Video Transcript
Find the measure of angle 𝐸𝐶𝐷.
First, we identify 𝐸𝐶𝐷 on our diagram and mark it as 𝜃. The lines through point 𝐵 and 𝐶 are tangent to the circle, and they meet at an external point. Therefore, the length of line segment 𝐴𝐶 is equal to the length of line segment 𝐴𝐵 by the property of two tangents and an external point, which means the measure of angle 𝐴𝐶𝐵 is equal to the measure of angle 𝐴𝐵𝐶. We’ll label this measure as 𝑎 for the time being. In triangle 𝐴𝐵𝐶, the sum of all the angles is 180 degrees. Therefore, 180 degrees equals 76 degrees plus 𝑎 plus 𝑎. If we subtract 76 degrees from both sides of the equation and combine the two 𝑎-terms, we have 104 degrees equals two 𝑎. Dividing both sides by two, 𝑎 equals 52 degrees. The measure of angle 𝐴𝐶𝐵 and the measure of angle 𝐶𝐵𝐴 equals 52 degrees.
We’re now able to say something about the measure of angle 𝐶𝐷𝐵. The measure of angle 𝐶𝐷𝐵 is equal to the measure of angle 𝐵𝐶𝐴 by the alternate segment theorem, which makes the measure of angle 𝐶𝐷𝐵 equal to 52 degrees. From there, recall that segment 𝐷𝐵 is parallel to the ray 𝐴𝐸. Segment 𝐷𝐶 intersects two parallel lines. The measure of angle 𝐶𝐷𝐵 is equal to the measure of angle 𝐸𝐶𝐷 by the properties of alternate interior angles. That is the property that tells us when two parallel lines are intersected by the same line, the measure of the alternate interior angles will be equal to each other. Based on that, the measure of angle 𝐸𝐶𝐷 equals 52 degrees.
