Video Transcript
Find the measure of angle 𝐸𝐷𝐶.
First, we identify the angle 𝐸𝐷𝐶 on our diagram. At first, it doesn’t seem clear which path we’ll take to finding the measure of angle 𝐸𝐷𝐶. A good strategy here is just to annotate the diagram with the information we already know and see if it becomes clear how to find the measure of angle 𝐸𝐷𝐶. Noticing that line segment 𝐸𝐶 and line segment 𝐵𝐶 are equal in length, triangle 𝐸𝐵𝐶 is isosceles. Therefore, the measure of angle 𝐶𝐸𝐵 is equal to the measure of angle 𝐸𝐵𝐶.
Next, we notice that ray 𝐴𝐵 is a tangent to the circle at 𝐵 and ray 𝐴𝐶 is a tangent to the circle at point 𝐶. We can see that chord 𝐸𝐵 intersects the tangent at point 𝐵. And similarly, the chord 𝐸𝐶 intersects the other tangent at point 𝐶. Based on these facts, we can apply the alternate segment theorem, which tells us that in any circle, the angle between a chord and a tangent through one of the endpoints of the chord is equal to the angle in the alternate segment. This means the measure of angle 𝐶𝐵𝐴 equals the measure of angle 𝐶𝐸𝐵 and the measure of angle 𝐵𝐶𝐴 equals the measure of angle 𝐶𝐸𝐵.
Because we know that triangle 𝐸𝐵𝐶 and triangle 𝐴𝐵𝐶 share two angles that are equal to each other, we can say that these two triangles are similar, which makes the third angle in triangle 𝐸𝐵𝐶 equal to 52 degrees. Since we’re working with triangles, we know that all the interior angles must sum to 180 degrees. Letting our two unknown angles be equal to 𝑎, we can say two 𝑎 plus 52 equals 180, two 𝑎 equals 128, and 𝑎 equals 64.
We filled in our diagram up until this point. To move forward, we now need to move from just looking at these interior angles to looking at the arcs. Notice that angle 𝐸𝐷𝐶 subtends the major arc 𝐸𝐶. Based on the inscribed angle theorem, we can say the measure of an inscribed angle subtended by an arc is half the measure of this arc.
So if we find the measure of the major arc 𝐸𝐶, one-half of that will be the measure of angle 𝐸𝐷𝐶. The major arc 𝐸𝐶 is made up of the minor arc 𝐸𝐵 and the minor arc 𝐵𝐶. The minor arc 𝐸𝐵 is subtended by the angle 𝐸𝐶𝐵. Therefore, the minor arc 𝐸𝐵 is going to be equal to twice the measure of angle 𝐸𝐶𝐵, two times 52 degrees, which is 104 degrees.
Similarly, the minor arc 𝐵𝐶 is subtended by the angle 𝐵𝐸𝐶. Therefore, this minor arc will be equal to two times the angle which measures 64 degrees, 128 degrees. To find our major arc 𝐸𝐶, we need to add these two smaller arcs together. Major arc 𝐸𝐶 equals 232 degrees. As we’ve already said, the measure of angle 𝐸𝐷𝐶 is equal to one-half the measure of the major arc 𝐸𝐶. One-half of 232 degrees equals 116 degrees.
