Given that the measure of angle 𝐶𝐴𝐵 is 64 degrees, find the measure of angle 𝐶𝐵𝐷.
And then we’ve been given a circle with center 𝑀 and four points 𝐴, 𝐵, 𝐶, 𝐷 on the circumference of that circle. We’re told that the measure of angle 𝐶𝐴𝐵 is 64 degrees. So we add that to our circle. Similarly, we’re trying to find the measure of angle 𝐶𝐵𝐷. That’s this angle here.
Now, we might notice that angle 𝐶𝐵𝐷 is the sum of two individual angles 𝐶𝐵𝐴 and 𝐴𝐵𝐷. So if we can find the measure of angles 𝐶𝐵𝐴 and 𝐴𝐵𝐷 individually, their sum will tell us the angle 𝐶𝐵𝐷. Well, the key to finding these two individual angles is to spot that the line segment 𝐵𝐴 passes through point 𝑀, the center of the circle. So 𝐵𝐴 is a diameter of this circle.
Then, we can use a special version of the inscribed angle theorem to calculate the measure of angle 𝐴𝐶𝐵 and 𝐵𝐷𝐴. This tells us that angles subtended by the diameter are equal to 90 degrees. Since angles 𝐵𝐶𝐴 and 𝐵𝐷𝐴 are subtended by the diameter, they are both equal to 90 degrees. Then, we might notice that 𝐴𝐵𝐶 and 𝐴𝐵𝐷 are triangles. And angles in a triangle sum to 180 degrees. So let’s define angle 𝐶𝐵𝐴 to be 𝑥 degrees and 𝐴𝐵𝐷 to be 𝑦 degrees. Thinking purely about triangle 𝐴𝐵𝐶, we know that 𝑥 plus 90 plus 64 is the interior angle sum. And then we know that this is equal to 180. This sum simplifies to 𝑥 plus 154. So we can say that 𝑥 plus 154 equals 180. Then, we subtract 154 from both sides. So 𝑥 is equal to 26 degrees.
With that in mind, now let’s perform a similar process to find the value of 𝑦. To find the value of 𝑦, we spot that arc 𝐵𝐷 is congruent to arc 𝐷𝐴. This means line segments 𝐵𝐷 and 𝐷𝐴 are also congruent to one another. The triangle therefore is isosceles, and so angle 𝐵𝐴𝐷 is also equal to 𝑦.
We can therefore say that 90 plus two 𝑦 equals 180. Subtracting 90 degrees from both sides, we get two 𝑦 equals 90. And then we divide through by two. So 𝑦 is equal to 45 degrees. We now know the measure of angles 𝐶𝐵𝐴 and 𝐴𝐵𝐷. So the measure of angle 𝐶𝐵𝐷, which is the sum of these, is 26 plus 45. And that’s 71 degrees. So the measure of angle 𝐶𝐵𝐷 is 71 degrees.