Video: Finding the Measure of an Inscribed Angle given the Measure of Another Inscribed Angle Subtended by the Same Arc

Given that π‘šβˆ π΄π΅π· = 44Β°β€Ž, and π‘šβˆ πΆπΈπ΄ equals 72Β°β€Ž, find π‘₯, 𝑦, and 𝑧.

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Video Transcript

Given that the measure of angle 𝐴𝐡𝐷 equals 44 degrees and the measure of angle 𝐢𝐸𝐴 equals 72 degrees, find π‘₯, 𝑦, and 𝑧.

Let’s start by listing what we know. Angle 𝐴𝐡𝐷 equals 44 degrees, angle 𝐢𝐸𝐴 measures 72 degrees. These two chords intersect at point 𝐸. And that means we can say that angle 𝐡𝐸𝐷 and angle 𝐢𝐸𝐴 are vertical angles, which means their measure will be equal to one another. They are congruent angles. And in this case, that means that angle 𝐡𝐸𝐷 is also equal to 72 degrees. The points 𝐸, 𝐡, and 𝐷 form a triangle, which means that their three angles must sum to 180 degrees. And we can substitute what we know for these three angles into this equation. 72 plus 44 equals 116. 116 plus 𝑧 equals 180. So, we subtract 116 from both sides. And we find that 𝑧 equals 64 degrees.

We won’t be able to follow the same procedure to find π‘₯ and 𝑦. So, we’ll need to think about some of the circle theorems. If we look at inscribed angle 𝐡, we see that it has endpoints along the circle at 𝐴 and 𝐷 and that its intercepted arc is arc 𝐴𝐷. We could write them as arc 𝐴𝐷 intercepts angle 𝐴𝐡𝐷. But there’s another angle in this circle that also intercepts the same arc, and that would be angle 𝐴𝐢𝐷. Because both of these angles are subtended by the same arc, we can say that the measure of angle 𝐴𝐢𝐷 will be equal to the measure of angle 𝐴𝐡𝐷. And that means π‘₯ will be equal to 44 degrees.

And because all three angles need to sum to 180 degrees, we can tell that angle 𝑦 is going to be equal to 64 degrees. If we wanted to confirm this, we could see that angle 𝐢𝐴𝐡 intercepts arc 𝐢𝐡 and angle 𝐢𝐷𝐡 intercepts arc 𝐢𝐡. And so, we found that π‘₯ equals 44 degrees, and both 𝑦 and 𝑧 equals 64 degrees.

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