Lesson video

Hi, there.

Welcome to today's lesson.

My name is Mr. Peters, and in this lesson today we're gonna be extending our thinking about the understanding of the angles on a straight line and that they sum to 180 degrees and how we can apply this to solve a range of problems. If you're ready to get going, then let's get started.

So by the end of this session today, you should be able to say that I know that the angles in a straight line sum to 180 degrees and I can use this to solve a range of problems. Throughout this session today, 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 word is sum.

Your turn.

And the second word is opposite.

Your turn.

Let's have think about what these mean.

The sum is the result of adding two or more numbers.

And if something is opposite, we can say that is placed on the other side.

In this lesson today, we've got two learning cycles.

The first learning cycle is thinking about angles on a straight line and the second learning cycle is about identifying missing angles.

Let's get started with the first cycle, shall we? In our lesson today, we'll be joined by both Jun and Andeep.

As usual, they'll be sharing their thinking as well as any questions that they've got to help us develop and broaden our understanding of the concepts that we're looking at.

So our lesson starts here.

Jun and Andeep are discussing their holiday plans.

Jun says that he's going on holiday to Spain next week.

And we can see the distance from Spain here.

Because we live on an island here in the UK, there's a lot of water to go around to get to Spain, isn't there? Hmm, so how are you gonna get there, Jun? Ah, he says he's gonna fly there.

He's never been on a plane before.

Have you been on a plane before? It's quite an exciting experience.

You're right, it is great fun, particularly the takeoff where you start zooming off into the sky.

If we go on to think about aeroplanes a little bit more, we can start thinking a bit more about the takeoff that happens when they leave the ground.

When they do leave the ground, planes create an angle of anywhere between 5 degrees and 15 degrees for the takeoff.

There we go.

Hopefully now you can see an example.

This plane here has taken off at an angle of 10 degrees.

So if we know the angle at which the plane took off, we can now start to think about the remaining angle, this angle here.

We know that the plane started on a flat course, didn't it, and then it started rising towards the sky.

So the plane actually started on what was a straight line.

We know that the angles on a straight sum to 180 degrees from our prior learning, don't we? Exactly that, Jun.

That's because it's half of a full turn, isn't it? So, we could therefore say that 10 degrees plus something degrees is equal to 180 degrees.

Here we go.

We can see the 10 degrees that we already know and we can see that if we add the remaining angle onto that, that whole angle will sum to 180 degrees.

Hmm, this is like a missing addend problem to me.

So if we've got a missing addend here, we know we can use subtraction, can't we? We could now say that 180 minus 10 degrees will leave us with the size of the missing angle or the unknown angle.

In this case, 180 minus 10 degrees is going to equal to 170 degrees, isn't it? So the size of the remaining angle in this case is 170 degrees.

Hmm.

Great question, Jun.

What would happen then if the plane took off at an angle of nine degrees this time? Well, how would you work this out? You're right.

We could just follow the same process, couldn't we? We could say that 9 degrees plus something degrees is equal to 180 degrees.

And then we know we've got a missing addend problem, so we could subtract it again from the 180.

So 180 degrees minus 9 degrees is equal to 171 degrees.

So that was a process we could do.

However, let's take a moment here to stop and have a think.

What do you notice here? Hmm, I wonder what you noticed.

Jun's saying that both angles sum to 180 degrees, which is true.

Ah, and he is also noticed that when one of the angles is acute, which is the angle at which the plane took off at, the other angle is always obtuse.

That's a really interesting observation, Jun.

Ah, but Andeep thinks he's spotted something different this time.

Ah, that's great thinking, Andeep.

He says, "Each time one angle increases in size, the other angle decreases in size." Let's have a look more carefully at this.

At the moment we've got a 10-degree angle and a 170-degree angle.

However, if we increase the 10-degree angle to 11 degrees, we can see that the 170-degree angle has now reduced in size by one degree, it's changed to 169 degrees.

So if one of the angles increases by one degree, then the other angle must decrease by one degree.

Here's another example.

Oh, look, we had 11 degrees, but now it's a 12-degree angle.

That means the other angle must also decrease by one degree again, so that would be 168 degrees.

In every single example, all of these angles are summing to 180 degrees each time.

There's another example.

This time we've increased it to 13 degrees, therefore the other angle has decreased by one degrees again, so it's 167 degrees.

Yeah, exactly that Andeep.

So not only is it when one angle increases, the other angle decreases, they must increase and decrease by the same amount each time.

So for all the examples we've looked at so far, when one angle increases by one degree, the other angle must decrease by one degree.

Or we could say if one angle decreases by one degree, then the other angle must increase by one degree.

Hmm.

Great question, Jun.

Does that still work if it increases or decreases by more than one degree? Well, let's have a look, shall we? At the moment we've got 13 degrees and 167 degrees.

But look carefully at 13 degrees, what does it change by now? Ah, it's changed to 33 degrees.

So we can say that from 13 degrees now to 33 degrees, the size of the smaller angle has increased by 20 degrees, hasn't it? So what does that mean for the other angle? That's right.

The other angle has to decrease by 20 degrees.

Previously it was 167 degrees, but now it's 147 degrees.

So we can see it's decreased by 20 degrees as well.

Hmm.

Have a look at this example here.

What do you notice now? Hmm, what did you notice? Yeah, great thinking, Andeep.

You've noticed that it seems to be happening exactly underneath the line as well, doesn't it, as the diagonal line has intersected the horizontal line underneath as well, hasn't it? So from this, we can say that if we know that this angle here is 13 degrees, then we also know that this angle here is also 13 degrees.

That means the larger remaining angle must also here be 167 degrees.

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

The angles on a straight line sum to how many degrees? A, B, C, or D? Take a moment to have a think.

That's right.

It's B, isn't it? The angles on a straight line sum to 180 degrees.

And you're right, Andeep.

It's equivalent to two right angles, isn't it? Great thinking.

And have a look here now then.

Find the missing angle of a.

Take a moment to have a think.

Yep.

Well, there we go.

We know the angles on a straight line sum to 180 degrees and we know part of the angle here.

We've got a 140-degree angle, so we now need to find the missing angle.

We know that 140 degrees plus something is equal to 180 degrees, therefore, because this is a missing addend equation, we can rearrange this so it is a subtraction equation.

180 degrees minus 140 degrees is equal to 40 degrees.

So the missing angle a is in fact 40 degrees.

Well done if you got that.

Okay, time for you to have a practise now.

What I'd like you to do is calculate the missing angles for these three here.

And then once you've done that, I'd like to have a go at calculating the missing angles in these equations here, going downwards for each set of questions, starting with A, then sequence B, and then sequence C.

Good luck with those.

And I'll see you back here shortly.

Okay, welcome back.

Let's see how you got on.

So the missing angles here, then.

We know that for angle a, here we've got a horizontal line and we know that the angles on the horizontal line add up to 180 degrees.

The angle that we know is 42 degrees, so we can say 42 degrees plus angle a is equal to 180 degrees.

We can rearrange this for 180 degrees minus 42 degrees, and that gives us a total of 138 degrees.

So angle a is 138 degrees.

We've got a straight line here, which is almost vertical, but not quite.

However, we still know it's a straight line, so that means the angle on that is 180 degrees.

The known angle this time is 60 degrees.

So once again, we can say 60 degrees plus angle b, in this case, is equal to 180 degrees.

And now we can do a subtraction equation to represent this.

180 degrees minus 60 degrees is equal to 120 degrees.

So angle b is equal to 120 degrees.

And then finally for c.

Well, one of the lines is thicker here, but does that matter? It doesn't, does it? The thickness of the lines don't matter when we're creating an angle? And actually have we got two straight lines? We do.

So we can work with the thick straight line to identify that the angles on that would sum to 180 degrees.

So far we've got one angle, which we know is 81 degrees.

So we can say that 81 degrees plus angle c is equal to 180 degrees.

And then we can do 180 degrees minus 81 degrees, that would be equal to 99 degrees.

So angle c is equal to 99 degrees.

Okay, I'm gonna place the answers in now and then we'll have a think about what you noticed each time.

Okay, let's start with A.

What did you notice when you went through these? Well, that's right.

We always start with 180 'cause that's the whole, isn't it? And one of the angles that we were subtracting to find the missing angle was decreasing by one degree each time, wasn't it? Therefore, that must mean the unknown angle would increase in size by one degree each time.

So as soon as we worked out the first calculation, we could work out the other ones really quickly and easily by just adding one degree each time.

A similar thing was happening for sequence B, wasn't it? Here we knew one of the angles as the difference in this case, and those angles were increasing by one degree each time.

The size of the angle that was unknown must have been decreasing one degree each time.

For the first one on sequence B, we could have done 180 minus 105, which left us with 75 degrees, and therefore, as I said, it would increase by a degree each time because the difference was decreasing by one degree each time.

And then finally, for the last one, what did you notice here? Ah, that's right.

The difference this time was decreasing by five degrees each time.

That must mean the amount we were subtracting was increasing by five degrees each time.

So the first one was 180 degrees minus 60 degrees is equal to 120 degrees, and therefore each time we were subtracting a number that was five degrees more.

Well done if you've got all of those.

Okay, that's the end of cycle one.

Moving on to cycle two now: identifying missing angles.

Okay, so look at this example here.

What do you notice? How do we tackle this one this time? Well, you're right, Andeep.

We've actually got three angles on a straight line this time, haven't we? Hmm, that's more than we've had before, isn't it? And you're right, they still all sum to 180 degrees, don't they? So we can say that angle a plus the 129-degree angle plus the 10-degree angle all sum to be equal to 180 degrees.

Hmm, so why don't we now, to make this calculation easier, sum the two known angles together.

We've got 129 degrees and 10 degrees.

If we add these two together, that gives us a total of 139 degrees.

There we go.

So we've now been reduced to two angles, haven't we? And we can now say that angle a plus the 139-degree angle will be equal to 180 degrees.

And of course it's a missing addend equation now, so we can turn this into a subtraction, can't we? So 180 degrees minus 139 degrees will be equal to the missing angle a, and in that case it is in fact 41.

So missing angle a here is 41 degrees.

Well done if you managed to spot that for yourself.

Oh, here's a different example this time.

What do you notice now? Hmm, that's right.

We've got two straight lines this time.

So actually we need to identify which straight line we want to work with first here, don't we? If we use this straight line here, we can see that we've got a 140-degree angle and we've got the missing angle of a.

We know that the sum of this will equal 180 degrees altogether.

So we can say that 140 degrees plus angle a will be equal to 180 degrees.

We now know it's a missing addend problem, so we can rearrange it to use subtraction to support us.

We can say 180 degrees minus 140 will be equal to angle a, and in that case angle a will be equal to 40 degrees.

So we can place that on here now.

Okay, let's have a go here, check our understanding now.

Can you calculate the missing angle x in this example? Take a moment to have a think.

Well, we've got three angles here, haven't we, which will sum to 180 degrees.

So we can say angle x plus 33 degrees plus 70 degrees is equal to 180 degrees.

If we sum the two angles that we know together, we've now got angle x plus 103 degrees is equal to 180 degrees.

And now we've got one missing addend, we can write this as a subtraction equation.

180 degrees minus 103 degrees will be equal to 77 degrees.

So angle x is equal to 77 degrees.

Well done if you've got that.

Okay.

And for the next one then, we need to work out angle x and angle y.

How would you go at doing this? Take a moment to have a think.

Well, if we use a straight line for x to start off with, we can see that x plus 139 will be equal to 180.

We can use subtraction now.

180 minus 139 will be equal to x, therefore x is equal to 41 degrees.

And we can also now use our knowledge that x will be equivalent to y in this case because the y angle is a part of a straight line which has a 139-degree angle already.

And we know that we need 139 degrees plus another 41 degrees to be equal to the 180 degrees on the straight line, so angle x is equal to angle y here.

Okay, onto our final tasks for today then.

What I'd like you to do is find the missing angles for x each time here.

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

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

Okay, let's see how we got on then.

So for A, we can see that the angles on the straight line add to 180 as always and we know two of the angles here.

So we can say the x plus 78 degrees plus 30 degrees is equal to 180 degrees.

If we add the 78 and the 30 together, that gives us 108 degrees.

And then we can subtract that, can't we, from 180.

So 180 minus 108 degrees is equal to x, which in this case was 72 degrees.

For B, we can see we've got another three angles here, haven't we, which are working with a straight line which is nearly vertical.

So this time, again, we've got an angle of x, an angle of 30, and an angle of 78.

So here we can add x plus 78 plus 30 is equal to 180.

Then we can add those two known angles again, which gives us 108 degrees, and then subtract that from 180, which actually also gives us 72 degrees again.

Hmm, I'm starting to notice something here.

What about the last example? Look carefully.

Can you tell me what the angle of x is gonna be? Well, again, it's very similar to the previous example, isn't it? However, x has moved slightly.

If we use the value of x from the previous example that we had, then we know that that angle would be equal to 72 degrees.

Hmm, now we can subtract that 72 degrees from the 180 degrees, which would be the angle of the sum on the straight line.

And altogether that would give us 108 degrees.

So the x here is equal to 108 degrees.

Hmm, was there anything you noticed in particular? Well, that's right.

We were using the same values of the angles each time, but we were just moving things around slightly.

And you may have also noticed that the angle's opposite each other here were actually the same each time.

So that helped us to calculate all of these rather simply rather than having to go through the whole process of working them out procedurally each time.

Well done if you spotted all of that.

Okay.

And onto our last couple of examples then.

To work out the size of angle x here, first of all, we need to work out this missing angle.

So the size of this missing angle can be worked out by adding the two known angles together and then subtracting them from the 180, so that gives us a value of 74 degrees.

Now we know this angle of 74 degrees, we can work out this next angle here by, again, adding the known angles that we have.

That, once again, gives us 108 degrees.

And we can subtract that from 180 degrees, which gives us, again, 72 degrees.

And then to work out the value of this x, we can use the straight line which has got 34 degrees on it and 72 degrees on it.

Add these two together, that gives us 106 degrees.

And if we subtract that, we can see that that also gives us 74 degrees.

What did you notice about the examples there? Hmm, what did you notice that time? That's right.

The angles opposite each other were equal, weren't they, each time.

So could you now work out the remaining missing angle in example A? That's right.

That would be 34 degrees, wouldn't it? Well done if you managed to spot that.

So for example B then, I wonder if we can use what we've just learned.

We can see here that 45 degrees has an opposite angle, so we know that must also be 45 degrees.

We can now use the angles on the straight line to help us solve this.

We've got 95 degrees plus 45 degrees plus x will be equal to 180.

95 degrees plus 45 degrees will be equal to 140.

And we can now subtract that from 180, which leaves us with an angle of 40.

And then for the last one, can we work it out, angle x? We don't need to do any calculations, do we? We can just work it out straight away by knowing that the opposite angle of 49 will also be 49 degrees.

There we go.

Well done if you managed to get all of that too.

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

Hopefully you've come away making some new connections and a bit clearer in our understanding of the angles that sum to make 180 degrees on a straight line.

So we can summarise our learning today by saying that the angles on a straight line sum to 180 degrees.

We can say that where one angle is unknown, we can subtract the known angle from 180 degrees to find the missing angle.

And we can also say that angles around a point that are opposite each other are equal to one another.

Brilliant.

That's the end of our learning for today.

Hopefully you enjoyed it, I know I certainly did.

Take care, and I'll see you again soon.