Lesson video

Hello, my name is Mr. Thomas, I'm sure you're used to me by now, having seen a few of these lessons before, and welcome to this online lesson on angles in polygons, polygons and triangles.

Please take a moment, as always, to clear away any distractions.

That may well be your brother, your sister, or your pet, and find a quiet space where you can really focus on the maths.

It's going to be so important you focus on this because as always we're going to do something really powerful, we're going to learn lots in today's lesson, and we're really needing to focus.

Okay? Now, once that's all done let's get ready to go.

So have a go now at the try this.

Your first question now, what is the sum of the angles in the blue triangle? Should be pretty good to go with that one.

What is the sum of the angles in the quadrilateral formed by the orange and grey shapes? Again, not too tricky, assuming you've watched the previous videos.

And then what is the sum of the angles in the entire hexagon? Pause the video now and have a go at that task.

Okay, excellent.

I'm assuming that you've done that task and that we're going to go through it now.

So the sum of the angles in the blue triangle of course.

Well we know that.

The angles in a triangle sum to? Fantastic, 180 degrees.

The interior angles sum to 180 degrees.

So we can write that in there.

So that's 180 degrees there.

What is the sum of the angles in the quadrilateral formed by the orange and grey triangles? Well again, we know the interior angles of a triangle sum to? 180 degrees.

Of course.

So that is going to be 180 degrees, and then 180 degrees there.

So combining the two together of course.

Well 180, add 180, gives us? 360! You got it, well done.

So it's 360 degrees there.

Very good.

What is the sum of the angles in the entire hexagon? Well, a hexagon is how many sides? Six, right? So six sides.

And coincidentally we've got one, two, three, four, five, ah, six sides! So that's an irregular hexagon there.

We've got six sides for that one there, so it works.

So we can say it's going to be the sum of all of those triangles.

So we've got one, two, three, four.

Four lots of 180 degrees.

Or we could say that that's the same as adding 180, add 180, add 180, add 180.

So that gives us in total what? 720 degrees, right? So it's 720 degrees there.

Very good if you got that without my help, and even if you did, if you were quite quick off the mark or able to answer those questions, very, very good.

Let's go on then.

So we've got to split these shapes here.

Now, I'm thinking how could I split these? Well I could split them just in half but is that really any use? Like, we're not really going to get much out of that.

But the one really key thing that we need to do when we're splitting shapes, and what we saw before, is they all went from one specific point.

So they all went from, say, this point here and we formed, what did we form when we saw those split shapes with that hexagon just a moment ago? They were triangles, right? So if we want to create a triangle what I'm going to do is I'm going to do a dotted line here that then splits it off into one triangle there.

I've still got a triangle here that I need to split, so I need to go to another corner, another vertex, here.

And I split it off into a second triangle, a third triangle.

So I can then work out, well what are the total angles within that, well what is it, it's a one, two, three, four, five sided shape, which would mean it's a? Pentagon, right? So that's an irregular pentagon.

So I've worked out there are three triangles.

So three triangles in an irregular pentagon, so a five sided shape.

That's going to be really important further down the line.

So if you're going to watch a few more of Mr. Thomas' videos you're going to think, yeah, three triangles, five sided shape, cool.

Right.

Let's think about this next shape then, that's really crucial.

We're going to go from quite a few different points or one specific point? One specific point, right? So let's have a think about this.

This one here seems pretty accessible to every single corner that we can think off.

Because if I split it here that gives me one triangle.

Split it here again, I get two triangles.

And then up here I'm going to get a third triangle and then a fourth triangle.

So I've got four triangles in total and what sided shape is it? How many sides has that shape got? Let's have a think about that.

It's got one, two, three, four, five, six.

So it's a six sided shape.

I'm going to just pause for a moment for you to think and reflect on that last one, 'cause I think you may be getting the idea now.

What do we need to do? The very first step? We need to choose one specific point, don't we, to go from.

So I could choose up here for example.

And then what do I do after that? Do I just go to a random side, just like down to here? No, of course I don't.

I go to a vertex, don't I? I go to like a corner.

So I could do a dotted line going down to there and then ah, brilliant, I've split it already, I've got one triangle, and then two triangles.

So I've got two triangles in total.

And well how many sides does that shape got? How many sides does that shape got? It's got one, two, three, four sided shape.

It's got a four sided shape.

Little bit of lag there, never mind.

So it's a four sided shape.

Hopefully that's clear now, about what we need to do.

Let's move on.

So what I want you to do for this independent task is I want you to think about the sum of the total interior angles.

Now you got all sorts of shapes there that you may need to split, right? So you go from one specific corner and then draw it to another specific vertex as well at the same time, right? I'll leave you to get on with that now.

Pause the video and have a go please.

Excellent.

Let's have a go then at seeing why these answers are the case.

If I have a go at doing this one, splitting this shape off, I can split it off into one triangle, two triangles, three triangles, and then four triangles there.

Four triangles.

So four times 180 will provide me with my answer of 720 degrees.

This is the total interior angles.

What about this one? Well this is an interesting shape to begin with.

This is a.

Well how many sides have we got? One, two, three, four, five.

So I could go from here, couldn't I? That'd be a clever idea.

I could then reach over to here which would give me one triangle.

I could then reach over to here.

Two and then three.

So three triangles in total, times by 180 degrees, gives me that total of 540.

Very good for the interior angle.

But what about this shape here? Ooh! This looks a little bit nastier, not very nice.

But again, we can maybe make something of this.

We could start off here.

That may be quite clever.

We could start off there and go to there, so that would be one triangle.

We could then split it off into here, two triangles.

Split it there.

Three triangles.

Split it down to there.

Four triangles.

And then we've got like a weird quadrilateral here, right? Hmm.

Well I'm not sure about that splitting, was that a very good idea? Maybe we should split it some other way.

So let's have a think again.

Where would be a better place to maybe split this shape from? Have a think about that now while I start rubbing this out.

Where would be a good idea to separate the shape? Have you got an idea? 'Cause I'm thinking at the moment, once I've allowed my pen to load up, I'm thinking at the moment maybe somewhere like here.

That might be quite a nice place to split it from.

Or maybe even here.

Well we've done that.

Could it be here maybe? That seems good.

That seems like it could reach almost every corner, right? One of those vertexes.

So how about there then? So one, two, three, four, and five, and then six.

So that one works really nicely.

So we didn't need to use that one.

So that gives us one, two, three, four, five, and then six.

And six times 180 for that total interior angle gives us that 1080.

Very good.

Now you may have been really switched on and realised, actually, hold on Mr. Thomas, you're doing something here, some sort of wizardry of some sort.

Well I may have played a little trick on you here because actually this is pretty much the same shape, I've just scaled it down slightly.

It's a similar shape.

All those angles are exactly the same, I've just scaled it down slightly.

So that's exactly the same process as this one here, so we can give that a little tick to say that we've done that.

Okay.

So let's move on.

We've got an explore task now and I want you think about drawing three quadrilaterals, pentagons, and hexagons that are non-congruent.

So you're thinking ahead, scratching your head thinking, jeez, what's a quadrilateral, what's a pentagon, and what's a hexagon? And they've got to be non-congruent.

So four really key things there.

I want you to first find the sum of their interior angles and then tell me what do you notice.

Pause the video now if you're happy to do that, or if you'd like some support or an answer to the question feel free to carry on.

Awesome.

So I'm going to go through the answer or support you however you feel fit.

So we need to first think, well what does non-congruent mean? So non-identical, I don't want exactly the same shape, 'cause that would be pointless, right? I want to be able to analyse different shapes.

If I just had the regular shape of a regular quadrilateral, well what's a regular quadrilateral, what's that called? There's a special name for that.

It's a? Square, isn't it, right? So if I just had four squares that's not really going to be that interesting to analyse, that's the same thing repeated, right? Who cares, that's not that same thing.

So with that in mind what I want to do is I want do draw, say, four quadrilaterals of course, right? So I could go with a square.

That looks like a square.

I could then think, well, find the sum of their interior angles.

Well split it of course.

I've got 180 there, I've got 180 there.

So I've got 360, 'cause I've got two lots of 180 there.

So what about another shape that is a quadrilateral? Well I could go for something that looks like this.

And that is an example of a? Trapezium, isn't it, right? So a trapezium there.

And what I could do here is I could, again, split it off into two triangles.

A slightly wonky line, but never mind.

So two triangles there, which again is 360.

Right? I'm not going to do a third one 'cause I think you get the idea.

And if I tried a pentagon now.

So how many sides does a pentagon have? It's got five, isn't it? I've got this sort of hint down there.

So if I then did that, I could do one, two, three, four, and then five.

Okay.

So that would be a five sided shape, pentagon, I could then split it off.

One, two, and then three.

So that would give me a total of 540 degrees.

Three triangles, three times 180.

I could then try another shape, which would be a? Let's try a hexagon this time.

All of them are going to be 540 for a pentagon, but a hexagon? Let's see what that would be.

One, two, three, four, five, six.

That would be one, two, three, and then four.

So one, two, three, four.

So four triangles.

Four lots of 180, what would that be? 720, right? So it's 720 for that one there.

You're starting to notice something I hope, and that's something we're really going to explore next time, when we try and make a little more sense of what's going on here.

And with that, that brings me to the end of our lesson that we've done today.

We've learned loads from today's lesson.

I'm really, really happy and really proud of what you've done.

So please, keep up the great work and don't forget to do that post quiz.

It's really important you test your knowledge of what you've learned.

I've made those questions fairly easy, and there are some that are going to be a little bit challenging.

So please, make sure you do that so you can really make sure you've understood today's lesson to the best of your ability.

Okay? For now, take care, and I'll see you hopefully in the next lesson.