Given that the measure of angle 𝐸𝐶𝐷 equals 54 degrees and the measure of angle 𝐹𝐵𝐷 equals 78 degrees, find 𝑥 and 𝑦.
First, we can add the measure of the two angles we’re given into the diagram. To solve further, we will need to remember the alternate segment theorem. If we have points 𝐴 and 𝐵 that fall on a circle and point 𝐶 is the point where a tangent passing through the line intersects the circle, here the measure of angle 𝐶𝐵𝐴, we’ve labeled as 𝜃, and that will be equal to the measure of angle 𝐷𝐶𝐴. We’ll let the measure of angle 𝐶𝐴𝐵 be labeled as 𝛽, and this angle will be equal to the measure of angle 𝐸𝐶𝐵. Using this theorem, we can find that the measure of angle 𝐷𝐵𝐶 equals 54 degrees. The measure of angle 𝐷𝐶𝐵 equals 78 degrees. And from there, we’ll see the angle 𝐵𝐶𝐴 and angle 𝐶𝐵𝐴 both measure 𝑥 degrees.
Because 𝐵𝐶𝐷 forms a triangle, we know that the three angles inside sum to 180 degrees. And therefore, we can find 𝑥 by taking 180 and subtracting the other two angles 54 plus 78. And we see that 𝑥 equals 48. To find 𝑦, we’ll consider the triangle 𝐴𝐵𝐶. It’s made up of the angles 𝑥, 𝑥, and 𝑦. So 𝑥 plus 𝑥 plus 𝑦 must equal 180 degrees. We’ll plug in what we know. 48 plus 48 is 96. So we subtract 96 from both sides. And we find that 𝑦 must be equal to 84.
Our missing 𝑥- and 𝑦-values are 48 and 84, respectively. And we found this using the alternate segment theorem.