Given that 𝐵𝐶 is a tangent to the circle, find the measure of angle 𝐴𝐵𝐶.
So now to solve this problem, we’re actually going to use something called the alternate segment theorem. And the alternate segment theorem actually tells us that the angle at the tangent is equal to the opposite interior angle. So what does this actually mean? Well, I’ve drawn a sketch here to help us actually see.
We got angle 𝑎 which is our angle on the tangent. And according to the alternate segment theorem, this angle should actually be equal to the opposite interior angle — so in this case 𝑏. Okay, great, we’ve got a relationship. But how have we got that? And I think it’s important to know that. So what I’m gonna do is show you how we can actually prove that. So what I’ve done to actually help us prove it is actually drawn two identical triangles going from the centre of our circle out to the edge of our circle.
Well, the first thing we actually know is that actually the angle between the tangent and the radius is got to be equal 90 degrees. So therefore, 𝑎 plus 𝑥 is got to be equal 90 degrees. And what we also know is that 90 plus 𝑥 plus 𝑦 must be equal to 180 degrees because there’s 180 degrees in triangle. And now it’s three angles within because we’ve got the 90 degrees or the right angle in the middle. Well, from this, we can see that 𝑥 plus 𝑦 must be equal to 90 degrees because if we subtract 90 degrees from each side of the equation, we’re left with just 𝑥 plus 𝑦 equals 90 degrees.
Well, this actually allows us to determine a very important relationship because if we look carefully we can see that we have 𝑎 and 𝑦 that both must be equal to the same thing because both 𝑎 and 𝑦 plus 𝑥 both equal 90 degrees. So therefore, we can say that 𝑎 is equal to 𝑦. Okay, we now know that relationship and it’s going to become very useful in a second.
Let’s change our focus to 𝑏. Well, to help us find 𝑏, we actually have a relationship we know, which is that the angle at circumference is half the angle at the middle. So therefore, we might look at the sketch I’ve done to decide here. We could say that if the angle at circumference is 𝑥, then the angle of the middle is going to be two 𝑥. So therefore, we can say that the angle 𝑏 — because it’s the angle at the circumference — is going to be equal to half the angle at the middle or the angle at the middle is going to be two 𝑦. So we can say that 𝑏 is equal to half of two 𝑦, which therefore gives us that 𝑏 is equal to 𝑦.
And this is the key fact because it brings us back to what we looked at earlier, where we said that 𝑎 is equal to 𝑦. Well, if we got 𝑎 is equal to 𝑦 and 𝑏 is equal to 𝑦, then therefore 𝑎 must be equal to 𝑏. So we can say okay, great, the alternate segment theorem is right and we’ve proved it without working here.
Okay, now we know what the alternate segment theorem is and we’ve proved it. Let’s get back on and find the measure of angle 𝐴𝐵𝐶. Well, using the knowledge that we’ve just gained, we can say that the measure of angle 𝐴𝐵𝐶 must be equal to 78 degrees. And again, we have to give our reason. And the reason is because of the alternate segment theorem. And we’ve arrived at that because the angle at the tangent is angle 𝐴𝐵𝐶. And this must be equal to the opposite interior angle, which in this case is angle 𝐴𝐷𝐵 and 78 degrees.