### Video Transcript

Given that the measure of angle πΆπ΄π΅ is 76 degrees, find the value of π₯.

Letβs begin by adding to our diagram what weβve been told, that the measure of the angle πΆπ΄π΅ is 76 degrees. Now, angle πΆπ΄π΅ is here. So how does that help us calculate the value of π₯? Well, itβs important to realise that the lines joining πΆ and π΄ and the lines joining π΅ and π΄ 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 πΆπ΄π΅ is actually an isosceles triangle.

We recall then that the base angles in an isosceles triangle are equal in size. And weβll use the fact that angles in a triangle add up to 180 degrees. Weβre going to use these facts to calculate the size of angle π΄πΆπ΅. To do so, we subtract 76 from 180 degrees. And then, we divide this in two. And thatβs because angle π΄πΆπ΅ and π΄π΅πΆ are equal in size. 180 minus 76 divided by two is 52 degrees.

And now, we look really carefully at the shape weβve been given. We have a triangle inscribed in a circle. That means the radius at point πΆ. 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 π΅π·πΆ is 52 degrees.

So angle π₯ is 52.