Video Transcript
Given that the measure of angle ๐ถ๐ด๐ต is 76 degrees, find the value of ๐ฅ.
Letโs begin by adding to our diagram what weโve been told, that the measure of the angle ๐ถ๐ด๐ต is 76 degrees. Now, angle ๐ถ๐ด๐ต is here. So how does that help us calculate the value of ๐ฅ? Well, itโs important to realise that the lines joining ๐ถ and ๐ด and the lines joining ๐ต and ๐ด are tangents to the circle. We know that tangents to a circle that meet at a point are of equal length. And so we can see that triangle ๐ถ๐ด๐ต is actually an isosceles triangle.
We recall then that the base angles in an isosceles triangle are equal in size. And weโll use the fact that angles in a triangle add up to 180 degrees. Weโre going to use these facts to calculate the size of angle ๐ด๐ถ๐ต. To do so, we subtract 76 from 180 degrees. And then, we divide this in two. And thatโs because angle ๐ด๐ถ๐ต and ๐ด๐ต๐ถ are equal in size. 180 minus 76 divided by two is 52 degrees.
And now, we look really carefully at the shape weโve been given. We have a triangle inscribed in a circle. That means the radius at point ๐ถ. And so we can use the alternate segment theorem. Another way of saying this is that, in any circle, the angle between a chord and a tangent through one of the end points of the chord is equal to the angle in the alternate segment. And essentially, this means that the measure of the angle ๐ต๐ท๐ถ is 52 degrees.
So angle ๐ฅ is 52.
