Lesson video

Hi there.

Welcome to today's lesson.

I hope you're doing okay.

My name's Mr. Peters, and in this lesson today, we're gonna be thinking about how we can apply our knowledge that the angles at a point sum to 360 degrees, and we can then use this to solve a range of problems. If you're ready to get started, let's get going.

So, by the end of this lesson today, you should be able to say that, "I can explain that the angles at a point sum to 360 degrees, and use this to solve problems." During this lesson, we've got two key words we'll be referring to.

I'll have a go at saying them first, and then you can repeat them after me.

The first one is "point".

Your turn.

And the second one is "sum".

Your turn.

Let's think a bit more about what these mean in a mathematical context.

A point is an exact location.

It has no size, only position.

And the sum is the total when numbers are added together.

Today's lesson is broken down into two cycles.

The first cycle is angles at a point, and the second cycle is identifying missing angles.

Let's get started with the first cycle.

In this lesson today, we've got both Lucas and Sofia.

They'll be sharing their thinking and, as always, sharing any questions that they have, that will also help us to develop and further our thinking as we go.

Okay, so here you can see that we have a lot of circle segments.

Each of these parts can be combined together to create different circles.

Each of these segments also have the angle between the two lines drawn on to them.

I wonder how many different complete circles could you make? Why don't you take some time now to have a go with these ones, and come up with some for yourself? Lucas thinks he's managed to make his own circle.

Let's have a look at what you've done, Lucas, shall we? You've used two parts, haven't you? You used one part which had a 256 degree angle, and you used another part which had a 104 degree angle.

Well done you! A great start.

Sofia, how have you been getting on? She thinks she's found one too.

Have a look at this example.

What do you notice? That's right.

Sofia's used the same part twice, hasn't she? She's found two parts which have an angle of 110 degrees, and then she's also found the part which has an angle of 140 degrees, and they've managed to combine together to create a circle as well.

Well done you! And exactly that.

There were definitely no rules saying you couldn't do that.

So I wonder, are we able to create one which has got more than three parts? Hmm.

Sofia thinks she's made one with four parts.

Let's have a look at your example, Sofia.

Wow, what do we think? We've got one part which is an angle of 134 degrees.

We've got another part with an angle of 107 degrees.

We've got a further part with an angle of 23 degrees.

And finally another part with an angle of 96 degrees.

Hmm.

It looks like it's created a circle as well, Sofia.

That's amazing! Great work.

That's a really interesting point, Lucas.

All of these circles that we've created so far, they all look like circles, don't they? But how can we be a hundred percent sure that they are, in fact, perfect circles? Well, have a look at the examples we've created so far.

What do you notice about all of these examples? Take a moment to have a think.

Well, well noticed, Lucas, you're right.

All of the angles meet at a point, don't they? And that point is right in the centre of the circle, isn't it? And that's where all of these parts meet.

They meet right at the centre, don't they? Hmm.

Sofia thinks she's noticed something else.

Ah, there we go! Sofia thinks that all of the angles in the circle add up to 360 degrees.

And do you know what, Sofia? You're right.

They actually do.

All of the angles around a point sum together to make 360 degrees.

So we can say that all the angles at a point sum to 360 degrees.

I wonder if you could say that with me.

Are you ready? All of the angles at a point sum to 360 degrees.

So, let's identify where these angles meet then, shall we? Here we can see where all of the points on the angles are, and we can join these together now, just like this, to show you the point at which all of these angles join together.

So as we've been saying, where all of these angles meet is known as a point.

And we can say that all of the angles that meet at this point will always sum to 360 degrees.

And there you go.

We can use our arc to represent that as well, can't we? Okay.

Time for you to check your understanding now.

The sum of the angles that meet at a point sum to, A, B, C, or D? Take a moment to have a think.

That's right.

It's D, isn't it? All of the angles at a point sum to 360 degrees.

Okay, and have a look here then.

Which part is going to complete this circle? Take a moment to have a think.

That's right.

It's part B, isn't it? At the moment, we've got a segment with the 180 degree angle.

We've also got a segment with 110 degree angle.

And therefore, if we add those two together, that gives us 290 degrees.

We need 360 degrees altogether.

So a 70 degree angle would be needed to complete the circle.

Well done if you got that.

Okay, time for your task now then.

How many different circles can you create with the segments that have been given to you here? Can you create circles with either two, three, four, or even more segments? Good luck with that, and I'll see you back here shortly.

Okay, welcome back.

Let's see how Lucas got on then, shall we? Lucas thinks he would be able to create circles with three, four, and five different segments in here.

And if we were to add up all of these angles, they would always sum to 360.

So great work, Lucas.

Fantastic.

I wonder what you came up with? Okay, moving on to cycle two now then.

Identifying missing angles.

So have a look at this circle here now.

What are the angles that we already have? And look, we've got one of the angles that is missing, isn't it? How do you think you might tackle this? That's right, Lucas.

Now that we know that the angles at a point sum to 360 degrees, we can use this knowledge to help us solve and find a missing angle, can't we? We know that the angles at a point sum to 360 degrees.

So this would be regarded as our whole.

We're gonna draw a bar model to help us represent this.

The whole circle would have 360 degrees.

Let's have a think about the amount of parts that we have.

Well, we've actually got three parts, haven't we? One of those parts is just over half of the whole, isn't it? And then we've got a smaller part, and then we've got a slightly bigger part as well.

So we can see how we've separated that up into our bar model, underneath the 360.

Our whole has been divided into three parts, isn't it? And we know the size of two of these parts.

One of these parts has an angle of 182 degrees, and the smaller part has an angle of 50 degrees, doesn't it? So now we can use this to help us identify the size of the missing angle, can't we? So to do that, we can now find the sum of the angles that we know so far, can't we? At the moment, we've got 182 degrees, and we've got 50 degrees, and we can combine these together by adding them, which would give us a total of 232 degrees.

So now that we've combined these two together, we just now need to find the missing part, don't we? We're trying to find the difference.

We can use subtraction here to find the difference between the whole and the size of the angles that we already know.

We know that the whole is 360, and we can minus the angles that we know, which sum to 232 degrees, so 360 degrees minus 232 degrees will be equal to 128 degrees.

So the missing angle was in fact 128 degrees.

Well done if you managed to work through that as well.

Okay, gosh, this is slightly different this time.

How are we gonna work out the size of angle A this time? Have a look carefully at the shape.

What do you notice? Hmm.

You're right, Lucas.

It does look a little bit tricky, doesn't it? We've got four parts that look roughly the same size, don't we? And then we've also got two smaller angles.

One of those angles is an angle that we do know.

We know the size of that smaller angle as 30 degrees.

Hmm.

So I wonder what the size of the other smaller angle is.

Sofia's saying, "Hang on." What have you seen, Sofia? Oh, that's good thinking, isn't it, Sofia? You think both of those smaller parts combine to make a bigger part, don't they? There we go.

And now what do you notice? That's right.

The whole circle has been divided into equal parts, isn't it? That's great thinking, Sofia.

Well done, you! So if the whole is divided into five equal parts, then we can divide our 360 into five equal parts, can't we? We know that the angles at a point sum to 360 degrees, so that is our whole, and then we can divide that into five equal parts to find the size of each one of the angles.

360 ÷ 5 gives us a value of 72 degrees.

So each angle at the moment represents 72 degrees.

Hmm.

How can we use this then to help us? Well, if we go back to the segment that we had originally, do you remember we combined those two smaller angles together, and we now know the sum of those two smaller angles combines together to make 72 degrees.

So, now then, we can break that back down into those two parts.

Have a look here at my bar model.

We've taken one part, which was 72 degrees, and now we've broken it down into two further parts.

We know the value of one of those parts, don't we? One of those parts is 30 degrees.

So now we're looking for the value of a, the missing part.

We can do that by saying 30 degrees plus something is equal to 72 degrees.

We know that we can write a missing addend question as a subtraction equation, can't we? So let's turn this into a subtraction equation.

We've now got 72 degrees, which is the whole, and we can subtract 30 degrees from that, which leaves us with a total of 42 degrees.

So missing angle A was in fact 42 degrees.

Well done if you managed to work through that as well.

Okay, time for you to check your understanding again now.

Can you find the missing angle here? Take a moment to have a think.

That's right.

We can find the missing angle, can't we? By summing up the angles that we already have.

So we've got 127 degrees plus 97 degrees plus 56 degrees plus another angle, unknown angle, would be equal to 360 degrees.

So if we combine these three angles together, that gives us a total of 280 degrees.

And now we can therefore subtract this 280 degrees from 360, which gives us a total of 80 degrees.

So angle A is equal to 80 degrees.

Okay, and another quick check.

Can you find this missing angle here for me? Take a moment to have a think.

Okay, let's have a look then, shall we? Well, what do we know? Well, we know that the angles on a straight line sum to 180.

So actually, the large angle is 180 degrees.

We can also see we've got a right angle here.

We know the value of that.

That would be 90 degrees.

So if we add both of those to the 56 degree angle that we already know, that will give us the total of the angles that we know so far, and then that should help us to find the missing angle.

So we can say 180 degrees plus 90 degrees plus 56 degrees, plus angle A would be equal to 360 degrees in total.

We can then sum the three angles that we know together already.

That gives us 326 degrees, and now we can subtract that, can't we? From the whole of 360 degrees.

And that gives us a total of 34 degrees.

So angle A has a value of 34 degrees.

Well done if you got that too.

Okay, onto our second set of tasks for today then.

What I'd like you to do here is to find the missing angles for me in each of these examples.

And then for task two, I'd also like you to find the missing angles in each of these as well.

It might be useful to help you think about looking for equal parts each time.

Good luck with those two tasks, and I'll see you back here shortly.

Okay, welcome back.

Let's see how you got on, then.

So did you remember that the sum of all the angles needed to add to 360 degrees? Well, let's have a look and go through this then.

The unknown angle in this first example was 160 degrees.

We could work that out by subtracting the known angle of 200 degrees from 360 degrees.

Therefore, unknown angle A was equal to 160 degrees.

For the next example, we knew two of the angles, didn't we? We had 175 degrees and 78 degrees.

If we add these two together, that gives us 253 degrees.

And again, we can subtract that, can't we? From 360 degrees, the whole, which gives us the size of angle B.

So the size of angle B was in fact 107 degrees.

And for angle C, then, we can then subtract all of the known parts that we have from 360 degrees, can't we? But before we subtract all of those parts, why don't we sum together? That will make our life a little bit easier.

So we can add 148 degrees, plus 81 degrees, plus 49 degrees together.

That gives us a total of 278 degrees.

And now we can say 360 degrees minus 278 degrees will leave us the difference, which in fact is angle C, wouldn't it? So angle C, in this case, is in fact 82 degrees.

Well done if you got those three.

Okay, and then for question two then, did you manage to spot the equal parts each time? Well, we know this first one was very similar to the one that we looked at earlier on.

It was split into five equal parts, therefore each part has a value of 72 degrees.

Then we could use that knowledge to subtract the size of the smaller angle that we already knew, which was 44 degrees.

So 72 degrees minus 44 degrees gives us a total of 28 degrees.

So the missing angle here was 28 degrees.

For the second one, how many equal parts was it divided into? That's right.

If you combine the two small parts that we had, we could say that the whole was divided into four equal parts.

So 360 ÷ 4 gives us a total of 90 degrees.

Each one of those parts has a value of 90 degrees then, and then we can now subtract the size of the smaller known part from the 90 degrees.

So 90 degrees minus 39 degrees gives us a total of 51 degrees.

Well done if you got that.

And then finally, the last one.

How many equal parts is this divided into? Actually, if you look at the larger parts, we could have said those three smaller parts could have been combined to make a third, larger part.

So that would be three equal parts.

That means 360 divided into three equal parts gives us a value of 120 degrees.

And now we can use that knowledge again to help us find the size of the smaller part of angle C.

So, now we can see that actually, the 60 degrees plus the 30 degrees plus the missing part C degrees, adds all up to 120 degrees.

So why don't we subtract the 60 first.

120 degrees minus 60 will leave us with 60 degrees.

And then that just leaves us with those remaining two small parts.

So we can do 60, subtract the known angle of 30 degrees that we have, and again, that will leave us with 30 degrees.

So missing angle C was in fact 30 degrees.

Well done if you managed to get that too.

Okay, that's the end of our lesson for today.

Hopefully you've enjoyed that one, and you're feeling a lot more confident about finding the size of the missing angles around a point with the knowledge that we've learned.

To summarise our learning, we can say that the angles at a point sum to 360 degrees.

If there is one missing angle, you can find the size of that angle by subtracting the known angle from 360 degrees.

And finally, you can apply your knowledge of equal parts to help you identify equal angles that are unknown.

Thanks for joining me today.

I've really enjoyed that lesson.

Hopefully you did too.

Take care.

I'll see you again soon.