Lesson video

Hello there and thank you for choosing today's lesson.

My name is Dr.

Rowlandson and I'll be guiding you through it.

Welcome to today's lesson from the unit of circle theorems. This lesson is called the alternate segment theorem, and by the end of today's lesson, we'll be able to derive and use this particular circle theorem.

Here are some previous keywords that'll 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, and 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 investigating angles using circular geoboards, and this would allow us to spot patterns and relationships in particular cases.

Then the second half lesson, we're going to be generalising to all cases by introducing the circle theorem more formally.

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

Let's start by reminding ourselves what circle theorems are, and of some previous circle theorems that'll be useful during today's lesson.

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

For example, here we have two angles in a circle where one is at the centre, one is at a circumference, and both those angles are subtended by the same arc.

The angle at the centre is always double the angle at the circumference.

And here we have a radius and a tangent which meet at a point on a circle, and when this happens, the tangent and radius are always perpendicular to each other.

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

And knowledge of circle theorems can also be used to explore other properties of circles and deduce additional circle theorems, and that's what we're going to be doing in today's lesson.

We're going to use the circle theorems on the previous slide to deduce a new circle theorem.

Let's start by looking at a circular geoboard.

This one contains nine points, equally spaced around a circle and a point in the centre.

And here we have Jun who is working with this geoboard.

He says, "I'll start by making an angle at the centre and work out its size.

Then I'll add more line segments to the geoboard and explore what other angles I can work out." And as we do this, let's keep in the back of our mind that some of these angles may show a relationship with other angles.

Let's see what happens.

First, Jun draws two radii.

Now radii split the circle into two sectors, and we can see the sector with the highlighted angle is 3/9 of the overall circle, which means that angle is 3/9 of 360 degrees.

It's 120 degrees.

He then adds a chord and this creates two new angles, which are now highlighted.

Jun says, "Angles in a triangle sum to 180 degrees.

Also, this triangle is made from two radii, so it's isosceles." That means those angles can be worked out by doing 180 subtract 120 and divide them by two to get 30 degrees each.

Jun then draws a tangent and that creates a new angle between the tangent and the chord, which you can see highlighted.

How can we work out the size of that angle? Well, Jun says the tangent at any point on a circle is perpendicular to a radius at that point.

We can see that the tangent does meet a radius at that particular point.

So that angle is 90 degrees, which we can now see marked on the diagram.

So the unknown angle plus the 30 degrees makes 90 degrees.

That means we can work out the unknown angle by doing 90 subtract 30 to get 60.

Then Jun draws on two more line segments to create an angle at the circumference, and this angle is subtended by the same chord as we've just been using.

So how can we work out that angle? Well, Jun says, "The angle at the centre is twice the angle I just made at the circumference." That's one of our circle theorems. The angle at the centre is 120 degrees, so the angle at circumference must be 120 divided by two, which is 60 degrees.

Then Jun notices something.

He says, "The last two angles I found are equal to each other." They're both 60 degrees.

Now these are not the only two angles in this diagram which are equal to each other.

There are two angles which are both 30 degrees, but we would expect those two angles to be equal to each other because they're both in an isosceles triangle.

It's a bit more of a surprise that the angle between the chord and the tangent and the angle at the other end of the triangle, at the circumference, that those two are equal to each other.

That was perhaps a little unexpected.

Jun says, "I wonder if these angles are always equal or if it's just the case for this particular example." Well, we can explore that by looking at some more examples.

Here's one for you to try.

The figure shows a nine point circular geoboard and you've got an angle mark there.

Could you please find the value of A? Pause while you do this and press play when you're ready for an answer.

That angle is 2/9 of 360 degrees, so it's 80 degrees.

The value of A is 80.

So now we've got another angle made at the circumference.

Could you please find the value of B and justify your answer with reasoning? Pause while you do it and press play for an answer.

B is equal to 40 and that's because the angle at the centre is twice the angle at the circumference.

So now we've drawn a chord which has created triangles, and we've got an angle marked C degrees.

Could you please find the value of C and justify your answer with reasoning? Pause while you do it and press play for an answer.

C is equal to 50, and the way we justify that is by saying angles in a triangle sum to 180 degrees, and base angles of an isosceles triangle are equal.

So an isosceles, because two of its sides are radii, so the same length.

C is therefore 50, so the angle is 50 degrees.

And now we've drawn a tangent, just created another angle between the tangent and that chord.

It's marked D degrees.

Find the value of D.

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

D is 40, and that's because the tangent at any point on a circle is perpendicular to the radius at that point, which means D degrees plus 50 degrees must be 90 degrees.

So we can do 90 degrees, subtract 50 degrees to get 40 degrees for that angle.

So just like with the previous example, it seems that the angle between the chord and the tangent is equal to the angle at circumference, which is subtended by the same chord.

They're both 40 degrees.

So we've seen two cases where this happens.

I wonder if it happens in other cases.

Let's find out.

It's over to you now for task A.

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

You've got four circular geoboards with 12 points equally spaced around a circle and a point at centre.

And the diagrams C and D include a tangent.

Now in each one, you need to work out the size of the marked angle, but what you'll notice is as you go from A to B to C, each diagram is the same as the previous one, but with some extra line segments added.

So you can use your answers from the previous question to help you with the next one each time.

And then once you've done that, write down which angles which you have found are the same as each other.

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

And here is question two.

You've got three diagrams each with an angle marked on it, A, B, and C, and you need to find the size of those angles, justifying your answers with reasoning.

That means if you use a circle theorem or any fact about angles, write that fact down.

And just like with a previous question, you'll notice that when you go from A to B to C, it's the same diagram each time, but with more lines and line segments added.

So you can use your answers from previous questions to help you with subsequent ones.

And once you've done that, write down what you notice about your answer to part C.

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

And here is question three.

You've got two diagrams which each show a circle and a tangent, and you need to find the values of each unknown and justify your answers each time with reasoning.

Pause while you do it and press play for answers.

Okay, let's go through some answers.

In question one, the angle at the centre is 120 degrees.

The angle between the chord and the radius in part B is 30 degrees.

In part C, the angle between the chord and the tangent is 60 degrees.

And in part D, the angle at circumference, which is subtended by the same chord, is 60 degrees.

So which of the angles are the same? It is the angles in part C and D.

And then question two, the angle at the centre is twice the angle at the circumference.

So the angle marked A degrees is equal to 140 degrees.

And for B, that angle is in a triangle whose angle is sum to 180 degrees, and it's also a base angle of an isosceles triangle, so it must be 20 degrees.

And then for part C, the angle between the radius and the tangent is 90 degrees because the tangent at any point on a circle is perpendicular to the radius of that point, which means the angle marked C degrees must be 70 degrees.

Now, what do you notice about your answer to part C? Well, angle marked C degrees is the same size as the first angle you were given, which is also 70 degrees.

Then in question three, you're given an angle at the circumference which is 55 degrees, and you have to use that to work out the values of A, B and C.

Well, the angle at the centre is twice the angle at the circumference, so A degrees must be 110 degrees.

And angles in a triangle sum to 180 degrees, and the base angles of an isosceles are equal, so B must be 35, or B degrees is 35 degrees.

And then for C, the tangent at any point on a circle is perpendicular to radius of that point.

So the angle marked C degrees must be 55 degrees, the same as the angle you were given.

Then the second part of this question, you are working in the opposite direction.

You are given the angle between the tangent and a chord.

You have to use that information to work the values of D, E, and F.

Well, the tangent at any point in a circle is perpendicular to the radius at that point, so D degrees is 22 degrees.

The angles in a triangle sum to 180 degrees, and base angles and isosceles are equal, so that means the angle marked E degrees is 136 degrees.

And the angle at the centre is twice the angle at the circumference, which means F degrees is 68 degrees.

It's the same as the angle you were given at the start.

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 three diagrams, which each show a circular geoboard containing points equally spaced around a circle and a tangent.

Each diagram contains a triangle with all three of its vertices on the circumference circle, and one of its vertices meets the tangent.

And there are two angles highlighted in each diagram.

One is between the tangent and a chord, and the other is the angle and the triangle, which is subtended by the same chord.

And in all of these cases, the two highlighted angles are equal to each other.

The triangle also makes another angle against the tangent.

If we look at those, we can see that we have two equal angles each time here as well.

Jun says, "We could explore more cases by using dynamic geometry software." That'll help us get a sense of whether or not it's just these particular cases where it happens, or whether it happens in other cases as well.

If you have access to this slide, you can click on a link at the bottom which takes you to a GeoGebra file that contains an interactive version of the diagram we're about to work with.

The diagram shows a triangle inscribed inside a circle.

A tangent to the circle is drawn at one of the vertices of the triangle, and then when we look at the angles, we can see that the angle between the tangent and a chord is equal to the angle at the circumference that is subtended by the same chord.

In other words, these two highlighted angles are equal.

In this case, they're both 50 degrees.

And this is a circle theorem that is referred to as the alternate segment theorem.

The triangle actually makes two angles against the tangent, which means there are two pairs of angles where this circle theorem applies at the tangent.

We can see the other pair of angles highlighted now, which are both 70 degrees.

One is between the tangent and the chord, which is highlighted, and the other is the angle that is subtended by that chord on the circumference.

And because a triangle has three vertices, we could draw the tangent at any of those three vertices and the circle theorem will still apply, for example, here and here.

And this circle theorem doesn't just apply to this particular triangle.

We could also adapt this triangle and see the angles still remain equal to each other, like this.

And this is how it would look on the GeoGebra file.

So let's check what we've learned.

Here you've got a circle with a triangle inscribed inside and a tangent.

Find the value of X, please.

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

The answer is 74.

The angle marked X degrees is equal to the angle which is marked 74 degrees.

How about on this diagram? What is the value of X? Pause while you write down your answer and press play when you're ready to see what the answer is.

Well, the angle marked X degrees is equal to one of those two angles.

The way you find out which angle is by looking at the chord which the X degree angle is against, and look for which angle is subtended by that chord at the circumference.

So X is 64.

Here we have a pair of examples which demonstrate the alternate segment theorem.

In each one, we have a triangle with all three of its vertices on a circumference of a circle, and one of its vertices meeting a tangent.

And you can see which angles are equal to each other in each diagram.

Here's Sofia who makes a really good point now.

She says, "Geoboards and dynamic software only demonstrate that these angles are sometimes equal." They don't prove that they are always equal.

These tools, geoboards and dynamic software, are really good for exploring lots and lots and lots of examples, and each time you can observe the fact that the angles are equal.

But in order to prove that they are always equal, we need to stop looking at specific cases and start looking at more generalised cases, and that's just by using algebra.

Lucas says, "We could use algebra and other circle theorems to prove that they are in fact always equal." We can do this by breaking it into a few small steps, and you're going to help me with it.

Here we've got a diagram with three points marked around a circle, A, B, and C, and a point at the centre, O.

Let's let the angle C, A, B equal X degrees, so rather than it being a specific number, it's a variable.

Please could you express the angle COB in terms of X? That's the angle marked in the centre.

And justify your answer with reasoning.

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

The size of angle COB is 2X degrees, and this is because the angle at the centre is double the angle at the circumference.

That's one of our circle theorems. Let's now add a chord from B to C.

Could you please express the angle OBC in terms of X and justify your answer with reasoning? Pause while you do it and press play for an answer.

The angle OCB is in a triangle with the two X degrees.

It's an isosceles triangle, so you can get an expression by doing 180 degrees subtract the two X degrees and divide by two, and that gives 90 minus X degrees.

And our justification is angles in a triangle sum to 180 degrees, and base angles in isosceles triangle are equal.

So if we add a tangent to this from D to E, could you please express the angle OBE in terms of X and justify your answer with reasoning? Pause while you do it and press play for an answer.

The angle OBE goes from the centre to the point of circumference where a tangent is to another point along the tangent.

The angle between the tangent and the radius is 90 degrees, so angle OBE is 90 degrees.

Our justification is the tangent at any point on a circle is perpendicular to the radius at that point.

And then finally, could you please express the angle CBE in terms of X? That's the marked angle which is between the chord CB and the tangent.

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

Well, the angle between the radius and tangent is 90 degrees, and we can see that part of that angle is 90 subtract X degrees.

So to write an expression for CBE, we can do it with a subtraction, and that would give X degrees.

And then when we simplify this diagram, we can see that both of these angles are expressed as the same thing.

They're both X degrees, which means no matter value X takes, no matter what size those angles are, they'll always be equal to each other.

Okay, so now for task B.

This task contains one question and here it is.

You've got a bunch of diagrams with unknown angles, and you need to find the values of the unknowns labels from A to V.

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

Okay, let's go through some answers.

A is 68 and B is 45, and these are both justified with the alternate segment theorem.

C is 56 and D is 57, and they're both justified with the alternate segment theorem.

And then with this diagram, E is equal to 57 and we can justify that using the alternate segment theorem.

It's equal to the angle which is given to you.

F is 90 degrees because the angle in a semicircle is always a right angle.

H can be justified in a couple of ways.

You could say H is 90 because the alternate segment theorem, H would be equal to F, or you can say that the tangent at any point on a circle is perpendicular to the radius at that point, therefore it's 90.

And then G is 33, and again, you can justify that in a few different ways.

You could say because angles in a triangle sum to 180 degrees, so you can subtract the answer for F and E from 180 degrees.

Or you can use that angles forming a straight line sum to 180 degrees as well.

And with this diagram, I is equal to 76, and that can be justified with the alternate segment theorem.

It's equal to the angle which is given to you.

J is equal to 76 because base angles of an isosceles triangle are equal, so J is equal to I.

K is equal to 28 because angles in a triangle sum to 180 degrees.

Then L is 76, and there's a couple of ways you can get that.

You can either use the alternate segment theorem and say that L is equal to J, or you can say that angles forming a straight line sum to 180 degrees and use 76 degrees and the answer from K.

This diagram is very similar to the last one in that it's an isosceles triangle, which meets a tangent, and we have a 76 degree angle.

So our working will be quite similar to last time, but the position of the sides which are equal to each other and the angles which are equal to each other are different, so our working will be just slightly different.

M is 76 because the alternate segment theorem.

N is equal to O because this is an isosceles triangle.

And they're both 52 degrees because angles in a triangle sum to 180 degrees.

P will be 76, and you can either use the alternate segment theorem to show that, or you can use that angles that form a straight lines sum to 180 degrees.

And here we have another diagram with an isosceles triangle meeting a tangent at 76 degrees, but again, the position of the equal sides are in different places.

So, R is 76 because of the alternate segment theorem.

N is 76 because base angles of isosceles triangle are equal.

S is 28 because angles in a triangle sum to 180 degrees.

And U is 28 because either the alternate segment theorem or angles that form a straight line sum to 180 degrees.

And then finally with this question, you've only got one angle to work out, but that can make it a bit trickier because you need to use several steps to get there, and you need to decide now what those steps are.

Here's one way we can do it.

We could label this angle here as V degrees because alternate angles in parallel lines are equal.

Can also express this angle as V degrees because of the alternate segment theorem.

And now we've got a triangle with 48 degrees and two angles marked as V degrees.

That triangle must be isosceles because two of the angles are equal.

And that means we can use this fact, angles in a triangle sum to 180 degrees, and do 180 subtract 48 divided by two to get 66 for our value of V.

Fantastic work today.

Now let's summarise what we've learned.

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

Algebra and geometric reasoning can be used to prove the alternate segment theorem.

And theorems can be thought of as puzzles to solve where your job is to show how to get the results.

And in order to use this theorem, you may need to draw a diagram or add information to an existing diagram in order to work things out.

Well done today.

Have a great day.