Video Transcript
If line 𝐸𝐷 is a tangent to the circle at the point 𝐵 and the measure of angle 𝐸𝐵𝐶 equals 40 degrees, what is the measure of angle 𝐴𝐶𝐵?
Let’s think about what we know. Line 𝐸𝐷 is tangent to the circle at 𝐵. The measure of angle 𝐸𝐵𝐶 equals 40 degrees. This information is not indicated on the figure, so we can go ahead and add that here. And from the graph, we see that the measure of angle 𝐴𝐵𝐶 is 65 degrees. And our missing angle is angle 𝐴𝐶𝐵.
Can we draw any conclusions from the information we were given? But to do that, we’ll need to think about the alternate segment theorem, which tells us that the angle between a tangent and a cord is equal to the angle in the alternate segment. And that means in our circle, the measure of angle 𝐴𝐶𝐵 will be equal to the measure of 𝐴𝐵𝐷. We’re saying that the measure of angle 𝐴𝐶𝐵 will be equal to the measure of angle 𝐴𝐵𝐷 by the alternate segment theorem.
Because we know that 𝐸𝐷 is a line, that line will measure 180 degrees. And that means the measure of angle 𝐸𝐵𝐶 plus the measure of angle 𝐴𝐵𝐶 plus the measure of angle 𝐴𝐵𝐷 must equal 180 degrees. 40 degrees plus 65 degrees plus the measure of angle 𝐴𝐵𝐷 equals 180. So, 105 degrees plus the measure of angle 𝐴𝐵𝐷 equals 180 degrees. If we subtract 105 degrees from both sides, we see that the measure of angle 𝐴𝐵𝐷 equals 75 degrees. And if 𝐴𝐵𝐷 equals 75 degrees, then 𝐴𝐶𝐵 equals 75 degrees.
And by the alternate segment theorem, we can also say that the measure of angle 𝐶𝐴𝐵 equals 40 degrees. And if we wanted to check this, we could check that these three angles inside the triangle 𝐴𝐵𝐶 add up to 180 degrees, which they do. And the final answer here is that the measure of angle 𝐴𝐶𝐵 equals 75 degrees.
