Given that the measure of angle 𝐶𝐴𝐵 is 76 degrees, find the value of 𝑥.
Let’s begin by adding to our diagram what we’ve been told, that the measure of the angle 𝐶𝐴𝐵 is 76 degrees. Now, angle 𝐶𝐴𝐵 is here. So how does that help us calculate the value of 𝑥? Well, it’s important to realise that the lines joining 𝐶 and 𝐴 and the lines joining 𝐵 and 𝐴 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 𝐶𝐴𝐵 is actually an isosceles triangle.
We recall then that the base angles in an isosceles triangle are equal in size. And we’ll use the fact that angles in a triangle add up to 180 degrees. We’re going to use these facts to calculate the size of angle 𝐴𝐶𝐵. To do so, we subtract 76 from 180 degrees. And then, we divide this in two. And that’s because angle 𝐴𝐶𝐵 and 𝐴𝐵𝐶 are equal in size. 180 minus 76 divided by two is 52 degrees.
And now, we look really carefully at the shape we’ve been given. We have a triangle inscribed in a circle. That means the radius at point 𝐶. 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 𝐵𝐷𝐶 is 52 degrees.
So angle 𝑥 is 52.