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Hello, and welcome to this lesson on angles in review, angles in polygons.

Really important that you're able to focus fully on this lesson, so make sure those phone notifications are silenced and that you are concentrating fully in that quiet space that you may have found in order to access the work to the best of your ability.

As always, we're going to do some very, very powerful maths, so make sure that you are concentrating well.

Make sure you've got that pen and paper in front of you so you can do all your workings, and you may even need a calculator if you want to speed up the process.

So with that in mind, let's continue with Mr. Thomas' lesson.

So, if you would all try this.

What I would like you to do, is I'd like you to think about how you can prove the idea of what the interior angles in a quadrilateral add up to.

Seems quite simple.

Is it? Have a go at it, I'll give you five minutes.

Pause the video now, and have a go at it.

Very good.

Let's go through it then.

So do you remember the idea that we need to go from one specific vertex and we needed to split it into triangles? So building on that idea, that we learned in the previous video about the interior angles.

In a, in a triangle.

Some 280 degrees.

Well, 180 degrees here, 180 degrees here.

So we know the interior angles in a quadrilateral- in a quadrilateral, goodness me.

My uh, my words can't come out today.

Um, that would be equal to 360 degrees.

Very good.

So, thinking about how we can extrapolate that onto more complex shapes, where we've got more sides.

Well, we just go from one specific vertex.

So for this, one, two, three, four, five sided shape.

A pentagon.

And what we can do is split it into one, two, and three.

So 180, 180, and 180.

So we get a total of 540 degrees.

It's my interior angles in that polygon.

I can then, for this six sided shape, all right? This hexagon.

I can do one, two, three, and four.

So 180, 180, 180, 180.

So then that gives me, if I add those all together, what would that give me, if I add all those 180s together? That would give me of course, 4 lots of 180, what would that give me? Would give me 720, wouldn't it? So you're starting to get the idea and we use this formula to help us, don't we? We can see all these triangles fine.

But actually, there's a really really helpful formula, because it's quite a lot of hassle in order to be able to do these all the time, right? It's very mechanical, uh starting to get a little bit bored of the fact that I've got to do that every single time.

What we notice, though, is that we've got 6 sides here, for example.

Right? And we've got four triangles.

So I know the amount of triangles that I have is, n minus 2, amount of triangles.

And I need to multiply, of course, by 180.

Don't necessarily have to have that multiplying but it's just helpful to know what that means.

So I follow that formula, 180, times by 'n' subtract 2, where is 'n' is the amount of sides of the polygon.

So, if I know this has one, two, three, four, five, six sides.

I know it's going to be six subtract two it's going to be four times 180.

And that gives me, of course, 720 degrees.

And I know that's right because I've already done a six sided shape there.

Yes, very good.

Let's move on then.

So for your independent task, here, what I'd like you to do is to use that sum of the interior angles, to have a go now at working at what those missing angles would be.

So for example, that's going to be a quadrilateral, sums for 260 degrees, so you know that missing angle is going to be the difference between the total sum there, and then of course that missing angle would be the difference between that 260 subtract um, the total.

Same idea for these sorts of questions over here.

So pause the video now, I will give you 8 minutes to do that.

Off you go.

Very good, let's go for it then.

So if we add up all those angles there, so far, we get 66 plus 96 plus 63.

So that gives us 225.

Now you could have a very long bar model modelling that, but for the purposes so far, I'm just going to do 225.

We know the total exteriors, sorry interiors, is going to be 360.

So I'm going to have to do 360 subtract 225 and that will give me A.

So when I do that, I know that 'a' is going to be equal to, what would 'a' be equal to? It will be equal to 135, very good.

So 'a' is going to be equal to 135 degrees.

'b' is a very similar process, I know it's a four sided shape here, so what I need to do for this one here, is I need to add up all those so far, so if I add those all together, I get uh, 150, 171? Yes, 171.

So, I need to do 360 subtract 171, and that gives me 'b'.

So what I get if I do that is, what do I get? Nice and loud please, shout it out.

What is it going to be? 189, right? Very good if you got that.

If you didn't, think about why.

So, 'b' is going to be equal to 189 degrees.

This final one's a little bit more complex, but we know it's going to be a one, two, three, four, five, six sided shape.

We've done that before so it's going to be a irregular hexagon.

So it's not a regular, 'cause regular means that it's all the same sides.

Uh, all equal sides.

Irregular means that's it's not necessarily all of those equal sides.

So what it's going to be, it's going to be four, cause six minus two, six sides times 180.

So, times that by 180 and we know that's going to be 720 degrees in total.

So if I add up all those uh, all these angles here that I have circled.

What I get is 98 plus 142 plus 113 plus 107 plus 130, that gives me 590.

So, what I need to do, 720 degrees subtract 590 degrees, and that gives me of course 'a'.

So therefore I know that 'a' is going to be equal to, what is 'a' going to be equal to? Very good.

130 degrees.

Nice.

Mark your work right or wrong and do corrections if you do get them wrong.

So what I would like you to think about now is explore task.

It says Antoni and Xavier are arguing about who is correct.

Who do you think is correct and why? So I've got Xavier saying 'It is possible to draw a six sided polygon in which all interior angles are 108 degrees.

Let's see.

'It is possible to draw a five sided polygon in which all interior angles are 108 degrees.

' Who knows? Pause the video now, I'll give you five minutes to have a think about and play around with that now.

Cool.

Let's go through it then.

So, the six sided shape then.

So let's, let's do it.

So all the interior angles going to be equal.

That means it has to be regular.

So it means that all the sides have to be the same length.

Now, I'm not going to be able to do it totally accurately.

But we get the idea.

If we do a six sided shape, it's going to be something like this.

Two, three, four, five six.

Okay, so that's my regular polygon there.

Okay.

Um, if I do that then, I need to think about what triangles I can split it into or I can split it or I can use the formula.

So I can do 180 multiplied by n subtract two, so six minus two which gives me four.

So, 720 degrees.

I then need to split those angles equally between it all.

So I need to do 720 divided by six, right? Now, if I do 720 divided by six, what do I get? Can you tell me, what do I get? I get 120 degrees, wouldn't I? So if I wanted to split them all equally, I'd get 120 degrees for each of them, wouldn't I? Etcetera, Etcetera, Etcetera.

Now, that clearly is wrong it's not possible then, as a result of doing that.

Let's check uh, sir Antoni's argument then.

Well, this one is going to be a five sided polygon, so it's going to be a pentagon.

So one, two, three, four, and then five.

So, that of course is a different uh, length of line to what we've had previously so we're just going to do a double dash.

Now, what I can say about this one is I'm going to do 180, multiplied by n minus two, well in this case it's going to be five minus two, which is three.

So that would be 540 in total.

Now I need to difference between one, two, three, four, and five angles.

So I need to do 540 divided by five.

If I do that, what I get is? Ah, seen something haven't you? It's going to be 108 degrees.

So I know that each individual angle here is going to be 108 degrees.

So actually, we've proved Antoni to be correct.

Very good if you got that without me having to help you.

But even if I did help you, very good if you understood it.

Well done.

So that brings us to the end of our session here today.

I just want to say a big congratulations if you've managed to keep up.

Make sure that you smash that exit quiz and that you stay tuned for the next one in the angle series.

For now, take care, and I shall be seeing you soon.

Bye bye.