Lesson video

Hi, I'm Mr. Chan.

And in this lesson, we're going to learn about the Circle Theorem that looks at angles in a cyclic quadrilateral.

Let's begin this lesson by looking at this example.

We're asked to work out the size of angles x and y.

So we can use the circle theorem that we've looked at previously, the angle at the centre is twice the angle at conference.

So what we can see here is that we've got 130 degrees at the centre, and angle x is at this conference.

So that means angle x must be 65 degrees.

Similarly, angle y is at this conference and the 230 is at the centre.

So what we can say there is y must be equal to 115 degrees again, because the angle the centre is twice the angle at the circumference Now, if we were to work out what x add y is they would equal 180 degrees.

Moving on to this example, now.

Again, we're asked to work out the size of angle x and the size of angle y.

So using the fact that the angle at the centre is twice the angle at the conference, we can say the angle x equals 98 degrees, we've got an angle at the centre subtended by the same arc of 196 degrees, angle x is subtended by the same arc, so that would be half of the 196, 98 degrees.

And also, angle y must be 82 degrees.

For the same reason that the angle of the centre is twice the angle at the conference.

We've got 164 degrees in the centre y must be 82 degrees.

Again, let's think about working out x, add y, we add those two angles together, we going to get 180 degrees.

What do you notice? Let's look at this Circle Theorem in more detail.

This Circle Theorem looks at angles in a cyclic quadrilateral.

A cyclic quadrilateral is just a four sided shape where all four vertices are on the circumference of a circle.

So we've got one shown here.

So we're asked to work what angle x add angle y is.

So using a previous Circle Theorem that we've learned, the angle at the centre is twice the angle at the conference, we can see at the centre there, we've got an angle that would be two X, because the angle the centre would be twice the angles conference.

And similarly, the other angle at the centre would be to y because the angle at the centre would be twice the angle at the conference which is y.

Now if we were to add those two angles together at the centre, we know they would equal 360 degrees, because angles around the point, add up to 360 degrees.

Now if we were to divide the equation on both sides by two, we would end up with x, add y equals 193.

So we've answered the question there.

But more importantly, what we can see is that the angles at the opposite sides of that cyclic quadrilateral add up to 180 degrees.

Now, if we introduced the other two angles in that quadrilateral, a and b there must also equal 180 degrees.

Because again, angles in the quadrilateral add up to 360 degrees in this case.

So this circle theorem leads on to the fact that opposite angles in a cyclic quadrilateral sum to 180 degrees.

Here's a question for you to try.

Pause the video to complete the task, resume the video once you're finished.

Here are the answers.

So in this first question, we've got a cyclic quadrilateral, I can say that because all four vertices of the quadrilateral are on the circumference of the circle.

Now, what we've just covered in the examples tell us the opposite angles in a cyclic quadrilateral must add up to 180 degrees.

So I can say that angle a is opposite the 91 degrees, that makes angle a 89.

And angle b is opposite the 102 degrees, that makes angle b 78 degrees.

And the reasons are opposite angles in a cyclic quadrilateral added to 180 degrees.

Here's some more questions for you to try.

Pause the video complete the task, resume the video once you're finished.

Here are the answers.

So let's look at a couple of these questions in particular.

Let's look at Part C.

Now in Part C, I can figure out what angle y is relatively easily by using opposite angles in a cyclic quadrilateral added to 190 degrees.

So angle y must be 77 degrees.

How do we find angle x? Well, looking at the notation on the diagram, we can see two parallel sides, those would be indicated by the arrows.

But what we do know about parallel lines is that co-interior angles also add up to 180 degrees.

So what we know about angles in parallel lines, angle x is co-interior to the 103 degrees.

So angle marked x must also be 77 degrees.

In question D.

Now angle x it lies in a straight line with the angle inside the cyclic quadrilateral there.

So the angle on the inside of angle x where angle x lies is opposite the 126 degrees.

So I can figure out the angle there, and then work out angle x, which also happens just to be 126 degrees.

Here's some more questions for you to try.

Pause the video to complete the task.

resume the video once you're finished.

Here are the answers question three.

Let's look at Part B.

Now in Part B, all we've been given is the exterior angles for the cyclic quadrilateral.

So in order to work out angle x and angle y, we use the fact that angles on a straight line add up to 180 degrees, and that will enable us to find the angles inside the cyclic quadrilateral.

So we can work those out and then use the fact that opposite angles in a cyclic quadrilateral add up to 180 degrees and we get the answers x equals 112 degrees and y equals 108 degrees.

Here's a prove question for you to try.

Think about the examples that we covered at the beginning of the lesson to help you pause the videos have a goal resume the video once you're finished.

Here's the proof for this circle theorem.

Now this circle theorem proof does rely upon a previous circle theorem, that the angle at the centre is twice at the angle this conference.

We did begin this lesson with some examples around that topic, so if you're unsure, have a look at those.

That's all for this lesson.

Thanks for watching.