The diagram shows a circle of center 𝑀 and diameter 𝐴𝐵. 𝑀𝐷 and 𝐴𝐶 are parallel. The size of angle 𝐴𝐵𝐶 is 62 degrees. Find angle 𝐴𝐶𝐷.
Always say that whenever you’re doing this kind of question, the first thing you need
to do is actually mark on any angles you can using the information that you’ve
got. Well, the first angle that we can mark on is actually the angle 𝐵𝐶𝐴. And we can actually say that this is a right angle. And we can do that because 𝐴𝐵𝐶 is a triangle inscribed in a circle, where 𝐴𝐵 is
So therefore, 𝐵𝐶𝐴 has got to be equal to 90 degrees. This is a relationship we know since it’s one of our circle theorems. The key thing here is that any value that you do find you need to give reasoning
for. And that’s what we’ve done. We gave a reasoning for why it’s actually a right angle.
The next angle we can actually mark on is the angle 𝐵𝐴𝐶, 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 we’ve got is a triangle 𝐴𝐵𝐶.
Well, we’ve got the right angle, which is 90 degrees. Then we’re also given 62 degrees. Add these two together; we get 152 degrees, and 28 degrees will give us 180
degrees. So we’ve actually calculated that and checked it. So we know that the angle 𝐵𝐴𝐶 is 28 degrees.
So we can also actually mark on angle 𝐴𝑀𝐷 cause we can say that angle 𝐴𝑀𝐷 is
also gonna be 28 degrees. And we know that 𝐴𝑀𝐷 is gonna be equal to 28 degrees because 𝐴𝑀𝐷 and 𝐵𝐴𝐶 are
alternate angles on parallel lines. This is sometimes known as Z angles, and I’ve 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 that’s angle 𝐴𝐶𝐷. And we can actually do that using another relationship.
Well, the relationship that we have that’s 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 I’ve actually shown this in a little sketch with 𝜃 and two 𝜃. So we can actually see that the angle at the center of the circle is twice, because
it’s two 𝜃, the angle actually on the circumference, which is 𝜃. But how does this apply to our diagram?
Well, we do actually have this relationship in our diagram. I’ve actually marked in pink on our diagram where this relationship is. And actually, it looks slightly different because it’s off center, and this can be
actually what catches students out. So be careful of this because it’s such a common error.
So therefore, what we can say about our diagram is that 𝑀 is the center of the
circle and 𝐶 is on the circumference of the circle. So therefore, angle 𝐴𝑀𝐷 is going to be twice angle 𝐴𝐶𝐷. And this actually comes from the circle theorem that we just looked at.
Okay, great! So let’s use this now to actually calculate what 𝐴𝐶𝐷 is going to be. So therefore, angle 𝐴𝐶𝐷 is gonna be equal to 28 divided by two because if 𝐴𝑀𝐷
is twice 𝐴𝐶𝐷, then therefore angle 𝐴𝐶𝐷 is going to be half of angle
𝐴𝑀𝐷. So therefore, we can say that angle 𝐴𝐶𝐷 is gonna be equal to 14 degrees.