### Video Transcript

Given that π΅πΆ is a tangent to the circle, find the measure of angle π΄π΅πΆ.

So now to solve this problem, weβre 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, Iβve drawn a sketch here to help us actually see.

We got angle π 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 β so in this case π. Okay, great, weβve got a relationship. But how have we got that? And I think itβs important to know that. So what Iβm gonna do is show you how we can actually prove that. So what Iβve 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, π plus π₯ is got to be equal 90 degrees. And what we also know is that 90 plus π₯ plus π¦ must be equal to 180 degrees because thereβs 180 degrees in triangle. And now itβs three angles within because weβve got the 90 degrees or the right angle in the middle. Well, from this, we can see that π₯ plus π¦ must be equal to 90 degrees because if we subtract 90 degrees from each side of the equation, weβre left with just π₯ plus π¦ 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 π and π¦ that both must be equal to the same thing because both π and π¦ plus π₯ both equal 90 degrees. So therefore, we can say that π is equal to π¦. Okay, we now know that relationship and itβs going to become very useful in a second.

Letβs change our focus to π. Well, to help us find π, 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 Iβve done to decide here. We could say that if the angle at circumference is π₯, then the angle of the middle is going to be two π₯. So therefore, we can say that the angle π β because itβs the angle at the circumference β is going to be equal to half the angle at the middle or the angle at the middle is going to be two π¦. So we can say that π is equal to half of two π¦, which therefore gives us that π is equal to π¦.

And this is the key fact because it brings us back to what we looked at earlier, where we said that π is equal to π¦. Well, if we got π is equal to π¦ and π is equal to π¦, then therefore π must be equal to π. So we can say okay, great, the alternate segment theorem is right and weβve proved it without working here.

Okay, now we know what the alternate segment theorem is and weβve proved it. Letβs get back on and find the measure of angle π΄π΅πΆ. Well, using the knowledge that weβve just gained, we can say that the measure of angle π΄π΅πΆ 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 weβve arrived at that because the angle at the tangent is angle π΄π΅πΆ. And this must be equal to the opposite interior angle, which in this case is angle π΄π·π΅ and 78 degrees.