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

Given that 𝑚∠𝐴𝐵𝐶 = 27°, and 𝑚∠𝐵𝐷𝐴 = 62°, find 𝑚∠𝐵𝐷𝐶 and 𝑚∠𝐵𝐶𝐷.

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Video Transcript

Given that the measure of angle 𝐴𝐵𝐶 equals 27 degrees and the measure of angle 𝐵𝐷𝐴 equals 62 degrees, find the measure of angle 𝐵𝐷𝐶 and the measure of angle 𝐵𝐶𝐷.

The first angle measure that we’re given is the measure of angle 𝐴𝐵𝐶, which is 27 degrees. The second angle measure is that of angle 𝐵𝐷𝐴, which we’re told is 62 degrees. The first angle we’re asked to calculate the measure of is that of angle 𝐵𝐷𝐶, which is the whole angle at the top of this figure. The second angle we need to calculate is that of angle 𝐵𝐶𝐷, which is on the right of the figure.

The first thing we might observe is that we have this bow tie shape within the circle, which may indicate that we’ll be using inscribed angles theorems. One of these theorems tells us that inscribed angles subtended by the same arc are equal. This angle of 𝐶𝐵𝐴 is inscribed by the arc 𝐶𝐴. And so it will be equal in measure to any other angle inscribed by the same arc. And so the measure of angle 𝐶𝐷A will be the same size. It will be 27 degrees. The first angle we needed to calculate was the measure of angle 𝐵𝐷𝐶. And so it will be the sum of these two angles, 62 degrees plus 27 degrees, which gives us an answer of 89 degrees. And so that’s the first angle measure found.

Now, let’s see how we work towards finding the other angle measure. Because we know that inscribed angles subtended by the same arc are equal, then the measure of angle 𝐷𝐴𝐵 will be equal to the measure of angle 𝐷𝐶𝐵. However, we don’t know the measure of this angle either. Let’s see if we can calculate either of these angles. To do this, let’s observe that we have a pair of congruent line segments. The line segment 𝐴𝐷 is equal to the line segment 𝐴𝐵. And so when we consider these line segments as part of the triangle 𝐴𝐵𝐷, then we know that 𝐴𝐵𝐷 will be an isosceles triangle.

Isosceles triangles have two equal sides and two equal angle measures. And so the two angle measures that are equal will be the measure of angle 𝐴𝐷𝐵 and the measure of angle 𝐴𝐵𝐷. These will both be 62 degrees. We can now use the triangle 𝐴𝐵𝐷 and the fact that the interior angles in a triangle sum to 180 degrees to find the measure of angle 𝐷𝐴𝐵. We can write the equation that the three angles in the triangle, that’s 62 degrees, 62 degrees, and the measure of angle 𝐷𝐴𝐵, must sum to 180 degrees. 62 degrees plus 62 degrees simplifies to 124 degrees. Subtracting 124 degrees from both sides gives us that the measure of angle 𝐷𝐴𝐵 is 56 degrees.

Now, we know that this angle subtended by the arc 𝐵𝐷 will be equal to the measure of angle 𝐵𝐶𝐷, which is subtended by the same arc. And so we can give the answer for both angle measures. The measure of angle 𝐵𝐷𝐶 is 89 degrees, and the measure of angle 𝐵𝐶𝐷 is 56 degrees.