𝐴, 𝐵, and 𝐶 are points on a circle. 𝐵𝐷 is a tangent to the circle. Write down the size of angle 𝑥. Give a reason for your answer.
This is a circle theorem question. And there is a little bit of a clue about how much work we actually need to do to answer it in the question. The words “write down” indicate that there’s little to no working out we need to do. In fact, we should be able to work out the size of angle 𝑥 just by looking at our diagram.
There are a few unnecessary bits of information. We actually don’t need to worry about the angles 𝐶𝐴𝐵 and 𝐴𝐵𝐶. The only one we’re interested in is this one at 𝐴𝐶𝐵. Our angle 𝑥 is actually the same as this angle. It’s 80 degrees. And this is called the alternate segment theorem. This only works when we have a triangle inscribed in a circle with a tangent at one of its vertices.
So in this example, these two angles would be equal as with these two. Using the same rule, we can also see that this angle here is the same as angle 𝐶𝐴𝐵. It’s 65 degrees. 𝑥 is 80 degrees. And it is enough to simply say the reason is because of the alternate segment theorem.