Lesson video

Hello.

And welcome you to this lesson about angles in polygons, finding missing angles in polygons.

Please take a moment as always to clear away any distractions you could have.

Maybe make sure that your phone is turned off and that you are silencing all those app notifications.

If you're still keeping it on.

So that we're all ready to go.

And that we're happy going forward.

We're about to do some very powerful maths.

So listen in, focus as carefully as you can, so you can understand everything as I'm going on.

Without further ado, let Mr. Thomas' lesson commence.

So if you'll try this, what I'd like you to do, is I'd like you to consider these statements.

Whether they're true, sometimes true or never true.

Pause the video now.

And have a go at that task.

Amazing.

Let's go through it then.

So, our first statement is a quadrilateral has a reflex and an obtuse angle.

Can we remember back, perhaps to a previous video, whereby we're going through a quadrilateral has four sides, of course? And a reflex angle is anything between 180 and 360 degrees.

And an obtuse angle is anywhere between 90.

And, what was it? 90 and 180 degrees, wasn't it? Yeah.

Very good.

So that would mean, that I could have a 90 degree angle, so let's draw it out a 90 degree angle there.

that's the lowest it could possibly be, a right angle, still technically obtuse.

And then I could have a reflex angle looking like that and then I could connect to up to there.

Couldn't I? So there's my reflex angle, or is it? Hmm.

Does that work? Does that reflex angle? That's another obtuse angle, isn't it? So it can't possibly be that.

Well let's draw another one.

So, that's our right angle, 90 degrees.

And then I'd have to go down to here and then all the way up.

Ooh goodness me, it's go all the way up here.

Yeah, so that's one, two, three, four sides.

It has a reflex angle there.

A very strange looking shape.

Actually, yes, that is true.

So that is, that is sometimes true.

Not every case, but sometimes.

So you could get a square, for example, that would break that and it would just be, there'll be no reflex angle there.

So it's only sometimes.

A right angle triangle has two acute angles.

Let's draw out a right-angle triangle, looks like that.

It's always going to have those acute angles there.

So that's always true.

There we go.

What about the middle statement? The sum of the interior angles in an octagon is double the sum of the interior angles, in a quadrilateral.

Well, interior angles in a quadrilateral you should know like the top of our head by now.

What's that going to be? That's going to be 360 degrees, right? Didn't hear you say it loud enough.

Say it again.

360 degrees.

Good.

So some of her.

of course, I can't hear you shoutin' at a screen for goodness sake.

The sum of the interior angles in an octagon, Well, if you don't know that off the top of your head, it's going to be eight minus two times by 180 of course.

So that would be six times 180.

What would that be? Six times 180.

Can you tell me that? We should be pretty good at these by now, I reckon.

What would that be? That would be 1080, right? And that clearly isn't so that, is never true.

All the angles in a polygon are obtuse, well a polygon, definition of that is a shape with three or more sides.

Shape with three or more sides.

So we've clearly seen down here, they're not always going to be obtuse.

So, that's not.

that's sometimes true because this case is where it's broken.

But then if you get into, remember I've discussed before we've got like a megagon and we go all the way around, right.

A thousand sides.

You're going to have really, really obtuse angles as you're going around that.

So sometimes true.

A regular hexagon has an interior angle of 120 degrees.

Okay.

Let's try it.

So a regular hexagon six sides.

And I can form four internal angles, sorry four internal triangles.

So times that by 180, what would that give us? That will give us 720 degrees right? Now I've got six angles that needs to be spread out.

So 720 divided by six, What would that give me? Come on 720 divided by six.

What is it? Don't be lazy now.

It's going to be.

Very good, 120.

So that is always true because we're talking about a regular hexagon here.

So they're always going to be equally distributed in a regular hexagon.

So always true for those there.

So mark your work right or wrong, and do any required corrections.

corrections that are required.

Very good.

A little bit of a tongue twister there seemingly.

Let's move on then, so for our connect, we want to use the sum of the total interior angles in a polygon to find B in that polygon below.

Now what we're going to have to do is we're going to have to see, well, first of all, there are one, two, three, four, five sides.

This is a pentagon.

We know as much as that, we also know, one, two, three, four of our angles, but we don't know our fifth internal angle.

but we do know that, that's a point.

And we know the angles around a point sum to.

360 degrees, right? So if I know the angles around a point sum to 360 degrees, what I can do then is do 360 minus 141.

If I do that, I get.

well, let's just work it out, knowing there I do one, and then two.

So I know that's going to be 219.

And I know that to be roughly true because if I extend that further, that'd be a straight line, 180.

Bit beyond that reflex angle.

Cool.

Makes sense, 219 degrees fine.

So I've worked out all my interior angles apart from B now.

So this one's like the stumbling block now.

I don't really know what to do, but we know the total interior angles of pentagon, it forms, three internal triangles, doesn't it? So I'm going to have to do five minus two, which gives me that three times by 180.

So three times 180.

What does that give me? Didn't hear you, come on.

Say it louder please, 540 degrees.

Very good.

So 540 degrees.

That's what my total should be.

So I've got so far, 219.

I need to add 107, 93, and 27.

And I'll hopefully get an answer of, well add those together, I get 17 plus the.

Sorry, 16 plus three, plus seven, gives me 10.

So that will be 26 altogether.

So that gives me that bit there.

Scratching my head thinking, well, what do I do next? we'll do one plus zero plus nine, 10, 11, 12, 13, 14, four, and then three there, so I'm left with 346.

Now, are we sure that's right? Have I done something wrong? Hmm.

It sounds like I may have done.

Do you want to think about what my working? What have I done wrong there? What have I done wrong? I didn't carry the one, did I? So I need to carry the one there and I get 446, right? So that is the.

that is my total so far.

I now need to do 540 subtract 446.

Now, if I do that, what do I get? Well I get, I can quite clearly see actually, that's not too tricky.

I've got a hundred and then six less than that.

So it's going to give me 94.

So I know that B is going to be equal to 94.

So we've got there eventually.

And we can see just by inspection, that that looks like to be almost a, a right angle.

So we know we're right there.

If it was not drawn necessarily to scale, but there is some idea of a scale there.

So we know we're roughly right there.

What I'd like you to do now is to use this time to have a go at the independent task.

Now you need to use the sum of the total interior angles in a polygon to find the unknowns in those polygons below.

I'm going to give you 10 minutes now to have a go at doing that.

So pause the video and have a go that now.

Great.

So we've got our independent task answers now, so let's go through it.

So we've got our total interior sum here of the angles would be, one, two, three, four, five, six.

We know that's a six sided shape, which is a.

hexagon, right.

So I know I'm going to have to do four cause it's going to be six minus two times 180.

And that gives me of course 720 degrees.

So I know the totally sum of all of those is 720 degrees.

I then also know that the total so far, when I add all these together is going to give me, what's that going to give me? What's it going to give me? 590 degrees.

So I know so far I've got 590, so I need to subtract 590 from 720.

That gives me 130 degrees.

So I know a is going to be equal to 130 degrees.

And a sense check just to see that that's not a straight line.

And then that's slightly beyond right angle, a right angle of 90 degrees is going to be correct.

So I'm happy with that.

Let's mark our work, do our corrections, if we've got it wrong.

So we've got our angle marked b, and we know that this is a regular hexagon.

We've already established what a regular hexagon, the internal angles of that would be.

But just to prove it, it's going to be, again, six minus two, which is four times by 180.

So when I do that, I get 720 again.

What's unique about this is I need to do 720 divided by six, because they're all going to be the same angle.

So 720 divided by six.

What would that give me? 120 degrees.

Very good.

So that will give me 120 degrees for that one there.

And again, sense check it.

See that that is roughly speaking.

Yeah.

Cool.

That's a 120 degree angle.

Yeah.

Seems about right.

Now this one's a little bit harder.

This one you've got, much like we had with the connect task, we've got angles around a point and that one there is going to be 360 minus 65.

So that'll be two, nine, five, 295.

What about the other one then? That one is going to be 360 minus 252.

Now, if I do that, I get 108 degrees.

And then I can clearly see that I've got a one, two, three, four, five, six, seven, eight sided shape.

So I know it's going to be called an.

octagon, right? So this is an octagon.

So eight minus two times by 180 means that I'm now going to have to do six times 180.

And that gives me, what does that give me? 1080, doesn't it.

So if I add all of my internal angles together, I know I'm going to get 1080.

So I need to do 34, add 295, add 130, add 59, add 130, add 108.

I get 756 thus far.

That's not my answer though.

I've got c and c.

So I need to do 1,080 subtract 756.

Now, if I do that, I get 324.

Now that's not my answer to c.

Think about why for just a moment why that's not.

it's because I've got two, lots of c, haven't I? So we need to divide that, answer there by two.

So if I divide it by two, what I'm left with is 162.

Now my writing and working out isn't the neatest it could possibly be, but you get the idea.

You've got a lot more space than I have to work with here.

So I know that my answer is going to be 162.

Very good.

I hope you got that right.

If you did, brilliant.

Two thumbs up for you from Mr. Thomas.

Okay, for your explore task, what I'd like you to do is I'd like you to suggest angle sizes for these shapes.

I'm going to give you another 10 minutes for this task, if you need some help I'll be on the next slide, and I'll be helping you, if not pause the video now and I will help and you can go through it.

Awesome, Right.

So I'm going to assume you need some help or you want to go through the answers.

So I can see that this is a one, two, three, four, five, five sided shape, so I know it's a pentagon.

and I can also tell that these are regular.

This is a regular pentagon rather.

So I know that these are going to be equal sides.

I know the angles for a pentagon are going to be equal throughout the whole thing.

So what would it be? Well, it would be, 180 times by three would give me the internal angle, which would be 540 divide that by five would give me, what would that give me if I divide that by five? 108, right? Degrees.

So I know each of those angles there that would be 108.

108 degrees, et cetera, et cetera, et cetera.

Now, continuing that on, one thing we want to be really, really aware of here is that we've got, we've got.

we've got angles here, which are reflex angles.

So for example, I know that this is a 90 degree angle for the right angles.

But then I want to think about what could this be? Now that looks to be about 270 degrees.

I can see it's about three quarters of the way through a full turn.

So 270 degrees then.

I could then think, well, 90 plus 90 gives me 180.

Plus 270, what would that give me if I add those together? What do I get? Can you tell me before I've worked it out? What's it going to be? Sounds like you beat me to it.

450 degrees, right? And we can see that these are acute as a result of doing that.

They have to be because the total sum is going to be one, two, three, four, five sided shape.

The pentagon'd be 540.

So they're going to have to be, they look to be equal.

So I'm going to have to have 45 degrees there cause I've got 90 degrees leftover.

So you've got the idea now.

So this whole task is all to do with that idea of suggesting angles that it could possibly be while still retaining the idea that this is a reflex angle, for example, 270 degrees to quadrat all together.

So I've got to have fairly acute angles now for these ones.

There's these two here are reflex, right? These are acute.

These ones here, Yeah? So it's so important that you keep those the same.

And you think about them throughout.

This would be a regular one, two, three, four, five, six.

So that's a regular hexagon for example.

So I'm not going to go through absolutely every single one of them, because I'd been here for absolutely forever.

I don't want to take up too much of your time, but I can allow you to explore that task even further, if you wanted to.

And with that, unbelievably, it takes us to the end of the lesson.

So well done.

You've done really, really well to keep up with that.

It's a really interesting concept there that we've covered.

Please, don't forget to do that exit quiz that you, I know you can smash and absolutely belt it out the park.

Until next time I'll be seeing you later.

Take care.

Bye bye.