Lesson video

Hello there, and thanks for joining me today.

My name is Dr.

Robson and I'll be guiding you through today's lesson.

Let's get started.

Welcome to today's lesson from the unit of "Circle theorems." This lesson is called "The opposite angles of a cyclic quadrilateral sum to 180 degrees," and that is a circle theorem, which by the end of today's lesson, we'll be able to derive and use.

This lesson will introduce a new keyword, which is cyclic quadrilateral.

A cyclic quadrilateral is a quadrilateral where all four of its vertices lie on a single circle and we'll see plenty of examples of that during today's lesson.

The lesson is broken into two learning cycles.

In the first learning cycle, we're going to be investigating angles using circular geoboards, and that is so we can explore patterns and relationships between angles for particular cases.

And then, in the second-half lesson, we're going to generalise this by introducing the circle theorem for all cases.

But let's start off with investigating angles with circular geoboards.

Here we have a diagram that shows a circular geoboard with nine points equally spaced around the circle and a point in the centre.

A cyclic quadrilateral is a quadrilateral where all four of its vertices lie on a single circle.

And we have an example of that on the screen here.

Here's an example of something that is not a cyclic quadrilateral.

It is a quadrilateral because it has four vertices and four sites, but it's not a cyclic quadrilateral.

And the reason why, it's not cyclic is because all four of its vertices are not on the circumference.

One of vertices is at the centre.

Here we have Aisha who has a cyclic quadrilateral on a nine-point circular geoboard and she wants to calculate the size of its interior angles.

Now, Aisha's gonna talk us through this in a moment, but perhaps pause the video and think about how you might go about doing this yourself and then press play when you're ready to see what Aisha does.

Let's see what she does.

She says, "I'll start by drawing two radii to opposite vertices of the cyclic quadrilateral.

And then I'll work out the size of the two angles at the centre." These two radii split the circle into two sectors.

The sector on the right, the minor sector is 3/9 of the overall circle.

So the angle in that sector at the centre of the circle would be 3/9 of 360 degrees, which is 120 degrees.

Then to work out the over angle, we could use the fact that angles around a point sum to 360 degrees, so we could do 360 subtract 120 to get 240 degrees for that over angle.

Then Aisha marks these two angles here and she says, "The angles at the circumference are half the size of the angles at the centre," or in other words, the angle at the centre is double the angle at the circumference.

That means, we can work out these two interior angles of the cyclic quadrilateral by dividing the angles at the centre by 2.

We could do 120 divided by 2 to get 60 degrees for one of them, and we can do 240 divided by 2 to get 120 degrees for the other.

So now we have two of the angles of the cyclic quadrilateral.

What about the other two? Aisha says, "I could then repeat the same process by drawing radii to the other pair of opposite vertices." We could get the angle at the centre by doing 4/9 of 360 degrees to get 160, and then we could subtract that from 360 to get the other angle at the centre that is 200 degrees, and then we could divide both those angles by 2 to get the angles at the circumference.

That would give us 80 degrees and 100 degrees.

And now we have all four interior angles of this cyclic quadrilateral.

Aisha says, "A similar process could be used for other cyclic quadrilaterals that are on a circular geoboard." She says, "We could also look out for other potential relationships between the angles too." We know one relationship for this is that angles in any quadrilateral sum to 360 degrees, but I wonder if there are any other relationships between angles when it's a cyclic quadrilateral.

Let's explore.

Here's one for you to work with.

The diagram shows a circular geoboard with nine points equally spaced around a circle and a point in the centre.

You've got an angle marked a degrees.

Could you please work out the value of a? Pause while you do it and press play for an answer.

a is 4/9 of 360, which is 160.

So could you use that answer to work out the value of b and then justify your answer with reasoning? In other words, write down what circle theorem or angle factor you use to get the value of b.

Pause video while you do it and press play for an answer.

b is equal to 80 and that's because the angle at the centre is twice the angle at the circumference.

Or you can say the angle at the circumference is half the angle at the centre, mean the same thing.

Once you've got that, could you now work out the value of c? Pause video while you do it and write down any angle facts you use along the way and press play when you're ready for an answer.

c is equal to 200.

You get that by doing, 360 subtract 160 because angles around a point sum to 360 degrees.

And now work out the value of d and justify your answer with reasoning.

Pause while you do it and press play for an answer.

d is equal to 100.

I can use the fact that the angle at centre is twice angle circumference and do 200 divided by 2.

And now, could you work out the values of e and f? And this time, you'll have to break this down into small steps yourself.

Pause while you do it and press play for an answer.

e is equal to 100 and f is equal to 80 and you can justify it either by phone, the same steps we did earlier to work out the first 80 and 100 degrees, they can see on the diagram.

Or you might have spotted that this quadrilateral is in fact an isosceles trapezium by counting the pegs around the circle.

So it's symmetrical, which means e is equal to 100 and f is equal to 80 in that way as well.

Okay, it's O2 now for task A.

This task contains four questions and here is question one.

Pause video while you work through this and press play when you're ready for question two.

And here is question two.

It's the same cyclic quadrilateral as in question one, but you need to work out different angles this time.

Pause video while you do this and press play when you're ready for question three.

And here is question three.

It's a different cyclic quadrilateral.

You need to work out the size of the four interior angles, but this time, you need to break the problem down to small steps for yourself.

Pause video while you do it and press play when you're ready for question four.

And here is question four.

You've got two diagrams with unknown angles labelled from a to d, which you need to find the values of.

And what you may want to do is draw some extra line segments on the diagram to help you.

You don't have to, but you may want to.

Pause video while you do it and press play when you're ready for answers.

Okay, let's go through some answers.

For question one, a is equal to 120, it's 4/12 of 360, b is equal to 240.

That's because angles are around a point sum to 360 degrees.

c is equal to 60 and that's because the angle at the centre is twice the angle at circumference.

And d is equal to 120 for the same reason.

Then for part e need to work out what the values of a and b sum to, those are two angles at the centre of the circle, well, they sum to 360 and you can see that by adding up your values from a and b, but also angles around a point sum to 360 degrees.

Then for f, you had to work out what the values of c and d sum to.

They are the opposite angles in that cyclic quadrilateral.

Well, if you add together 60 and 120, you get 180.

Then in question two, you had the same cyclic quadrilateral and you had to work out the size of the four marked angles.

Well, one of the angles at the centre will be 210 degrees because it's 7/12 of 360 degrees.

The other angle at the centre will be 150 degrees.

You can get that by using angles around a point sum to 360 degrees or you may have worked those out in the opposite order.

Once you have those two angles at the centre, you can use the fact that angle at the centre is equal to twice the angle at circumference and divide 'em both by 2.

Get each of the angles at the circumference.

And that'd be 105 degrees and 75 degrees.

In part b, you had to say what the two angles in the centre sum to.

Well, 210 plus 150 is 360, or you use the fact that angles around a point sum to 360 degrees as they do.

And then you have to write down what the opposite two angles in a cyclic quadrilateral sum to.

Well, if you do 105 plus 75 you get 180.

You may have noticed with questions one and two that you found yourself writing the same thing each time.

The two angles at the centre of the circle, sum to 360 degrees and the two opposite angles of the cyclic quadrilateral sum to 180 degrees.

To get a sense of why you could think about what processes you used to get from the angles at the centre to the angles at the circumference.

We'll come back to this later.

And for question three, you have a new cyclic quadrilateral and you have to work out the size of all four its interior angles.

We could use a similar process, draw on a pair of radii, work out the angles at the centre and have them to get the angles at a circumference.

When you draw on the other two radii, what you actually have here is a diameter.

So this time the other two angles are 90 degrees.

Once you've done that in part b, let write down what is each pair of opposite angles of the cyclic quadrilateral sum to.

Well, 72 plus 108 is 180 and 90 plus 90 is 180.

So the opposite angles of the cyclic quadrilateral here sum to 180 degrees.

And with question four, you have to work out the values of a to d.

With a diagram on the left, a would be equal to two times 70 'cause the angle at the centre is twice down at circumference.

You then subtract your answer for that from 360 to get the value of b, 'cause angles are around a point sum to 360, and then you could divide that by 2 to get the value of c because the angle at the centre is twice the angle at the circumference.

Now, by the time, you get to d, you might be pretty convinced about the opposite angles in a cyclic quadrilateral sum to 180 degrees.

So you could work out the value of d by doing 180 subtract 82.

However, you would be quite right to not be fully convinced by this yet because it hasn't been proven to be true in all cases.

We've just seen it happen in lots of cases.

So you ever in doubt, you could always do the same process again.

Work out the angles at the centre and then use them to work out the value of d as 98.

You're doing great so far.

Now, let's move on to the next part of this lesson where we're going to introduce the circle theorem.

Here we have two diagrams, which each shows a circular geoboard containing points equally spaced around a circle and a cyclic quadrilateral in each.

In both of these diagrams, we can see that the opposite angles in the cyclic quadrilaterals sum to 180 degrees.

For the diagram on the left, we can see that 120 degrees plus 60 degrees is 180 degrees, and we can also see that 105 degrees plus 75 degrees is also 180 degrees.

With the diagram on the right, we can see this again.

72 degrees plus 180 degrees is 180 degrees, and the two right angles also sum to 180 degrees.

Now, geoboards are great, but they are quite limited in terms of how many different examples you can explore.

Jacob says, "We could explore more cases by using dynamic software." For example, here we have a diagram containing a cyclic quadrilateral, and at the bottom of this slide, there is a link to a GeoGebra file that contains an interactive version of this diagram.

What we can see here, it's at the opposite angles in this cyclic quadrilateral sum to 180 degrees, we have 70 degrees plus 110 degrees, and we have 100 degrees plus 80 degrees.

And this is always the case, regardless of where the vertices of the cyclic quadrilateral are on the circle.

If we move some of these vertices around like this, we'll see that the opposite angles still sum to 180 degrees.

We can keep moving them around and you'll still sum to 180 degrees.

Let's check what we've learned.

Here we have a cyclic quadrilateral.

Could you please find the values of x and y? Pause the video while you do it and press play when you're ready for answers.

Well, the opposite angles of the cyclic quadrilaterals sum to 180 degrees, so x will be equal to 180 subtract 105, which is 75, and y would be equal to 180 subtract 56, which is 124.

Here we have another cyclic quadrilateral, and you've got three angles labelled a, b, and c.

Which one is equal to 88 degrees? Pause video while you choose and press play for an answer.

The answer is b.

That's because the opposite angles in a cyclic quadrilateral sum to 180 degrees.

So the circle theme that we are focusing on in today's lesson is that opposite angles in a cyclic quadrilateral sum to 180 degrees.

And we're seeing lots of examples during this lesson where that happens.

And here are two examples on the screen.

But here's Jacob who's going to make a really good point.

"Geoboards and dynamic software only demonstrate that opposite angles of a cyclic quadrilateral, sometimes sum to 180 degrees." They don't prove that they always do.

We've seen lots of examples of where it happens and we can use these tools to explore lots of more examples, but they don't prove that it's the case every time.

In order to do that, we need to start thinking about using a bit of algebra to generalise what's going on.

Aisha says, "We could use algebra and other circle theorems to prove that they always do." And we can do that by breaking the proof into small steps.

And you are going to help me.

Here we have a diagram container circle with points A, B, and D on a circle.

You may be wondering where points C is.

We'll see that later.

Let's let angle DAB equal x degrees.

So rather being a particular number, it is a variable.

Could you please express the angle DOB in terms of x? That's the angle at the centre and justify your answer with reasoning.

Pause video while you do that and press play for an answer.

Angle DOB would be equal to 2x degrees and that's because the angle at the centre is twice the angle at the circumference.

So could you please now express the reflex angle BOD in terms of x? And justify your answer with reason, again.

Pause video while do it and press play for an answer.

The reflex angle BOD will be equal to 360 subtract 2x degrees, and that's because angles are added point sum to 360 degrees.

So you do 360 degrees, subtract the expression 2x degrees.

And then we've got add point C to make a cyclic quadrilateral.

Could you please express the angle B, C, D in terms of x and justify your answer with reasoning? Pause video while you do it and press play for an answer.

Well, the angle at the centre is twice the angle at the circumference.

So we could take the 360 subtract 2x and divide it by 2 to get the angle B, C, D as 180 subtract x degrees.

And that now provides our proof because no matter what possible value x takes, we get the angle DAB and we can subtract it from 180 degrees to get the angle B, C, D.

Therefore, the opposite angles in a cyclic quadrilateral sum to 180 degrees and it can show that algebraically by taking the expressions that we have for those two angles and adding them together.

180 subtract x degrees and then plus x degrees gives us 180 degrees.

Okay, it's over to you now for task B, this task contains four questions and here is question one.

You need to find the values of the unknowns labelled a to g in these diagrams and justify your answers with reasoning.

That means, write down any circle theorems you use or any angle facts you use to work out any missing angles along the way.

Pause video while you do it and press play for more questions.

And here are questions 2 to 4.

Pause video while you do these and press play when you're ready to look at answers.

Okay, let's take a look at some answers.

In this first diagram, a is equal to 93 and b is equal to 114.

And that's because opposite angles in a cyclic quadrilateral sum to 180 degrees.

In this diagram, could first work out c, by using the fact that opposite angles in a cyclic quadrilateral sum to 180 degrees and c would be 68.

And then we can use the fact the angles in a triangle sum to 180 degrees and base angles in isosceles triangle are equal, to work d out as 56.

And then for this diagram, we could first work out the missing angle at the top there, the angles and a triangle sum to 180 degrees and base angles and isosceles are equal, so that means that missing angle will be 106 degrees.

And then we could work out e by using the fact that opposite angles in a cyclic quadrilateral sum to 180 degrees.

So e must be 74.

In this question, we could start by working out this unknown angle as 112 degrees because angles and triangles sum to 180 degrees and base angles in isosceles triangle are equal.

And then we could work out the opposite angle in the cyclic quadrilateral with that one by ignoring that chord that goes through the middle of that cyclic quadrilateral.

I would get 68 degrees for that one.

And then we have one of the angles in that isosceles triangle that contains f.

We could work out f by doing angles in a triangle sum to 180 degrees and base angles of an isosceles are equal.

So f would be equal to 44.

And then with this one, we could first work out this angle here, which is opposite the 60 degree angle by doing opposite angles in a cyclic quadrilateral sum to 180 degrees.

And then there's a lot of different ways, we can get the value of g.

We could use that co-interior angles between parallel lines sum to 180 degrees, for example, and get g as 60 that way.

And then with question two, you were presented with a quadrilateral that isn't inside a circle, but you need to work out whether or not it could be.

Could all four of vertices lie on a single circle? Well, if they could, that would make it a cyclic quadrilateral, which means it's opposite angles would sum to 180 degrees.

One way you can approach this is by working out the missing angle first, by doing the fact that all angles inside a quadrilateral sum to 360 degrees and then you can get 91 degrees for that missing angle.

And once you've done that, you can check whether or not the opposite angles sum to 180 degrees, they do.

Therefore, that means, yes, it could be a cyclic quadrilateral, and that means, yes, all four its vertices could lie on a single circle.

Now, you didn't have to actually work out the missing angle at 91 degrees an alternative justification could have been to say that 102 degrees plus 78 degrees equals 180 degrees.

And that means the other angles must also sum to 180 degrees because that's all what's left in the quadrilateral and therefore, it could be a cyclic quadrilateral.

Now, we're question three.

We have Alex who says, "If a cyclic quadrilateral is a kite, then he must have at least two right angles." And you have to explain why his claim is correct.

Well, a kite has at least one pair of opposite angles, which are equal.

And opposite angles in a cyclic quadrilateral sum to 180 degrees.

So if opposite angles are equal and sum to 180 degrees, then they must be 90 degrees, each one must be 180 divided by 2.

And then with question four, you had to work out whether it was possible for a cyclic quadrilateral to be a parallelogram without also being a rectangle or a square and justify your answer with reasoning.

Well, a parallelogram has two pairs of opposite angles that are equal.

And opposite angles in a cyclic quadrilateral sum to 180 degrees.

So if a pair of opposite angles are equal, each one must be 90 degrees, 180 divided by 2.

If all four angles are 90 degrees, then it's either a square or a rectangle, which means no, it's not possible for a cyclic quadrilateral to be a parallelogram without also being a rectangle or a square.

Fantastic work today.

Now, let's summarise what we've learned.

A theorem is a statement that can be demonstrated to be true by using accepted mathematical operations and arguments.

A cyclic quadrilateral is a quadrilateral where all four of its vertices lie on a single circle.

Algebra and geometric reasoning can be used to prove that the opposite angles of a cyclic quadrilateral sum to 180 degrees.

And theorems can be thought of as a puzzle to solve.

And what your job is, is to show how you get the results.

And in order to use this theorem, you may need to draw a diagram or add extra information or extra line segments onto an existing diagram.

Well done, today.

Have a great day!.