Lesson video

Hello, and thank you for choosing this lesson.

My name is Dr.

Ronson, and I'm excited to be helping you with your learning today.

Let's get started.

Welcome to today's lesson from the unit of Geometrical Properties and Pythagoras' Theorem.

This lesson is called "Congruent or Similar or Neither." And by the end of today's lesson, we'll be able to answer that precise question about shapes.

We'll be able to determine whether shapes are similar, congruent, or neither.

Here are some previous keywords that will help us during today's lesson.

So you may wanna pause the video at this point if you want to remind yourselves the meanings of any of these words.

This lesson contains two learning cycles.

In the first learning cycle, we're gonna be focusing on distinguishing between similarity and congruence to be able to figure out how are these two terms different to each other.

And in the second learning cycle, we're gonna be using that to check for similarity and congruence.

Let's start off with distinguishing between similarity and congruence.

Let's begin by revisiting the definitions of congruent and similar.

When two shapes are congruent, the angles are the same in each shape and the lengths are the same in each shape.

When two shapes are similar, the angles are the same in each shape and the lengths are in the same proportions.

Let's see that through an example.

Here we have a right-angled triangle, and the other angles are 37 degrees and 53 degrees.

Its lengths are 3 centimetres, 4 centimetres, and 5 centimetres.

This shape is congruent to shape A because it has all the same lengths and all the same angles.

This shape is similar to shape A because although it has all the same angles, it doesn't have the same lengths, but those lengths are in the same proportions as shape A.

If we took the lengths from shape A and doubled them all, we would get 10, 6, and 8 centimetres, like we can see here.

One is an enlargement of the other.

If two shapes are congruent, then they are also similar.

If you think about how this shape we can see on the screen here compares to shape A, we can see that all the angles are the same, and also the lengths are in the same proportions as shape A because they're the same.

So it fits the definition of both congruence and similarity.

So it's congruent to shape A and also similar to shape A.

However, if two shapes are similar, then they are not necessarily congruent.

Here we have a shape that is similar to shape A because all the angles are the same and the lengths are in the same proportions.

However, because those lengths are not the same as shape A, it's not congruent.

It is incongruent to shape A.

So here we have an example of a pair of shapes that are congruent and also similar, and also an example of a pair of shapes that are similar but not necessarily congruent.

Similar shapes are only congruent if the scale factor between them is 1.

So, all pairs of congruent shapes are also similar, but only some pairs of similar shapes are also congruent.

It kinda looks a bit like the diagram here, congruent sits inside similarity.

It's a special case of similarity.

Now, you might be thinking, every single time you have a pair of congruent shapes, do you need to say that they are both congruent and similar? Well, stating that shapes are congruent also implies that they are similar as well.

Therefore, it is sufficient to state that the shapes are congruent without stating that they are also similar.

And also, a pair of shapes can be neither congruent or similar as well.

So let's take a look at some examples.

Are these shapes congruent, similar, or neither? Let's have a little think.

Jun says, "It doesn't matter that the orientations are different.

They both are parallelograms with the same lengths and the same angles.

So they are congruent." It hits the definition of congruence.

How about these two then? Are these similar, congruent, or neither? Laura says, "They are not congruent because the lengths are different.

One shape, however, is an enlargement of the other shape, so they are similar." We can see that the angles are all the same in those two parallelograms. And also, if we take the 8 centimetres and the 10 centimetres and times them both by a half, that scale factor of a half there, we get 4 centimetres and 5 centimetres.

So one is an enlargement of the other.

They are similar.

How about these two here? Are these similar, congruent, or neither? Lucas says, "The angles are different, so that means they are neither congruent or similar." Remember that both of those definitions for similar and congruent both included that the angles must be the same.

So the angles are not the same, they are neither congruent or similar.

And how about this pair of shapes? Are they congruent, similar, or neither? Well, Sofia says, "The angles are all the same, but the lengths are different.

Does that mean they are similar?" Because for similar shapes, the angles need to be the same, the lengths don't need to be the same.

But Aisha says, "The lengths are not in the same proportion, so they are not similar." And we can see that by looking at these lengths.

On the right-hand parallelogram, the base 8 is the same as the other sloping length there, 8.

You could even think about how the multiplier between those two numbers is 1.

But on the left parallelogram, that's not the case.

The multiplier is not 1.

We don't times 10 by 1 to get 8, so they're not in the same proportion.

So that means they are neither congruent or similar.

So it's your turn now as we check what we've learned.

The two shapes here are, blank.

Fill in the blank.

Are those two shapes congruent, similar, or neither congruent or similar? Pause the video while you make your choice, and press play when you're ready to continue.

These shapes are congruent.

They have the same angles and the same lengths.

They're in different orientations, but that's absolutely fine for congruence.

And how about these two shapes? Are they congruent, similar, or neither congruent or similar? Pause the video while you make your choice, and press play when you're ready for an answer.

These shapes are neither congruent or similar.

Yes, they have the same angles, and okay, the lengths are different, so they could be similar, except they are not in the same proportions, so they're not similar.

They are neither congruent or similar.

And how about this one? Are these shapes congruent, similar, or neither? Pause the video while you make your choice, and press play when you're ready to continue.

These shapes are similar, and we can see that because the angles are the same and, yes, the lengths are different, but the lengths are in the same proportions.

If you take all the lengths from the top trapezium and divide them by 2 or times them by a half, you get the lengths on the bottom trapezium there.

Okay, it's over to you now for task A.

This task contains two questions, and here is question one.

We have an assortment of shapes here.

They are labelled A to F, and for shapes B to F, some of those are congruent to shape A, some of them are similar to shape A, and some of them are neither congruent or similar to shape A.

On the right of this question, we have a diagram similar to what we saw earlier.

What you need to do is compare each shape to A and write the letters from B to F inside that Venn diagram we can see there.

If you think it is congruent to A, put it in that middle circle in the centre that's labelled congruent to A.

If you think it's similar to A but not congruent, put it in the outer circle labelled similar to A.

And if you think they are neither similar or congruent, put it in that outside space there labelled neither congruent or similar to A.

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 a dotted grid with two points labelled on, L and C, where a line segment join them together.

That's our starting point because what we're gonna do is we're gonna plot points A to L on this grid so that we end up with a diagram that contains five connected rectangles.

Now, some of these rectangles will be congruent to each other and some will be similar to each other.

The information on the left of this question describes what these relationships are like between these rectangles and also the points on the grid.

So use the information to plot points A to L on the grid to draw those five rectangles.

Pause the video while you have a go at this, and press play when you're ready for some answers.

Okay, let's see how we got on then.

In question one, let's go through each of these shapes one at a time.

With shape B, it is neither congruent or similar to A.

Yes, they have the same angles, but the lengths are not the same and they're not in the same proportions, so they're neither similar or congruent.

Shape C is similar to shape A because the lengths are not the same but they are in the same proportions and it has the same angles.

Shape D is neither similar nor congruent, and we can rule those both out straight away because it doesn't have the same angles, so it can't be similar or congruent to shape A.

Shape E is congruent to shape A because it has the same angles and also the same lengths.

It's just in a different orientation.

And shape F is similar to shape A.

It has the same angles, and those lengths are in the same orientation.

If we take the lengths from shape A and multiply 'em by 1.

5, we get the lengths in shape F.

Then question two.

Let's work through these bits one at a time.

So it says LC is a horizontal line with length 2 units.

And we can see that.

That was our starting point.

Point F is below point C, and we know that CF is equal to two lots of LC, so point F must be here.

And we know also that LCFI is a rectangle, so we just need to plot that final point for I to create that rectangle, and there we have it.

Then we have some information about another rectangle.

We have that KLIJ is congruent to LCFI.

Now, that first rectangle, KLIJ, that has an L in it and also has an I in it, so it must go through those two points, L and I, and then be congruent and have the same lengths as the other rectangle we've already drawn.

So it must go here.

That goes through point L and point I, and it has the same lengths.

And then the next rectangle, ABCL is congruent to LCFI.

Again, look for points that they share in common.

We can see both those rectangles go through point C and also through point L, so it must be above it and be congruent like this.

And then we have rectangle CDEF is similar to LCFI but not congruent, so it's gotta be an enlargement of it of some kind.

We can see it goes through points C and points F, which we've already drawn.

Now, if we look at the length from C to F, that is four units.

If the shape we are about to draw is similar but not congruent, then its longest length cannot be four.

Its shortest length has to be four, which means its longest length has to be eight, and then the other two points go here.

This rectangle CDEF is an enlargement of rectangle LCFI by a scale factor of two, and they go through points C and F.

And then finally, FGHI is similar to LCFI but not congruent.

Let's apply the same logic again.

This rectangle we're about to draw goes through points I and F, which you've already drawn there.

Now, that length is two units, and that is the shortest length of LCFI.

What we're about to draw is not congruent to it, so our shortest length cannot be two.

That must be our longest length, which means to keep it in proportion, the shortest length must be one, and it goes here.

Now, we could have variations in this solution if we start overlapping rectangles.

For example, if those last two points went there.

However, D and E are fixed due to the space on the grid.

Fantastic work so far.

Now we've got our heads around the difference between similarity and congruence.

Let's apply that and learn a little bit more about how to go about checking for similarity and congruence with shapes.

Let's start by revisiting the difference between these two things again, congruence and similarity.

If two shapes are congruent, then the angles are the same, the angles are in the same order, and also the lengths are the same.

If two shapes are similar, then the angles are the same, the angles are in the same order, and here's the difference, the lengths are in the same proportions.

Now, yes, there's a difference between congruence and similarity, but there's also, they have something in common as well.

With both of these definitions, the angles need to be the same for 'em to be true.

So Sam says, "If two shapes have different angles, then we can rule out both congruence and similarity." So when we are checking for congruence and similarity, it makes sense to check the angles first because if the angles are different, we can rule out both of those categories, congruence and similarity, and we know that they are neither of those.

So let's apply that as we go forward.

A table can also be used to check that two shapes have the same angles in the same order.

Tables are not always necessary, but they can be a systematic way of laying our information out, especially when things get quite complicated.

So for example, here we have two shapes.

Now, we can see visually that they are not similar, but let's go about applying a systematic method to it so we can see how the method works.

If we take the angles for shape A and put 'em in the table, we have 100 degrees, and if we go clockwise round now, we have 100 degrees again, and 80 degrees and 80 degrees.

Now let's do the same with shape B.

Andeep says, "Here in shape B, there is a 100 degree angle and an 80 degree angle, and they are next to each other, so these could go here in the table at this point where we can see there is a 100 degree angle and 80 degree angle in shape A." So so far, the angles line up between shape A and shape B.

But the other angles then don't match.

Shape A has consecutive angles which are both the same, but shape B doesn't.

Hmm.

So the fact that those angles are not in the same order means that they are not congruent and also not similar.

Let's apply the same effort again with this pair of shapes.

The angles for shape A are already in the table.

For shape B, Izzy says, "Here are two consecutive 100 degree angles." And that's great because we have that in shape A as well.

So Izzy says, "I can put these in the table first and fill in the rest." And then going round, she says, "I can fill in the rest of the table in such a way that all the angles match up." So we can see that they are the same angles and also in the same order, so they could be either congruent or similar.

Izzy says, "I'll need to know some lengths before I can determine whether they are congruent or similar." So the angles have helped us decide that they are either similar or congruent, and the lengths will help us from there tell which one it is.

So now we know the angles are the same, let's look at the lengths.

On this table here, we can see there's a row for shape A lengths and a row for shape B lengths.

And what you can see is the boxes are a little bit out of line with the angle boxes.

That'll make sense as we start to fill it in.

Izzy says, "The edge between the two 100 degree angles is 6 metres." So we put the 6 metres in the box that is between those two 100 degree angles above it.

And then Izzy says, "We can fill in the rest of the table in the same way." So for example, the next two angles that have a empty box underneath them is 100 degrees and 80 degrees.

So look at the length that is between 100 degrees and 80 degrees, and that is 8.

And continue filling in the same way.

And we get this.

And then we can look at shape B.

Izzy says, "The edge between these two 100 degree angles is also 6 metres.

And I can fill in the rest of the table in the same way." And we get this.

What we can see now is, the lengths are the same in the two shapes.

We can even think about how the scale factor from shape A's lengths to shapes B's lengths is one, so they are congruent.

Let's compare that to a different case.

Here, once again, all the angles are the same and in the same order, but we can see the lengths are not the same between shape A and shape B.

But what we can see is that there's a scale factor that is consistent throughout all those corresponding lengths.

Izzy says, "In this case, all the lengths in shape B are double the corresponding lengths in shape A.

So they are similar." Shape B is an enlargement of shape A.

That means they're similar.

So let's check that we can apply that ourselves now.

Here we have a shape, and we have a table which has some details in about that shape.

All the angles are filled in.

One length is filled in.

There's a couple of blanks in the lengths column, and one of those lengths is labelled X.

What I need to do is locate that length on the diagram and find the value of X.

Pause the video while you do that, and press play when you're ready for an answer.

The answer is X equals 9.

We can see that by looking for the 87 degree angle and the 74 degree angle, and finding the edge that connects those, we can see it's 9 metres long.

So the value of X must be 9.

And we can do the same to fill in the rest of the table as well.

So here's another question where we are presented only with the table of information about two shapes, A and B.

And what you need to do is decide, are shapes A and B congruent, similar, or neither congruent or similar? Pause the video while you do this, and press play when you're ready for an answer.

These shapes are similar.

We know that they are at least congruent or similar because they have the same angles in the same order.

But because the lengths aren't the same, we should look for proportions.

We can see that if we take all the lengths from shape A and multiply 'em by 3, we get the lengths from shape B.

So how about this time? Information about shapes A and B are on the table again.

Are those shapes congruent, similar, or neither congruent or similar? Pause the video while you have a go, and press play when you're ready for an answer.

These shapes are congruent.

The angles are the same and in the same order, and also the lengths are the same as well.

Okay, it's over to you now for task B.

This task contains two questions, and here is question one.

This question involves three shapes, shape A, B, and C.

We can see shape B and shape C on the screen here.

For shape A, we can't see a diagram of it, but we do have details about it in the table.

We have the angles for shape A in the table.

What you need to do is fill in the table for shapes B and C, matching up the angles, if you can, with shape A.

And then once you've done that, use that information to answer parts B and C.

Pause the video while you do that, and press play when you're ready for question two.

And here is question two.

This question involves two shapes, shape A and shape B.

We can see shape B, but we can't see shape A.

However, we do have information about it in the table.

We have its angles, and we also have the length of the edges between each of those angles.

What you need to do is fill in the table for shape B and then use that information to answer question B here.

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

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

Here's question one.

We had to fill in the table for shapes B and C, match up the angles if we could.

For shape B, we could match all those angles up like for like with shape A, but for shape C, we could only match up these three angles.

Yes, 127, 72, and 135 appear consecutively in shape C.

However, the other two angles, 90 degrees and 116 degrees, they are out of order.

They're the other way round for shape C, so we can't quite put them in exactly the same order as we did for shapes A and B.

Can put 'em here.

And then with question B, which shape is neither similar nor congruent to shape A, it must be shape C because the angles are the same but they're not in the same order.

So because they're not in the same order, they are neither congruent or similar.

And then with question C, is it possible to determine congruence for the other shape? Explain your answer.

Well, even though the angles are all the same and in the same order, no, we cannot determine congruence because we don't know whether or not the lengths are the same.

It might be that shape B is an enlargement of shape A.

Then question two.

If we fill in the table for shape B, we get this, 12, 8, 4, and 16.

And then it says are shapes A and B congruent, similar, or neither? Well, we know they have the same angles and in the same order, so they're either congruent or similar.

But the lengths are not the same.

The lengths are in the same proportions, and all the lengths in shape B are double the lengths in shape A, so they are similar.

So we could summarise that by saying, shape B is similar to shape A.

They have the same angles, and the lengths in shape B are all double the lengths in shape A.

Shape B is an enlargement of shape A with a scale factor of two.

Great work today.

Now let's summarise what we've learned.

If two shapes are congruent, then they're also similar as well.

However, if two shapes are similar, it doesn't necessarily mean that they are also congruent.

It might be that one is an enlargement of the other by a scale factor other than one, because if the scale factor is one, they are congruent.

Congruent shapes have a scale factor of one.

A ratio table can be helpful as a tool for checking multipliers between lengths and also between shapes as well.

And the multiplicative relationship between sides of the object is preserved in its image with similar and congruent shapes.

Thank you very much for that.

Have a great day.