Lesson video

Hi there, and welcome to today's lesson.

My name's Mr. Peters, and today we're gonna be thinking about the angles that we can find within quadrilaterals.

This is a really useful lesson as it goes on to help us think about how we can support our reasoning when trying to find missing angles in lots of different shapes.

If you're ready to get started, then let's get going.

So by the end of this lesson today, you should be able to say that I can explain why the angles in a quadrilateral sum to 360 degrees.

We've got a few key words we're gonna be referring to throughout in this lesson.

I'll have a go say them first and then you can repeat them after me.

Are you ready? The first word is quadrilateral.

Your turn.

The second word is point, your turn.

And the final word is sum, your turn.

Let's have a think about what these mean.

A quadrilateral is a polygon with four straight sides and four vertices.

A point is an exact location.

It has no size, only position.

And finally, the sum is the result of adding two or more numbers together.

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

The first cycle, we'll think about angles in quadrilaterals, and the second cycle we'll think about identifying missing angles.

Let's get started with the first cycle then.

In this lesson today, we are joined by both Izzy and Sam.

They'll be sharing their thinking, as well as any questions that they may have that will help us to develop our thinking along the way.

So our lesson starts here today then.

True or false, the sum of the angles in the quadrilateral sum to 360 degrees.

What do you think? Take a moment to have a think for yourself.

Hmm.

In order to try and reason this, Izzy has decided to start thinking about a quadrilateral that she knows really well.

She knows that a square is a type of quadrilateral and a square has four right angles within it.

We can see this on the square on the left-hand side of the screen.

Each one of the angles in the corner is a right angle because it makes a square corner.

Sam is also saying that rectangles are a type of quadrilateral too.

What do you notice about the angles in a rectangle here? That's right.

There are also four right angles in a rectangle, aren't there? Just like a square.

And we know that because a square is in fact a special type of rectangle, isn't it? So if we know that a square has four right angles and a rectangle also has four right angles, then we know each angle has a value of 90 degrees.

There we go.

We know that a right angle has a value of 90 degrees, therefore each angle in either a square or a rectangle must also be 90 degrees too.

Yeah, and great thinking, Sam, we know that four lots of 90 is equal to 360.

We can use unknown facts to help us with that, can't we? If we know that four lots of nine is equal to 36, then four lots of 90 must be equal to 360.

So we can say that the angles in either a square or a rectangle, sum to 360 degrees altogether.

Hmm, great thinking, Izzy, I wonder that too.

Do all types of quadrilaterals have angles that sum to 360 degrees too? I wonder if we could explore that a little bit more.

So let's have a look at some other quadrilaterals then shall we? As Izzy is saying, we know that all quadrilaterals have four sides, don't they? And here's an example of one.

We know that this is a quadrilateral because it has four straight sides and it has four angles as well.

However, as Izzy is saying, problem is she doesn't know the size of each of these angles, does she here? Hmm, Sam thinks she's got an idea.

I wonder what that is, Sam? You're using some of your previous knowledge, aren't you, Sam? That's fantastic.

You are saying that you know that the angles around a point sum to 360 degrees, don't they? And we should know that from some of our prior learning.

So you're right.

We don't actually need to know the exact angle sizes here, do we, to work this out? In fact, as Sam's suggesting, we could just rip off each of the corners of the shape and place them around the point and see if they sum to 360 degrees, couldn't we? So here we go.

Let's rip off each of the corners.

There we go.

And let's label each corner, remembering that we can also name each corner a vertex, can't we? So let's take each vertex and place it around a point.

There we go.

Well, did you notice, that's right, where they all meet up at the point we've made one complete circle, haven't we? So because all of the angles from the quadrilateral meet together at a point and can be arranged in a way where there are no gaps.

And we can say that the angles in the quadrilateral that we had sum to 360 degrees, there we go.

We can see it shows one complete circle, 360 degrees.

Here's another example of a quadrilateral.

Let's check, is it a quadrilateral, first of all? Yes it is.

We know it's got four straight lines, don't we? And we also know it is got four angles.

And we know that quadrilaterals don't always have to have equal sides, do they? So let's use Sam's strategy then, shall we? As Izzy is suggesting, we need to rip off each of the vertices.

Here we go.

And now we need to try and start placing them around a point as well.

There we go.

Has that done the same thing? It has, hasn't it? So this example of a quadrilateral also works as well.

So this is helping to build our generalisation, Sam, isn't it? You're right in saying that the total sum of the angles in the quadrilateral sum to 360 degrees, doesn't it? And there we go.

Okay, time for us to check our understanding again now then.

The sum of the angles that meet at a point is A, B, C, or D.

Take a moment to think for yourself.

That's right, it's C, isn't it? The total sum of the angles around the point is equal to 360 degrees, isn't it? And have a look here then.

We've got some images here.

Can you tick the ones that are quadrilaterals? Take a moment to have a think.

That's right.

It's both C and D, isn't it? Why isn't it A or B? Well, we know that quadrilaterals have to be 2D shapes with four sides and four angles.

If we have a look at A, for example, we can see that it is in fact a 2D shape, but it doesn't have four sides or four angles does it? It actually has five sides, and therefore it would also have five angles as well.

And for B.

Well, B's got four sides, doesn't it? And it looks like, oh no, it doesn't.

There we go.

We can see that it's not fully joined up, is it? So we can't actually, in fact, call it a quadrilateral because all of the size haven't joined up to make the 2D shape, has it? So we can say that B in fact isn't a quadrilateral because it's not a closed shape at all.

So therefore we can say that C and D are the only quadrilaterals in these examples.

Okay, time for us to have a practise now then.

We're gonna continue our investigation around the question of true or false.

The angles in a quadrilateral sum to 360 degrees.

We found lots of examples that do so far.

I'm wondering, could you draw three quadrilaterals of your row and check to see whether the total sum of all the angles in your quadrilaterals, also sum to 360 degrees.

You might like to compare your findings with a friend's findings as well.

That might help also to generalise your thinking.

Good luck with that task and I'll see you back here shortly.

Okay, welcome back.

Let's have a look at an example that Izzy has come up with then.

Here is Izzy's quadrilateral.

We know it's quadrilateral because it has four sides and it also has four angles and it is a closed shape.

Let's mark on the angles here.

A, B, C, and D.

And if we were to rip each of the angles off and place them around a point, oh look, we can see that the sum of the angles here would also be equal to 360 degrees.

There we go.

Great work, Izzy.

I wonder what other quads you came up with.

Have you been able to now answer the question for yourself? Well done if you have.

Okay, onto cycle two now.

Identifying missing angles.

So let's have a look here first of all.

How could you identify the missing angle of this quadrilateral? Well, Izzy is saying that using our knowledge of knowing that all of the angles in a quadrilateral sum to 360 degrees, we can use this to help us going forward.

Let's use a bar model to help represent this.

We know that the quadrilateral is made up of four angles that sum to 360 degrees.

So 360 degrees can be the whole in our bar model.

We know we need four parts, don't we? Because we've got four different angles.

And how many of these angles do we know already? That's right.

We know three of the angles, don't we? Let's place these in the parts.

We've got one angle, which is 130 degrees.

We've got another angle, which is 120 degrees, and we've got another angle, which is 82 degrees, leaving us with one angle of angle A, which we don't know, do we? So at this point, Izzy is suggesting we could find the sum of the three angles that we do know.

So if we were to add 130 plus 120 plus 82 degrees altogether, that would give us a total of 332 degrees.

There we go.

We can now place that on our bar model.

And that will leave us with needing to find the difference, won't it? We now need to find the difference between the whole, which was 360 degrees and the amount of the three angles that we've worked out so far, which would be 332 degrees.

That would leave us with the size of the missing angle, angle A.

So what operation do we need to find the difference? That's right.

We can use subtraction to help us find the difference.

So now we can do 360 degrees minus 332 degrees, which leaves us with angle A, which in fact, 28 degrees.

There we go, solved.

Great thinking, Izzy.

That was really good application of your own prior knowledge to help us solve this problem.

Well done you.

Have a look at this example now.

How could we find the missing angle for this one? Take a moment to have a think for yourself.

Ah, Izzy thinks that's a really easy one.

Oh, I wonder why? Oh, of course, that's right, it's a rectangle, isn't it? We know that a rectangle is made up of four right angles, isn't it? At the moment we can see that we've got three right angles, all of which are 90 degrees in size.

Therefore the missing angle will also be 90 degrees, won't it? Exactly that Sam, no calculation needed at all.

However, what could we do to the arc in the top left hand corner? That's right, we could change that arc, couldn't we? Into using two small lines to show that it is also a square corner or a right angle.

Let's have a look at this example here.

Watch the shape carefully.

What do you notice? What did you see? Ah, Izzy says that it's still a quadrilateral.

So all of the angles must still sum to 360 degrees.

What was it to start off with? That's right, it was a rectangle to start off with, wasn't it? And we know that a rectangle would have four right angles.

So the two right angles at the bottom stayed the same, didn't they? They didn't change.

So we can keep them as they were.

However, the two angles at the top have changed slightly, haven't they? Because one of the sides on the right-hand side has got longer and therefore the top side has had to change slightly because of this.

Exactly that Sam, so we can see that the two bottom right angles, that part of the shape hasn't changed.

So as Sam's saying, we can still mark those on as 90 degree angles.

However, the previous ones, which were in fact 90 degrees have now changed, haven't they? We can see that the angle on the top left of the shape has now increased slightly, hasn't it, is now greater than a right angle.

And the angle at the top right hand corner of the shape has in fact decreased slightly, is slightly less than a right angle.

So let's have a look at our shape now.

And if the known angle at the top left-hand side was given to us, which in this case it is, it's now 99 degrees, how could we work out the missing angle? Ah, that's an interesting approach, Izzy.

You're right.

We know that the bottom two right angles are 90 degrees, don't we? And therefore, what was the top two right angles, sum to 180 degrees.

'cause that was two right angles as well.

So the top two angles must sum to 180 degrees.

So we can now find the difference between 180 degrees and 99 degrees 'cause we know the size of one of the angles.

So we could say that 180 degrees minus 99 degrees would be equivalent to 81 degrees.

So we could say the missing angle in this case is in fact 81 degrees.

Sam thinks she's noticed something that's even quicker.

I wonder what that is, Sam.

Both angles were right angles before they were 90 degrees.

That's right, Sam, they were.

And therefore that was equal to 180 degrees as we've already mentioned.

Yeah, brilliant Sam.

And now you're saying that we know that one of the angles has increased by nine degrees and that's right.

'cause it's changed from 90 degrees to 99 degrees, the top left angle, hasn't it? Oh, of course.

So that means that the other angle has to decrease in size by nine degrees, doesn't it? Of course it does.

If we're going to keep the sum of the angles the same and we increase one of the angles by a certain amount, then we're going to also need to decrease the other angle by a certain amount, aren't we as well? Great thinking, Sam.

So to work this out, we could have just subtracted the nine degree change that happened to the top-left angle.

And we could have just subtracted that from the top-right angle to find the size of the angle, which of course was 81 degrees as well.

Sam, that's fantastic thinking.

What a great way of reasoning the size of the missing angle in this case.

Why didn't you think of that? Well, next time Izzy, maybe you could have a go at thinking about Sam's strategy to help you calculate a missing angle if it works for you.

Here's one more example now, what do you notice this time? Izzy thinks that some of these angles look the same.

What do you think? Ah, nice.

Izzy thinks that these two look the same, and Izzy also thinks that these two angles look the same as well.

Nice thinking Sam, we could try and fold the shape to see if we can check that.

Should we have a go? That's not quite worked has it, Sam? Because we can't overlay the angles on top of each other to check that they're the same size here, can we? Izzy's got an idea.

Yep, we could do that, can't we? We could rip off the corner to check.

Should we have a go at that? Here we go.

Now we've ripped off the corners and let's see if they're the same then.

Here we go.

We said the A and C were the same, didn't we? So let's have a look.

Oh look, they overlay each other perfectly, don't they? And what about B and D? Do they overlay as well? Oh, brilliant, they overlay exactly over each other, don't they? So they're exactly the same angle, aren't they? So that's right, Sam, we can say that the opposite angles in the parallelogram are in fact equal, aren't they? So looking at this parallelogram now then what does that mean for us? Of course, Izzy, that's right, isn't it? If we know that one of the angles is 104 degrees, then the opposite angle will also be 104 degrees.

And if we know that this angle here was 76 degrees, then we also know that this angle here would also be 76 degrees.

That was easy enough for you two, wasn't it? Great thinking.

Okay, time for you to check your understanding now.

Can you find the missing angle A for me? Take a moment to have a think.

Okay, in order to solve this one then, we could have added up the sum of the angles that we have already, which gives us a total of 247 degrees.

And then we could have subtracted this from the total sum of the angles of a quadrilateral, which is 360 degrees, leaving us with a total of 130 degrees as the size of the missing angle.

Well done if you've got that.

And have a look at this one here, how could you tackle this by finding the size of angle B? Okay, well you could have taken a similar approach.

You could have said that the sum of the angle so far that we have is equal to 259 degrees, 'cause we've got a 90 degree angle and 90 degree angle and a 79 degree angle.

And then we could have subtracted that from 360 degrees, which of course would've given us the total amount of 101 degrees.

But as Izzy's saying, we didn't need to work with 360 degrees here, did we? We knew that we had two right angles already in this shape.

So in fact, we know that the other two angles must sum to 180 degrees.

So we could have just subtracted 79 from 180 degrees, which would've given us the total amount.

Or because we know both of them need to sum to 180 degrees, that in fact 79 degrees is 11 degrees less than a right angle of 90 degrees.

So we could have just added that 11 degrees back onto the size of the other angle, which again would've given us our total of 101 degrees.

Well done if you come up with using one of those slightly different strategies there.

Okay, onto our final tasks for today then.

What I'd like to do for task one is to find the missing angles for A, B and C.

And then for task two, I'd also like you to find the missing angles for A, B, and C as well.

And I'd also like you to have a think about using one of those strategies that we've been using in this lesson today to find the missing angles.

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

Okay, so how did you go about tackling these then? Well, we could have used our knowledge here to know that the total sum of the angles in a quadrilateral sum to 360 degrees, don't they? So we've got three known angles.

So we could add these three angles together, then we could subtract the total amount to find the difference, couldn't we? From the 360 degrees, which would be the sum.

So in order to do that, angle A would in fact be 142 degrees, if if we were to use that strategy.

If we were to follow the same strategy for the second shape, then angle B would in fact have been 70 degrees and a similar strategy for shape C would've also left us with an angle here of 80 degrees this time.

However, did you notice anything like Sam's asking? Well, in fact, we didn't need to do that three times, did we? We could have just done it once because actually the shape is exactly the same.

The shape has just been rotated in two different orientations.

And the angles, as you may have picked up, were exactly the same in each of the shapes.

So to work out the missing angles for B and C, we could have actually just used our knowledge of the angles in shape A.

Well done if you managed to spot that for yourself too.

Let's have a look at the shapes here and B then.

How did you go about tackling these? Well, for this shape here, we can see that the angles here look very similar, don't they? So in fact, there would've been 105 and 75 degrees as the angles that are missing.

C and D once again look like very similar sized angles here, don't they? So one way we could have done this is to add the known angles that we have so far, of 251 and 53 degrees together, and then divide what's remaining by two as these two angles for C and D would be exactly the same.

That would leave us with 28 degrees for both of these angles.

And for angle C and and for angles, E and F in shape C, we can clearly see that these angles again look very similar, don't they? So we can say the angle E was 124 degrees and angle F was in fact 56 degrees.

However we've estimated from these so far, and then we'd probably need to check in a certain way to be able to prove that they are exactly the same.

Izzy saying that one way that she did this was to fold the shapes in half each time, so the trapezium could have been folded in half with a vertical line right down the middle.

And doing this would've proven that those two angles were exactly the same as the other two angles.

We could have done the same for B, drawing a vertical line straight down the middle, which again would prove that both C and D are the same size angles.

And then finally, for E, we could have drawn two lines here.

We could have drawn a vertical line down the middle, which would help prove that the 56 degree angles were the same, or we could have drawn a horizontal line down the middle and folded the shapes over these lines, which should have proven that the 124 degree angle was exactly the same.

And then finally for C, where we can see that A is in fact a rectangle, and we know that rectangles have four 90 degree angles, so missing angle A was in fact 90 degrees, wasn't it? We can see that for B, the angle is increased by nine degrees on the left.

So that means the angle on the right-hand side would need to decrease by nine degrees.

That would leave us with 81 degrees.

And then for the last one, we can see that the angle, in fact, at the top-right is 63 degrees.

If we were to use a similar strategy here, we could say that 63 degrees is in fact 27 degrees less than 90 degrees.

So we could add that 27 degrees back on to the left-hand angle, which would give us a total of 117 degrees.

Equally, we could have said that the two angles at the top would need to sum to 180 degrees.

That's because we've got two angles at the bottom, which are also equal to 180 degrees.

And we know that the angles in a quadrilateral sum to 360 degrees altogether.

So if one of the angles is 63 degrees, that means the other angle would need to be 117 degrees in order for those two angles to sum to 180 degrees altogether.

Well done if you managed to get that for yourself too.

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

To summarise what we've been thinking about, we can say that the angles in the quadrilateral sum to 180 degrees.

If one angle is unknown, then we can calculate the missing angle by subtracting the known angles from 360 degrees.

And finally, you can use the properties of shape to help identify angles that have the same values too.

Thanks for joining me today.

Hopefully you're feeling a lot more confident about understanding the sum of the angles in a quadrilateral and how to find any missing angles within a quadrilateral.

Take care and I'll see you again soon.