### Video Transcript

The diagram shows a circle of center π and diameter π΄π΅. ππ· and π΄πΆ are parallel. The size of angle π΄π΅πΆ is 62 degrees. Find angle π΄πΆπ·.

Always say that whenever youβre doing this kind of question, the first thing you need
to do is actually mark on any angles you can using the information that youβve
got. Well, the first angle that we can mark on is actually the angle π΅πΆπ΄. And we can actually say that this is a right angle. And we can do that because π΄π΅πΆ is a triangle inscribed in a circle, where π΄π΅ is
a diameter.

So therefore, π΅πΆπ΄ has got to be equal to 90 degrees. This is a relationship we know since itβs one of our circle theorems. The key thing here is that any value that you do find you need to give reasoning
for. And thatβs what weβve done. We gave a reasoning for why itβs actually a right angle.

The next angle we can actually mark on is the angle π΅π΄πΆ, cause this is 28
degrees. And we get this because 180 minus 90 minus 62 equals 28. And we knew to use this calculation because the angles in a triangle actually add up
to 180 degrees. So what weβve got is a triangle π΄π΅πΆ.

Well, weβve got the right angle, which is 90 degrees. Then weβre also given 62 degrees. Add these two together; we get 152 degrees, and 28 degrees will give us 180
degrees. So weβve actually calculated that and checked it. So we know that the angle π΅π΄πΆ is 28 degrees.

So we can also actually mark on angle π΄ππ· cause we can say that angle π΄ππ· is
also gonna be 28 degrees. And we know that π΄ππ· is gonna be equal to 28 degrees because π΄ππ· and π΅π΄πΆ are
alternate angles on parallel lines. This is sometimes known as Z angles, and Iβve actually shown this with a little
sketch to the side.

Okay, great! But what else can we actually mark onto our diagram? Well, now what we can do is actually move on to the angle that we want to find in the
question, and thatβs angle π΄πΆπ·. And we can actually do that using another relationship.

Well, the relationship that we have thatβs actually gonna help us to find this angle
is this one here. So what we can see is that actually the angle subtended by an arc at the center is
twice the angle subtended at the circumference. And Iβve actually shown this in a little sketch with π and two π. So we can actually see that the angle at the center of the circle is twice, because
itβs two π, the angle actually on the circumference, which is π. But how does this apply to our diagram?

Well, we do actually have this relationship in our diagram. Iβve actually marked in pink on our diagram where this relationship is. And actually, it looks slightly different because itβs off center, and this can be
actually what catches students out. So be careful of this because itβs such a common error.

So therefore, what we can say about our diagram is that π is the center of the
circle and πΆ is on the circumference of the circle. So therefore, angle π΄ππ· is going to be twice angle π΄πΆπ·. And this actually comes from the circle theorem that we just looked at.

Okay, great! So letβs use this now to actually calculate what π΄πΆπ· is going to be. So therefore, angle π΄πΆπ· is gonna be equal to 28 divided by two because if π΄ππ·
is twice π΄πΆπ·, then therefore angle π΄πΆπ· is going to be half of angle
π΄ππ·. So therefore, we can say that angle π΄πΆπ· is gonna be equal to 14 degrees.