𝐢, we can subtract 137 from both sides of the equation. And we get the measure of arc 𝐡𝐢 is 43 degrees. And here’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 𝐢𝐡 equals 43 degrees, arc 𝐷𝐴 also equals 43 degrees.

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

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