Lesson video

Hello, I'm Mr. Langton, and in this lesson, we're going to be solving geometric problems using linear equations.

Now, don't worry, it's not as scary as it sounds.

Just take a minute to grab something to write with and something to write on.

Make sure you're in a quiet spot where you won't be disturbed.

And then when you're ready, let's begin.

So we'll start off with our try this activity.

Angle a is double the size of angle b.

What angle relationships do you know and what else can you work out? So pause the video and have a go.

When you're done, unpause it and we'll go through it together.

And start pausing in three, two, one.

Okay, so I hope you had some time to have a go.

We're going to start off by labelling what we know.

Now, this angle here, I'm drawing that.

That's vertically opposite to a, and so that one is also a degrees.

And similarly down at the bottom, this angle here is virtually opposite to b, so that one is b degrees.

Now let's see.

Right, this one here is a corresponding angle with a there.

Those two angles correspond, which means that that angle is also a degrees.

And because vertically opposite angles are equal, that one there is a degrees.

Now, I can see a couple of formulae there that I can put together.

I can say a plus b plus a plus b is going to be 360 degrees because they're angles around a point and they add up to 360.

So 2a plus 2b is 360 degrees.

Now, is there anything else I can label? Angle b here is a corresponding angle there to that one, and that one there is also a b degrees because they're vertically opposite.

Other things that we can say, then.

We can see that here, if I were to sketch that one just up above it again, we've got a there and b there, They're angles on a straight line, and angles on a straight line add up to 180 degrees.

So a plus B is 180 degrees.

We can see that in other places, as well, because here, and I'm just going to mark these two a and b there.

We've got some co-interior angles inside our parallel lines.

So these two together will also add up to 180 degrees, so that's another reason why a plus b is 180.

And there's probably a few more you can see.

Can you see your alternate angles here? So with b here, there's alternate to b there.

So b equals b, which we certainly hope it does.

So there's loads of things we can get from there.

Now, what I have missed so far, I've ignored is this first line.

Angle a is double the size of angle b.

So that means that a is equal to two lots of b.

And we can now start putting that into some of these formulas that we've come up with and some of these equations, and we can start finding the value of a and b.

So if we know that a plus b makes 180 degrees, and a equals 2b, we can replace a in this formula with 2b.

So 2b plus b is 180 degrees, or 3b is 180.

And that means that b must be 60 degrees.

And if b is 60 degrees and a equals 2b, then a must be 120 degrees.

We can use our geometric understanding to form and solve equations.

In this case, we've got a pair of corresponding angles that have been labelled here.

The corresponding angles are equal.

So we can write that as an equation.

40 take away x is equal to 3x plus 20.

And we can solve this to find the value of x.

If I add x to each side, then I'm left with 40 on the left-hand side is equal to 4x plus 20.

Now I'm going to subtract 20 from each side, which means I've got 20 is equal to 4x.

And if I divide each side by four, five is equal to x.

Now, on top of that, I can now substitute those values back into the formulas to work out the size of the angles themselves.

So this angle here is 40 take away x, or 40 take away five, which is 35 degrees.

We can double check that down there because it's quite important.

We've already said they're the same.

We started off saying they're the same, so let's check.

We've got three lots of five, which is 15, add on 20 is 35 degrees.

So yep, we can be very, very happy with x equals five.

Okay, so now it's your turn.

There are five questions there.

If you pause the video on the next slide, you can access the worksheets.

You can download it and have a go yourself.

If you've got any issues, if you're concerned, don't pause it just yet.

Let the video keep playing, and I'll go through a couple of questions with you just to give you a bit of a starting point.

Okay, so just a few hints, then.

We'll start off with this one here.

We've got three angles all situated on a straight line, and the sum of the angles on a straight line is 180 degrees.

So a plus 2a plus 3a is 180 degrees.

If we simplify that 6a is 180 degrees.

I'll let you solve that to find the value of a, but there's a starting point for you.

The angles in a triangle add up to 180 degrees.

So I'm going to presume that you can do that one as it is.

I'm going to have a go at this trapezium one, because it's quite tricky.

One thing to spot with a trapezium is it's got one pair of parallel lines, in this case, the top and the bottom.

And if you've got a pair of parallel lines, inside these two angles here form these co-interior angles, which we said earlier add up to 180 degrees.

So that means that 4d plus d makes 180 degrees.

And over on the right-hand side, d plus 20 plus f will also make 180 degrees.

So here's was a hint.

If you can solve this one first and get the value of d, you should then be able to do the other one.

Okay, so have a pause and finish off what you've done.

Okay, here are the answers.

You can mark your work now.

Okay, we're going to finish up with the explore activity.

Different sets of angles have been marked in different colours.

What can you work out about the sums of the sets? Now, I do need to be clear that just because the angle, three of the angles are marked in red, that doesn't mean that they're equal to each other, but I want to know what happens if I'm going to add together the three red angles.

Similarly, the green angles are not all equal to each other, but what can you tell me about the sum of the three green angles? And finally, the blue ones.

What can you tell me about the sum of those six blue angles? Pause the video and have a go.

When you're ready, we'll go through it together.

You can pause in three, two, one.

Okay, we'll start off looking at the red angles.

All three red angles are inside a triangle.

They're the three angles inside a triangle.

Well, we know the three angles in a triangle add up to 180 degrees.

So our red angles must total 180 degrees.

Now, the green angles are also going to add to 180 degrees.

Can you see why? Well, the reason is that this green angle here is equal to this red angle here and this green one here is equal to this red one here and this green one here is equal to this red one here.

So our three green angles must be equal to our three red angles.

I have three red ones that add up to 180 and the three green ones must add up to 180.

Remember, we're looking for the sums of those angles.

So finally, the blue ones.

There's probably a few ways that you could look at this, but the way I decided to look at it was look at the whole picture first.

So I'm starting off with four triangles.

Now, if the angles in one triangle make 180 degrees, then the angles in four triangles must add up to 720 degrees.

Now, we know that the red ones and the green ones are both 180.

So in total, that's 360 degrees.

So if we've taken away from 720 the 360 that we've already found, then the blue ones must also total 360 degrees.

And just finally, if you'd like to share your work without Oak National, then please ask your parent or carer to share it on Twitter, tagging @OakNational and hashtag LearnwithOak.

See you later.