Lesson video

Hello there, and thank you for joining me.

My name is Dr.

Rowlandson, 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 angles in the same segment are equal".

Now this is a circle theorem, which by the end of today's lesson we'll be able to derive and solve.

Here are some previous keywords that will be useful during today's lesson, so you might want to pause the video if you need to remind yourself what any of these words mean, then press play when you're ready to continue.

The lesson is broken into two learning cycles.

In the first learning cycle, we're going to be investigative angles using circular geoboards.

That's so that we can look for particular patterns and relationships as we work out missing angles in a circular geoboard, in the hope we might observe some cases where this circle theorem happens.

Then in the second learning cycle, we're going to be introducing the circle theorem more formally.

Let's start off with investigating angles with circular geoboards.

Before we start looking at today's circle theorem, let's pause and think about circle theorems in general, what they are, and what they tell us.

Circle theorems describe relationships between angles or lengths in figures containing a circle and lines or line segments.

Let's take a look at some circle theorems which you may already know.

If we have two angles that are subtended by the same arc, where one angle is at the centre of the circle, and the other is at a point on the circumference, then the angle at the centre is double the angle at the circumference.

Another circle theorem is that if we have a circle with a diameter, the diameter will split the circle into two semicircles, and the angle in a semicircle is always a right angle.

If we have a chord and a line or line segment that is perpendicular to that chord, at the point where they meet, the chord will be bisected by the line which is perpendicular to it.

And if we have a tangent and a radius that meets at a point on a circle, the tangent and radius will be perpendicular.

Circle theorems can be proven to be always true, and knowledge of circle theorems can be used to find unknown angles or unknown limbs in a figure containing a circle and lines or line segments.

Also, knowledge of circle theorems can be used to explore other properties of circles, and to deduce additional circle theorems. So during today's lesson, we will use some of the circle theorems we know in order to understand a new circle theorem.

Let's just look at one more thing before we look at today's circle theorem, and that is how to find missing angles on a triangle that is constructed on a circular geoboard.

So here we have a circular geoboard that contains 10 points, equally spaced around a circle, and a point in the centre.

And we have a triangle inside that geoboard.

One angle is marked x.

Let's find the value of x.

Now, you might already know what the value of x is based on maybe some previous circle theorems. If that's the case, let's just pretend that you don't know what it is, and consider how we could work out the angle based on the properties of the circular geoboard.

What we could do is we could draw a radius from the centre of the circle to the point where that angle is.

We now have three radii on this circle, which are all equal length.

That means we also have two isosceles triangles.

We can use that property to help us find missing angles.

Also, these radii split the circle into sectors.

So this angle here is 1/10th of 360 degrees because the sector which that angle is at the centre of is 1/10th of the whole circle, because there are 10 points around that circle.

Now we know that, we can work out the other two angles in that isosceles triangle.

We could do 180, subtract 36, then divide our answer by two to get 72 degrees for each angle.

And we can do the same now with the other isosceles triangle.

This angle here is 4/10ths of 360 degrees, because that sector is 4/10ths of the overall circle.

And then once we know it's 144 degrees, we can work out the other two angles by doing 180, subtract 144, and divide 'em by two to get 18 degrees for each angle.

Now we know those angles, we can work out the value of x by adding 72 and 18 together.

And that would give 90 degrees.

So here's one for you to try.

You've got the same circular geoboard, and another triangle now.

Please find the value of y.

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

The answer is 90 again.

And there's the working for how to get it.

Now both of these questions have something in common.

One of the sides of the triangle is the diameter.

In particular, it's a side which is opposite the angle we've just found.

Now a diameter splits a circle into two semicircles, and each time we have this angle in a semicircle, it's 90 degrees.

Here are some more examples of that.

In each figure, the angle at the circumference is subtended by the diameter, and in each case, the angle is a right angle.

This is because angles in a semicircle are always a right angle.

Ah, here comes Sofia! She says, "I wonder what would happen if the angles were subtended by a chord that was not the diameter." Well, it would look something a bit like this.

These figures show circular geoboards with nine points equally spaced around the circle and a point in the centre.

And when there's an odd number of points on a geoboard, you can't draw the diameter going from one point to another, like in these situations here.

In each of these diagrams, the angular circumference which is marked is subtended by a chord, but that chord is not the diameter.

And Sofia says, "These angles won't be 90 degrees because they are not in a semicircle." However, you may have noticed that all those angles are subtended by the same chord.

So Sofia says, "I wonder if they are related in any way." Let's investigate.

I'm going to do an example and then you'll go do one very similar.

In my example, I have one of the circular geoboards from the previous slide.

It's got nine points equally spaced around a circle, and a point in the centre.

We've got a chord drawn, and that chord subtends the angle x, and we want to find the value of x.

To do this, we could draw on two radii, and we could find this angle here at the centre.

That angle would be 4/9ths of 360 degrees, because the sector formed by those two radii is 4/9ths of the overall circle.

Four ninths of 360 is 160 degrees.

And then we could use another circle theorem to help us find the value of x.

We could say that the angle at the centre of the circle is double the angle at the circumference.

So that would mean that x is equal to 160 divided by two, which is 80.

So that means the angle is 80 degrees.

Here's one for you to try.

Find the value of y.

Pause while you do it, and press play when you're ready for an answer.

The value of y is 80, and here's your working for how to get it.

So let's now compare these examples side by side.

In both cases, the angle we found was 80 degrees, but also, in both cases, that angle is subtended by the same chord.

So let's check what we've learned.

We've got the same diagram that you're working with in the previous slide, where you've got a circular geoboard with an angle y subtended by a chord.

And then you've got three diagrams below where you have an angle which is marked, and each of those is subtended by a chord.

In which diagrams is the marked angle subtended by the same chord as the angle y? Pause the video while you choose, and press play when you are ready for answers.

The answer is a.

And the way you can see that is by looking at at which point on that circular geoboard the chord starts and ends.

Okay, it's over to you then for task A.

This task contains one question, and here it is.

Each diagram shows a circular geoboard containing nine points, equally spaced around a circle, and a point in the centre.

And each time it has an angle marked with a letter from a to j.

You need to find the values of a to j.

And then, once you've done that, look at all your answers and consider, which angles are equal to each other? And also, what else do these particular angles have in common on the diagram? Pause the video while you do this, and press play when you're ready for answers.

Okay, let's see how we get on.

Let's start by looking at a and b.

The angle at the centre for a is 4/9ths of 360 degrees, which is 160.

And then you can use the fact that the angle at the centre is double the angle at the circumference to do 160 divided by two is 80, so the value of a is 80.

And for b, when we draw on the radii to either end of that chord, the radii goes to the same two points on that circular geoboard.

So once again, the angle at the centre is 4/9ths of 360 degrees, which is 160.

And then if we divide that by two to get the angle at the circumference, we get 80 again.

So when we looked at which angles are equal to each other, a and b are equal to each other, and what they have in common is both angles are subtended by the same chord.

Let's now look at c, d, e, and f.

In each of these, when we draw two radii from the centre to the end points of the chord that subtends the angle, we are drawing that radii to the same two points on a circular geoboard.

And each time that creates a sector which is 3/9ths of the overall circle.

So the angle at the centre each time is 3/9ths of 360, which is 120 degrees.

And then we can use the fact that the angle at the centre is double the angle at the circumference, so to work out the values of each angle, we can do 120 divided by two to get 60 for c, d, e, and f.

So all these angles were equal to each other.

What else do they all have in common? Well, they are all subtended by the same chord.

Then we've got g, h, i, and j.

Once again, when we draw two radii from the centre of the circle to either end of the chord that subtends the angle, we are drawing that radii to the same two places on that circular geoboard, and that creates a sector which is 2/9ths of the overall circle, so the angle at the centre is 2/9ths of 360, which is 80 degrees, and then the angle at the circumference will be half of that each time, which will be 40 degrees.

So all four of these angles are equal to each other, and what do they all have in common? You guessed it.

They are all subtended by the same chord.

You're doing great so far.

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

Here we have four diagrams, which each show a circular geoboard containing nine points equally spaced around a circle, and a point in the centre.

And you'll notice that each of those diagrams contains an angle which is marked.

The marked angles are all equal to each other.

In this case, they're all 60 degrees.

Now Sofia notices something else.

She says, "All the marked angles are subtended by the same chord." So here we can see lots of cases where angles that are subtended by the same chord are equal to each other.

If we want to explore more cases, we could do so on a circular geoboard, but they're quite limited to how many different examples you can explore.

So Lucas says, "We could explore more cases by using dynamic geometry software." If you have access to this slide, at the bottom of this slide, there's a link to a GeoGebra file where you can explore this in a little bit more detail for yourself.

It looks something a bit like what you can see on the screen here.

So let's look at it together.

Sofia says, "Angle DAE and angle DBE are both subtended by the same arc." It's that arc that goes from D to E, which is highlighted at the bottom of the circle.

Lucas says, "If we draw a chord from D to E, we can see that angle DAE and DBE are also subtended by the same chord." So those two angles are subtended by the same arc, and are subtended by the same chord.

They are both, in this example, 64 degrees.

Sofia says, "Angle DAE and angle DBE are equal to each other." Lucas says, "I wonder what would happen if we change the positions of points D and E." Let's see what happens.

If we move point E, we can see that both of those angles have changed, but they are still equal to each other.

They're both 71 degrees.

If we move point D, we can see that they're now both 58 degrees.

If we move point E again, they both change to 30 degrees.

If move point D, they both change to 51 degrees.

Sofia said, "Angle DAE and angle DBE both change each time, but they remain equal to each other." Lucas says, "I wonder what would happen if we changed the position of point A." That's the point where the angle DAE is.

Would the angle DAE change if we did that? The angle is currently 61 degrees, but if we drag point A clockwise around this circle, how might that affect the angle? Well, let's take a look.

If we move the pointer here, we can see the angle is unchanged.

It's still 61 degrees.

If we move it to here, it's still 61 degrees.

It's still 61 degrees.

And it's still 61 degrees! Sofia says, "Angle DAE has not changed.

It has remained equal to angle DBE." Lucas says, "I wonder if it is always the case that angles subtended by the same arc or chord are equal." For example, let's move point A a little bit further.

What do you think will happen as we move point A even further clockwise around the circle? Perhaps pause the video and think about this, and press play when you're ready to continue.

Okay, let's see what happens.

We move point A a little bit further around the circle.

The angle is still 61 degrees.

It's still equal to the other angle.

But if we move that point a little bit further again, so it goes past point E, ah no, it's changed! It's now 119 degrees.

And if you move it down here, it's still 119 degrees as well.

So Sofia says, "Angle DAE has now changed.

It is no longer equal to angle DBE." Lucas says, "I wonder why the angle changed.

What are the conditions that makes angle DAE equal to angle DBE?" We saw lots of cases where that angle was 61 degrees all the way as we moved it clockwise around the circle.

But why did it then change when it passed point E? And if we continued moving that point clockwise around the circle, when might it change again? Pause the video while you think about this, and press play when you're ready to continue.

I wonder what you thought.

Let's take a look at this one more time with the dynamic geometry software.

We can see that both of the highlighted angles are currently equal to each other.

They are both 64 degrees.

But watch what happens to angle DAE as we move the point A clockwise around a circle.

The angle remains the same, and when we get to this point, we can see very clearly that those two angles are equal to each other.

And the angle remains the same until the point passes point E.

And now those angles are different to each other, until the point passes point D, and now they're the same as each other again.

The angles were equal to each other while they were in the same segment, and that is the circle theorem of this lesson.

The circle theorem is that the angles in the same segment are always equal.

When you've got a circle and you draw a chord, the chord splits a circle into two segments, and if you have two or more angles that are subtended by that chord in the same segment as each other, those angles will always be equal to each other.

Let's take a look at some examples.

In each of these diagrams we have points D, E, and we could draw a chord from D to E, and it would split the circle into two segments.

With the example on the left, once we draw in our chord, we can see that the two highlighted angles are in the same segment as each other.

They're both in that major segment above the chord, and they're both equal to each other.

But with the example on the right, once we draw our chord DE in, we can see that the two highlighted angles are in different segments to each other.

One of them is in the major segment above the chord, and the other is in the minor segment below the chord.

And we can see that those two angles.

So when we have two angles which are subtended by the same chord, they are equal while they're in the same segment, but they are not equal, or not necessarily equal, while they're in different segments.

Let's check what we've learned.

Here we've got a diagram with four points around a circle, and you've got two angles highlighted.

Angle CAB, and angle CDB.

True or false? Those two angles are equal to each other.

Pause while you write down an answer, and also write down an explanation for why, and press play when you're ready to see the answer.

The answer is true, and the reason why is, if we draw a chord from B to C, we can see that both of those highlighted angles are subtended by that chord and are in the same segment.

And our circle theorem says that angles in the same segment are equal.

So same diagram again, but two different angles are highlighted this time, angle ABD and angle ACD.

True or false? These two angles are equal to each other.

Pause while you choose, and write down a reason why.

Then press play when you're ready for an answer.

The answer again is true.

And the reason why is if you draw a chord from A to D, you can see that both those angles highlighted are subtended by that chord and are in the same segment, and our circle theorem says that angles in the same segment are equal.

So how about this one? We've got angle ABC and angle ADC are both highlighted.

True or false? These are equal to each other.

Pause while you choose and write down a reason why, and then press play for an answer.

The answer is false, and the reason why is, if we draw a chord from A to C, yes, we can see that both those angles are subtended by the same chord, but they are not in the same segment.

And also, they don't comply with any rules or relationships that would suggest that they are equal.

So no, they are not equal.

And here we have another one.

Angle ACB and angle OAB.

True or false? These two angles are equal.

Pause while you choose, and write down a reason why, and press play for an answer.

The answer is false, and the reason why is, both of the angles are not on the circumference.

One angle is at the centre, and the other is at the circumference.

And actually, that relates to a different circle theorem.

That is the angle at the centre is double the angle at the circumference.

And this time we have a slightly more complex diagram, with points A to G around a circle, and we have five angles highlighted.

Angle GAD is the angle in the top left of the circle at point A.

And what I want to know is, which angle or angles are equal in size to that angle? GAD.

Pause while you choose, and press play when you're ready for answers.

The answers are A and B.

All five of those angles are subtended by the same chord, GD.

And you can see that by the angle notation.

They all start with G, and they all end with D.

But if you draw on that chord, you'll see that only angles GBD and GCD are in the same segment as the angle GAD.

So, same diagram again.

Which angle or angles are equal to the angle GFD? That's the one in the bottom left of the circle.

Pause while you choose, and press play when you're ready for an answer.

The answer is D.

The angle GED is equal to the angle GFD because they are subtended by the same chord, and they are both in the same segment.

So our circle theorem for today's lesson is that angles in the same segment are always equal, and we can see two examples of that on the screen.

But here's Sofia, who makes a really good point.

She says, "Geoboards and dynamic software only demonstrate that angles in the same segment are sometimes equal." We can use these tools to see lots and lots of cases where they are equal, but they don't prove that they are equal in all cases.

Lucas says, "We could use algebra and other circle theorems to prove that they are always equal." So we're going to do that together now, and I'm gonna break it down into a couple of smaller steps, and you're going to help me with them.

Here we have a circle with three points around the circumference.

Points A, C, and D.

Now you might be thinking, "Where's point B?" We'll see that shortly.

And we've got a point in the centre labelled O.

Let's let angle DAC equal X degrees.

So rather than giving it a numerical value, we'll give it a variable value.

And then what I want you to do please is express angle DOC in terms of x.

That's the angle in the centre.

And justify your answer with reasoning.

Pause the video while you do that, and press play when you're ready for an answer.

The angle at the centre, which is angle DOC, would be equal to 2x degrees, and that's because the angle at the centre is twice the angle at the circumference when they are subtended by the same arc.

That's one of our previous circle theorems. So if we add another point to this circle, point B, and we create another angle, which is from D to B to C, could you please express angle DBC in terms of x, and write down a reason for your answer? Pause the video while you do that, and press play when you're ready for an answer.

Angle DBC is also equal to x degrees, and that's because the angle at the centre, which is 2x degrees, is double the angle at the circumference, so it must be x degrees.

So what we can see now is that those two angles which are on the circumference, angle DAC and angle DBC, they are both subtended by the same arc.

They're also both in the same segment, and they are both half the size of the angle at the centre.

Therefore, they are both equal to each other.

That means angles DAC equals DBC, and angles in the same segment are always equal.

It's proven.

Okay, it's over to you now for task B.

This task contains two questions, and here is question one.

We're working with the circle theorem that is, "angles in the same segment are equal".

So in part A, you are shown four diagrams where each diagram has two angles marked, and what you need to do is tick the figures where this circle theorem applies to those two marked angles.

And then in part B, I'd like you to come up with some examples yourself for diagrams where that circle theorem does apply.

Draw a circle, draw some line segments in it, and mark two angles, and make it so that those two angles comply with the circle theorem which is, "Angles in the same segment are equal." Pause the video while you do that, and press play when you're ready for question two.

Here is question two.

You've got a bunch of diagrams with angles marked with unknowns from a to o.

Could you please find the values of those unknowns and justify your answer with reasoning each time? So anytime you use a circle theorem, write down which circle theorem you use.

If you use any other facts relating to angles, write down those facts if you use them as well.

Pause the video while you do this, and press play when you're ready for answers.

Okay, let's take a look at some answers! In question one, part A, it's these figures where the circle theorem applies, where angles in the same segment are equal.

In both of those figures, the angles are subtended by the same chord, and they are in the same segment.

And then in part B, you have to draw some examples for yourself.

Well, your answers will vary to each other.

So to check, you could draw a chord that subtends both of your angles, and check that both angles are in the same segment, and check that they're both on the circumference.

You could also check by measuring the angles with a protractor.

They should both be equal to each other.

Then question two, we had to find the values of the unknowns and justify our answers with a reason.

That meant we have to either write down which circle theorem we used or which angle facts we used.

For a and b, each of those two angles are equal to other angles in that diagram.

We can use the fact that angles in the same segment are equal to say that a is equal to 63, and b is equal to 19.

And then in this diagram, the two unknown angles c and d are equal to the angles we are given.

We just have to choose the right ones.

Angles in the same segment are equal, so the angle which is labelled c is four degrees, and the angle labelled d is 41 degrees.

This diagram is ever so slightly different to the previous one, in that we are only given the size of one angle, four degrees, but we are told that two of the chords are parallel to each other by the feather markings on there.

So we can bring in other angle facts to help us with this as well.

In particular, facts about angles between parallel lines.

We can say that angles in the same segment are equal, to notice that angle e degrees must be 40 degrees, and then we can use the fact that alternate angles in parallel lines are equal.

Now f is alternate to the 40 degrees that we're given, and g is alternate to the angle labelled e.

So they all must be equal to each other.

They're all 40 degrees.

Then this diagram.

We could use the fact that angles in the same segment are equal to say that h is equal to 46, and then we could use the fact that angles in a triangle sum to 180 degrees to work out the value of i by subtracting the two angles in that triangle from 180 degrees.

And once you know that, there's quite a few different ways you can get the value of j, but you could say that angles in the same segment are equal, i is equal to j, so that means they're both 29 degrees.

Then this diagram, once again we can say angles in the same segment are equal.

But we have to match the right unknown to the right angle, and what helps is to draw the chord in so you can see which ones are in the same segment.

We can see k would be equal to 100, and l would be equal to 80.

In this diagram, there are a couple of different approaches you could take.

You can use the fact that the angle at the centre of the circle is double the angle at the circumference to go from 134 degrees to m being 67 degrees.

And then you could use the fact that angles in the same segment are equal, and notice that m must be equal to n, so they are both 67 degrees.

Or you could just use the first fact twice.

134 divided by two gives you m, and 134 divided by two gives you n as well.

In this diagram, we need to find the value of o, but we may need to find some other angles first and use some other angle facts along the way.

If DB is diameter, that means angle DCB is a right angle, because the angle in a semicircle is a right angle.

And then we can use the fact that angles in a triangle sum to 180 degrees to get DBC as being 47 degrees.

And then we can see that the angle marked o degrees is subtended by the same chord as the angle marked 47 degrees, and they're in the same segment, which means o must be 47.

Fantastic work today.

Now let's summarise what we've learned.

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

And we can use algebra and geometric reasoning to prove that angles in the same segment are always equal.

Theorems can be thought of as puzzles to solve, and that can be really fun.

But your job is to show how you get the results each time.

And in order to use this theorem, and any other circle theorem sometimes, you may need to draw a diagram or add information to an existing diagram.

Great job today.

Well done.