### Video Transcript

๐ด, ๐ต, and ๐ถ are points on a circle. ๐ต๐ท is a tangent to the circle. Write down the size of angle ๐ฅ. 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 โwrite downโ indicate that thereโs little to no working out we need to do. In fact, we should be able to work out the size of angle ๐ฅ just by looking at our diagram.

There are a few unnecessary bits of information. We actually donโt need to worry about the angles ๐ถ๐ด๐ต and ๐ด๐ต๐ถ. The only one weโre interested in is this one at ๐ด๐ถ๐ต. Our angle ๐ฅ is actually the same as this angle. Itโs 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 ๐ถ๐ด๐ต. Itโs 65 degrees. ๐ฅ is 80 degrees. And it is enough to simply say the reason is because of the alternate segment theorem.