WEBVTT
00:00:01.090 --> 00:00:08.220
Given that ray 𝐵𝐶 is a tangent to the circle below, find the measure of angle 𝐴𝐵𝐶.

00:00:08.950 --> 00:00:12.200
We’ll let the measure of angle 𝐴𝐵𝐶 be 𝜃.

00:00:12.830 --> 00:00:17.770
We see that the major arc from 𝐴 to 𝐵 equals 190 degrees.

00:00:18.330 --> 00:00:22.110
And this means that the minor arc will measure 170 degrees.

00:00:22.710 --> 00:00:27.310
We find this by subtracting 190 degrees from 360 degrees.

00:00:28.040 --> 00:00:34.030
In order to say anything further, we remember a corollary, angles of tangency and arc measures.

00:00:35.270 --> 00:00:53.990
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.

00:00:54.620 --> 00:01:03.660
The angle formed here, 𝜃, is a one-half the arc formed on the same side of the circle as that angle.

00:01:04.320 --> 00:01:09.390
Therefore, our angle 𝜃 is going to be one-half of 170 degrees.

00:01:10.060 --> 00:01:16.430
The measure of angle 𝐴𝐵𝐶 is 170 degrees divided by two, which is 85 degrees.