Lesson video

Hello there and welcome to today's lesson.

My name is Dr.

Rowlandson and I'll be guiding you through it.

Let's get started.

Welcome to today's lesson from the unit of geometrical properties and Pythagoras's theorem.

This lesson is called "Congruence in Shapes", and by the end of today's lesson we'll recognise that for congruent shapes both the side length and the angle sizes are preserved.

Here are some previous keywords that are going to help us during today's lesson, so feel free to pause the video at this point if you want to read over these words before pressing play to continue.

this lesson contains two learn cycles.

In the first learn cycle, we're going to focus on how to identify whether or not shapes are congruent.

And in the second learn cycle we'll be solving some problems involving congruent shapes.

Let's start off with identifying congruence, shapes which have the same lengths and angles are congruent.

In other words, if you have a pair of shapes that are congruent, you could perform either a reflection, a rotation, or a translation on one of them or a combination of those things to then map it onto the other one.

When shapes are not congruent, they are incongruent.

Let's take a look at a few pairs of shapes and think about whether or not they are congruent.

Here's one pair of shapes.

Do we think these shapes are congruent or incongruent? Well, the shapes have the same angles, but they are incongruent because the lengths are different.

We couldn't trace over the shape on the left and then put that tracing over the shape on the right and it have it match up perfectly so they are incongruent.

How about this pair of shapes? Do we think these are congruent or incongruent? Well, these have the same lengths, but they are incongruent because the angles are different.

So these are incongruent.

And how about this pair of shapes? Do we think they are congruent or incongruent? Now this time the shapes are in different orientations, but that's okay.

They are congruent because they have the same lengths and angles we could trace over that shape over the left and using rotation and bit of translation, we could put it onto the shape on the right.

So it doesn't matter that they're in different orientations, they are congruent because they have the same lengths and the same angles.

So let's check what we've learned, true or false.

The two shapes here are congruent.

Do you think that's true or do you think it's false? And choose a justification for your answer.

Pause the video while you have a go and press play when you're ready to continue.

This is false and the reason why is because they have different lengths.

Yes, their angles are the same, but they don't quite have the same lengths, so they are not congruent, they are incongruent.

True or false? These two shapes are congruent.

Do you think it's true or false? And choose a justification.

Pause while you have a go and press play when you're ready for an answer.

This is true.

It doesn't matter that they're in different orientations.

That's okay for congruence.

The what matters is the shapes have the same lengths and the same angles, shapes which have the same lengths and angles are congruent, but not all lengths and angles need to be known.

In order to determine congruence, we can use facts about angles and equal lengths in order to work out missing measurements on shapes, and that might help us determine congruence.

But even if every single length and angle cannot be worked out, we can still determine congruence if enough information is known to ensure that there are no other possibilities.

Let's take a look at that now together, here we have Lucas and Izzy and they were looking at this pair of shapes and Lucas says, is there enough information to determine whether these shapes are congruent? What do you think? Pause the video while look at these and have a little think about if there's enough information there and if not, what information would you need? And then press play when you're ready to continue.

Izzy says, I can see that all the angles are the same in the two shapes, but I don't know the length.

So there's not enough information to determine whether or not they are congruent.

It might be that the shape on the right is an enlargement of the shape on the left, probably only a slight enlargement based on how they look, but it might not be exactly the same lengths.

So they're not necessarily congruent.

They could be, but we can't determine it.

So Lucas says, what if I tell you all of the lengths now on this shape? Can you determine whether or not they are congruent? Now what do we think here? Pause the video while you think about it and press play when you're ready to continue.

Izzy says, I can see that all of the angles and lengths are the same in each shape.

So yes, I can now be sure that they are congruent.

So Lucas says, I've removed the angle for EFG.

Is there still enough information to determine congruence? What do we think now? Pause the video while you think about this and then press play when you're ready to continue.

Izzy says, I could work out angle EFG by subtracting the remaining three angles from 360 degrees.

So yes, I can still be sure that they are congruent.

So Lucas says, what if I also remove angle HEF? Is there still enough information to determine congruence? What do we think now folks, pause the video while you think about this and press play when you're ready to continue.

Izzy says, I can see that two of the sides are parallel.

I could use the fact that co-interior angles sum to 180 degrees and parallel lines to work out the two missing angles.

So even though we don't know those two angles initially, we could use that fact and be sure that HEF is 90 degrees and that EFG is 127 degrees.

We could work them out.

So yes, I can still be sure that they are congruent.

So Lucas says, right, what if I also remove angle GHE? Is there still enough information to determine congruence? What do you think now? Pause the video while you think about this and press play when you're ready to continue.

Izzy says, well, I could work out angle GHE by using co-interior angles between parallel lines.

So we know that we have 127 and we have the 53 degrees.

Then if I draw EF as six centimetres and I draw GH as 12 centimetres using those angles, then there's only one way to draw the remaining eight centimetre length.

So yes, this is still enough information to determine congruence, right? Lucas says, what if I remove the length of EH? Is there still enough information to determine congruence? Pause the video while you think about this one now and then press play when you're ready to continue.

Izzy says, like before, I can work out the co-interior angle and draw three edges that I know.

Then it doesn't matter if I know how long EH is because there is still only one place that you can go.

So yes, I can still be sure that they are congruent.

Lucas is getting pretty tired of this now and he says, what if I remove the parallel line markers? Is there still enough information to determine congruence? Pause the video while you think about this and press play when you're ready to continue.

Izzy says, that means I can't work out angle EFG, because remember, co-interior angles only sum to 180 degrees when they are on parallel lines.

So we don't know they're parallel.

We can't use that fact to work out that you have a co-interior angle.

There are now options for which way EF could go, which would affect the length of VH as an example here on a screen.

So no, I cannot determine for certain that they are congruent now.

One thing we can learn from this though is we don't need to know every single angle on every single length to determine congruence.

We just need to know enough information to be absolutely sure that the there are no other options for the remaining information that we don't know.

So let's check what we've learned then, find the size of the three remaining interior angles in this shape.

Pause the video while you do this and press play when you're ready to continue.

Angle DAB is 100 degrees, angle ABC is 80 degrees and angle BCD is 100 degrees and we can work those out using co-interior angles between parallel lines.

'cause this is a parallelogram, true or false, it is possible to work out all the remaining interior angles in this shape.

Do you think that's true or do you think it's false? And choose a justification, pause the video while you do this and press play when you're ready for an answer.

The answer is false.

That's because there is not enough information on the diagram to be able to use any angle facts for the missing angles.

Even though this shape does look like a parallelogram without those parallel line markings or enough angles to be sure about this parallel lines in there, we don't know for certain that it is a parallelogram.

So if we don't know for certain that they are parallel lines, we can't use co interior angle facts to work out other angles in there.

So there's not enough information there to help us.

Is there enough information to determine whether the shapes are congruent here and justify your reasoning? Pause the video, why you have a go at this.

Write yes or no and a sentence to justify your reasoning and then press play when you ready for an answer.

The answer is no.

Congruence cannot be determined without knowing any of the lengths whatsoever because for example, one shape could be an enlargement of the other shape.

How about now? Is there enough information to determine whether the shapes are congruent? Say yes or no and then justify your reasoning with a sentence.

Pause the video while you have a go and press play when you're ready for an answer.

The answer is yes, even though we don't know all the lengths on that right hand trapezium to begin with.

What we do know is that the left and right edges are equal to each other because they have those hash markings.

So the hash markings on the edges mean that the missing length is 13 centimetres.

Therefore we know that all the lengths and angles in each shape are the same, so they must be congruent.

Is there enough information now to determine whether the shapes are congruent? Say yes or no and justify your reasoning.

Pause the video why you have a go and press play when you're ready for an answer.

The answer is yes.

This is very similar to the last question in that we don't know the right hand edge, but we can work that out using the hash markings and see that it's 13 centimetres, but we don't know the other three angles to begin with.

However, co-interior angles can help us work out that the top left vertex must have an angle of 113 degrees and the fact that this shape is symmetrical, we have some symmetry here on the left and the right of this trapezium.

We know that the other two angles must be the same as the angles on the left.

So yes, we can be sure that all the angles and lengths are the same and they are congruent, over to you now for task A, this task contains two questions, and here is question one.

The shapes below contain three pairs of congruent shapes.

Can you match the shapes which are congruent? Now, you may use a ruler and a protractor to help determine congruence or you may use any other equipment you think might help you with this task.

Pause the video while you have a go at this and press play when you're ready for question two.

And here is question two.

We have eight quadrilaterals with shape A.

We can see all of the lengths and all the angles and we can see which lengths are parallel to which other lengths.

However, we don't have as much information on the remaining shapes from B to H.

You need to decide which shapes contain enough information to be able determine whether it is congruent to shape A.

Please tick the shapes where congruence can be determined.

Now the lengths are not drawn accurately, so you're not going to be able to determine this by measuring angles and lengths.

You need to use the information that is provided to you in this task, and all lengths are given in the same units.

So pause the video while work on this and press play when you are ready for some answers.

Okay, let's see how we got on with that.

For question one, you could do this by measuring all the length and measuring all the angles and seeing in which cases you have the same length and same angles in a pair of shapes.

Another way you could do it is you could use tracing paper and carefully trace over a shape and then place it on top of all the other shapes or try to place it on top of all the other shapes to see which one it can be congruent to.

Either way you do it.

Let's look at the pairings.

A is congruent to F, B is congruent to E, and C is congruent to D.

And now let's go through question two.

In this question, we had to look at all the shapes from B to H and consider whether they have enough information on them to determine congruence to shape A.

Let's work through these one at a time.

With shape B, we can see it's a parallelogram because we have those parallel line markings on those four edges and we can see two of the lengths, the four and the five.

And because it's a parallelogram, we know then that the top horizontal edge is five.

It must be the same as the bottom one, the base, and we know that the sloping length on the right must be the same as the one on the left because they are equal in a parallelogram.

So we can determine all of the lengths, but what we don't know is any of the angles.

This might just be a parallelogram with the same lengths but on a different incline.

So no, we don't contain enough information to determine whether that one is congruent.

For C, yes, we have all four angles and we can see it's parallelogram and we know that the top horizontal edge is the same as a base, that's five.

But what we don't know is what those two edges on the left and right are.

They could be four, but they could be more than four or less than four.

We don't know, so we don't know for certain that that is congruent to shape A, that does not have enough information.

For Shape D, we have the 68 degrees, and because it's on parallel lines in a parallelogram, we could use co interior angles to work out the three remaining angles in there.

So we know the angles are the same as shape A and we have the four and the five on the length.

And because it's a parallelogram, we know that the other two lengths will be the same.

So we can be sure about the angles and we can be sure about the length.

Therefore, yes, it has enough information to determine congruence to A.

With shape E, shape E is a bit like shape D to begin with in that we can use that 68 degrees on all the parallel line markings to work out the remaining three angles.

So we know the angles are the same as A, but shape E is also a little bit like shape C in that we know the base is five.

We know that the top horizontal edge is also five, but we don't know what those edges are on the left and right.

We don't know if they're four or there might be a little bit more than a four or a bit less than four.

So without that information, we cannot determine congruence.

For shape F, we don't have the parallel line markings this time, but can we still do it? Well, we've got three angles in that quadrilateral, and we know that all four angles of a quadrilateral sum to 360 degrees, so we could subtract those three angles from 360 to work out the fourth remaining angle, which is 68 degrees.

So we know the angles are the same, but also because we can see that the co-interior angles sum to 180 degrees, 68 plus 112 is 180 because those sum to 180 degrees, we then know that we have parallel lines because co interior angles only sum to 180 degrees when they're on parallel lines.

So now we can be sure that it is a parallelogram and at the top edge must be five.

And also the sloping edge on the right must be four.

So yes, we have enough information there to determine that all the angles and all the lengths are the same as shape A.

Now with G, we can see that the 68 plus 112 makes 180 degrees.

So that means the edge on the left with the four in it must be parallel to the edge on the right.

However, we don't know the length of that edge on the right, so it might be four, it might be more than four or less than four.

So it really boils down to whether or not we can determine if those two angles are the same.

Because we don't have those parallel line markings on the bottom edge and the top edge, we cannot use co interior angles there to work out those missing angles.

And we can't use angles summon to 360 degrees in a quadrilateral because we don't have three angles to subtract from 360.

So there's no way we can work out those two remaining angles.

And without that, we can't be certain that the angles are the same and we cannot be certain that those two remaining edges are equal to five and four.

So no, that does not contain enough information.

With H, we have a little bit more information.

Yes, we can't work out those top two angles because we don't have co interior angles necessarily.

And also we don't have enough angles to subtract from 360 degrees.

However, once we've drawn the four and the five and the four at those angles, what we can see here, well, there's only one remaining way we could possibly draw that fourth remaining edge.

Therefore, yes, we have enough information determined by it's congruent to shape A.

Well done so far.

Let's now move on to the second learn cycle, which is solving problems with congruent shapes.

Let's do an investigation together.

I'd like you to start by please drawing a rectangle.

Pause the video while you do this and press play when you're done.

Rest your pencil over the rectangle in such a way that it splits it into two shapes.

Now you can do it like I've done on the screen here, or you can rest your pencil in a different way, as long as it splits into two shapes.

Now look at your two shapes and decide are they congruent.

Before we go any further with this, let's hear from a few people.

Aisha thinks the two shapes will always be congruent.

Jun thinks the two shapes will never be congruent, whereas Sofia thinks sometimes to be congruent and sometimes they're not.

Who do you agree with, Aisha, Yun or Sofia, do you think will always be congruent, never congruent or sometimes congruent? And what's the reason behind your answer? Pause the video while you think about this and press play when you're ready to continue.

Let's start with what Aisha said.

Aisha said, I think the two shapes will always be congruent.

Now this is not correct, but it's not enough just to say something's not correct.

Let's also prove it's not correct as well and think about how we could prove it's not correct.

One way to prove a statement is incorrect is by placing a counterexample down.

In other words, because Aisha is saying they'll always be congruent, if we can find an example of when they're not congruent, it shows that always is not the case and that statement's not quite true.

For example, if we placed a pencil here so that we have a triangle and a trapezium, we can see that they are not congruent, therefore they're not always congruent like Aisha said, we could also place the pencil here and create two trapezia where the vertical lengths are not the same.

If you look at the vertical length on the bottom right of that bottom trapezium, we can see it's very short compared to any of the vertical length and the top trapezium.

So those are not congruent either.

Yun says, I think they can never be congruent.

Now this is not correct either.

If we want to prove that this is not correct by counter example, we want to find an example of when the R congruence, that shows that it never is not the case.

We could place the pencil like this so it splits it into two triangles.

And these triangles are congruent because they are two right angle triangles.

Those right angles came from the rectangle and also with a rectangle, opposite lengths are equal to each other as well.

So we know all the lengths are equal to each other in these triangles.

And also with rectangles, opposite lengths are parallel.

So we can use angles in parallel lines, particularly alter angles in parallel lines to show that the angles in these triangles are equal too.

So here we have a way of splitting it into two congruent shapes, and that shows that never is not quite true either.

We could even do it like this and create two trapezius, but just make sure pencil is placed just right so that the vertical lengths are equal.

In other words, if you look at the short vertical length on the bottom trapezium, that needs to be the same as the short vertical length on the top trapezium.

And if we can get that just right, it provides the same amount of difference left on either side of those vertical lines for the other lengths of a trapezium there too.

So we can be sure that those have the same lengths and use angles in parallel lines to show that the angles are the same as well.

So those are congruent.

So it seems like Sofia was right.

Sometimes the are congruent and sometimes not.

When we place a pencil over a rectangle to split it into two shapes, and we've got an example on the left of when they are congruent, an example on the right of when they're not congruent, but in what circumstances are the shapes congruent? Why are they congruent sometimes and not over times, can we fix it so that they are always congruent by the way we place our pencil, pause the video while you think about this and press play when your ready to continue.

Let's look at some examples again, of cases where they are congruent and cases where they are incongruent.

On the left, you've got four examples of ways to place that pencil down or ways to split the rectangle up and to create two congruent shapes.

And on the right we have examples of where they are incongruent.

If we look at those examples of congruent shapes, those pencils, they all have something in common.

There's something about the position of those pencils that they all have in common.

And on the right hand side, those incongruent ones, that's not the case for any of those.

Can we spot what it is? In all the cases where the pencil split it into congruent shapes, the pencil goes through the centre of the rectangle and all the incongruent ones, it doesn't.

So over to you now for task B, for each question, draw a rectangle and then split it according to the instructions.

If for example, in A, you want to split it into two incongruent shapes, but in B, you want to split it into two congruent shapes, pause a video while you have a go at this and press play when you're ready to go through some answers.

Okay, let's see how we got on with that.

For each of these questions, we could split the rectangle in lots of different ways to answer the question.

So we can't go through every single possibility, but what we can do is look at a few examples together.

In question 1A, we need to split the rectangle into two incongruent shapes.

Here are two examples.

On the left, we have two rectangles, but they are not the same size.

They don't have the same lengths.

On the right we have two different shapes.

We have a triangle and a trapezium, so those are incongruent.

In B, split the rectangle into two congruent shapes.

Well, here are some examples.

We have two rectangles on the left, and we have two trapezius on the right.

The key thing there is splitting it through the centre of the rectangle to create two congruent shapes.

For C, we need to split the rectangle into three incongruent shapes when all three shapes be incongruent to each other.

On the left we have an example of splitting it into three different shapes, a triangle, a Pentagon, and a rectangle.

On the right, we split it into three of the same shape.

They're all triangles, but those triangles are all different to each other.

You have different lengths and different angles, so they are incongruent.

In part D, need to split the rectangle into three congruent shapes.

Here are two examples of that.

In both examples, we split it into three rectangles and we just need to measure it correctly so that those length are all the same and we can see all our right angles in 'cause of rectangles.

And E split the rectangle so that two shapes are congruent and the other is incongruent.

Well, here are two ways we could do that.

On the left we've used three rectangles, but we've made sure that one of those rectangles is much smaller than the others, but the other two are the same.

So that means that horizontal line must be halfway up the rectangle.

And then in the other example, we've done the same.

Again, we've got that real narrow rectangle on the right, but we split the remaining shape in half by going diagonally.

You could even think about this as create your incongruent shape first and then use the remaining rectangle to split that into two congruent shapes like we did earlier.

And part F means to split the rectangle into four incongruent shapes.

And here are two examples of that.

On the left, we have four rectangles, but they all have different lengths.

On the right we have some different shapes.

We have a triangle, another triangle, a trapezium and a rectangle, and we can see those two triangles are not the same.

'Cause one's a right angle triangle, and if one is not necessarily a right angle triangle.

In part G, we need to split the rectangle into four congruent shapes.

Here are a couple of ways we could do that.

On the left, we've split into four rectangles, which all have the same width and length, and we've done that by going through the centre of that rectangle.

But there are other ways we could do that.

And on the right we split into four triangles.

They're all right angle triangles, and so long as that vertical cut is halfway along that rectangle, then all the triangles will have the same length as well.

And in H, split the rectangle into two pairs of congruent shapes.

Here are some ways we can do that.

On the left, we have a bit like the example we saw earlier.

We create a thin rectangle on the right and then a big rectangle on the left and split those in half.

So we have two pairs of congruent shapes, on the right we have split it into four triangles.

You see the top triangle and bottom triangle.

They are congruent.

They have all the same angles, and we can use vertically opposite angles to show that.

And alternate angles and parallel lines too, and the same as well with the triangle on the left and the triangle on the right as well.

Fantastic work with today's lesson.

Let's summarise what we've learned.

If one shape can fit exactly on top of another shape by using either rotation, reflection or translation or a combination of those, then the shapes are congruent.

That's because shapes which have the same lengths and angles are congruent.

Now, it doesn't matter about the orientations.

The orientations can differ so long as the lengths and angles are the same.

Now missing and length and angles can be worked out if possible to help determine congruence.

But so long as you have enough information to be absolutely sure they're congruent, then you don't need to know every single length and angle along the way.

You just need to know enough to be certain.

Great job today, thank you very much.