Video Transcript
Given that ray 𝐵𝐶 is a tangent to the circle below, find the measure of angle 𝐴𝐵𝐶.
We’ll let the measure of angle 𝐴𝐵𝐶 be 𝜃. We see that the major arc from 𝐴 to 𝐵 equals 190 degrees. And this means that the minor arc will measure 170 degrees. We find this by subtracting 190 degrees from 360 degrees. In order to say anything further, we remember a corollary, angles of tangency and arc measures. If 𝐴 is a point on a circle and 𝐵 is the point where the tangent passing through 𝐵 and 𝐶 intersects the circle, then the angle of tangency, angle 𝐴𝐵𝐶, is half the measure of the arc 𝐴𝐵 formed on the same side, as shown in this image.
The angle formed here, 𝜃, is a one-half the arc formed on the same side of the circle as that angle. Therefore, our angle 𝜃 is going to be one-half of 170 degrees. The measure of angle 𝐴𝐵𝐶 is 170 degrees divided by two, which is 85 degrees.
