Question Video: Finding the Measure of an Angle given Its Arc’s Measure Using Another Inscribed Angle Mathematics

Given that 𝑚∠𝐵𝐴𝐷 = 36° and 𝑚∠𝐶𝐵𝐴 = 37°, find 𝑚∠𝐵𝐶𝐷 and 𝑚∠𝐶𝐷𝐴.

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Video Transcript

Given that the measure of angle 𝐵𝐴𝐷 is equal to 36 degrees and the measure of angle 𝐶𝐵𝐴 is equal to 37 degrees, find the measure of angle 𝐵𝐶𝐷 and the measure of angle 𝐶𝐷𝐴.

Let’s begin by adding the angles that we know onto the diagram. The measure of angle 𝐵𝐴𝐷 is equal to 36 degrees, and the measure of angle 𝐶𝐵𝐴 is equal to 37. We are looking to calculate the measure of angle 𝐵𝐶𝐷, which is this one, and the measure of angle 𝐶𝐷𝐴, which is this one. We now observe that the first unknown angle, that’s 𝐵𝐶𝐷, is subtended by the same arc 𝐵𝐷 as angle 𝐵𝐴𝐷. And we know that inscribed angles subtended by the same arc are equal. So the measure of angle 𝐵𝐴𝐷 must be equal to the measure of angle 𝐵𝐶𝐷. But of course, we’ve now seen that that’s 36 degrees.

In a similar way, we observed that angle 𝐴𝐷𝐶 is subtended from the same arc as angle 𝐴𝐵𝐶. And so, these two angles are congruent. The measure of angle 𝐴𝐵𝐶 must be equal to the measure of angle 𝐴𝐷𝐶. And that’s 37. So, using the property of inscribed angles subtended from the same arc, we find the measure of angle 𝐵𝐶𝐷 is equal to 36 degrees and the measure of angle 𝐶𝐷𝐴 is equal to 37 degrees.