Lesson video

Hello, everyone and welcome to this lesson in the Transformations Unit.

I am Mr. Gratton and today we will be identifying whether an image is congruent, similar, or neither when compared to its object.

Pause here for a quick look at some of the keywords that you'll be needing for today's lesson.

First up, sometimes it is easier to identify when an image is not congruent to its object.

Let's take a look.

As a recap, the object is what you start on.

The image is a result of a translation, rotation, reflection, or an enlargement, a transformation.

Izzy's mindset of checking if they've performed that transformation correctly is absolutely great.

We should always be critical of how accurately we draw our shapes.

What are some properties of errors that we should be looking out for? Well, when an object is transformed by translation, rotation, or reflection, note enlargement is not included in this, the image will be congruent to its object identical in shape and size if it has been done correctly.

After transforming your object, you should always check for congruence.

If the object and its image are not congruent, then there are clearly some errors somewhere in what you have drawn.

Identifying the error makes it easier to correct your image.

Right, here's our first example of non-congruence.

Jun transforms this object into this image like so.

Pause here to think about or discuss why the object and image are not congruent to each other.

Well, it's pretty simple, an object and an image are not congruent if they are not the same shape.

If you start with a rectangle, then end with a triangle, something has definitely gone wrong.

This image, however, is more likely to be congruent to the original object because they are both rectangles.

However, this is not yet guaranteed until we check for other properties.

Right, for this quick first check, which of these object-image pairs are not congruent to each other? Pause here to choose the correct pair.

The answer is B.

That trapezium became a triangle so they are definitely not congruent.

Next up, what about this? Pause here to identify which object-image pair is not congruent.

The answer is A, whilst both object and image are quadrilaterals, the object is a parallelogram whilst the image is a rectangle with strictly 90 degree angles.

Next example, Izzy transforms this object into this image and claims, "They are congruent because they're both rectangles." But Izzy's comment is incorrect.

Pause here to think about or discuss why.

Even though they are both rectangles, the image is far longer and thinner than its object.

The object is four units by two units.

Whilst the image is six units by one unit.

The object and image that are the same shape are only congruent if all corresponding sides are also the same length.

For this next check, pause here to identify which of these object-image pairs are not congruent to each other.

And this time there may be more than one right answer.

The answers are A and C.

A has a much shorter height on its image compared to its object.

Whilst with C, the type of triangle has changed from a right angle triangle to an isosceles one which instantly makes them non-congruent.

And similarly pause here to identify the non-congruent pair or pairs.

And the answer is all of them.

Pause Here to think about or discuss explanations for why each pair is not congruent.

Next up, Sophia transforms this object into this image.

Sophia claims that, "They are both congruent because they are exactly the same shape." Both rectangles with sides that are proportional to each other.

However, the object and image are not congruent.

Pause here to think about why.

The object has lengths of four units and two units, but the image has lengths of two units and one unit.

Yes, the image is a fractional enlargement of the object.

However, an object and image are never congruent if one is an enlargement of the other, if the size of the image is then different from its object.

Once more, pause here to identify the non-congruent object-image pairs.

The answers are B and C.

Lucas measures the interior angles of this object, then transforms the object into the image and then measures some of the interior angles of the image.

From just the given information, this object and image are not congruent.

Pause here to think about why.

Even though both are parallelograms, the object and image have different interior angles.

An object and image are congruent only if they are the same shape and all of their corresponding interior angles are equal in size.

For this next check, by considering the angles given, pause here to identify the non-congruent object-image pairs.

Pairs B and C are non-congruent.

For C, both these triangles have angles of 45 degrees but they have different side lengths.

If the remaining angles of the image were measured or calculated, they would be different to those on the object.

And for this next check, by considering the angles given and the fact that the image may have been rotated, pause here to identify the non-congruent object-image pairs.

A and C are the non-congruent pairs.

And last up, let's have a quick look at the order of the angles inside a shape.

Alex transforms this object into this image and then measures some of the interior angles in both shapes.

Alex says, "They are congruent because they both have the same interior angles." Even though Alex is correct that both shapes have the exact same interior angles, the object and image are not congruent to each other.

Pause here to think about or discuss why.

Even though both are quadrilaterals with the exact same interior angles, they are not congruent because their angles are in a different order.

But what do I mean by order? Well, you can find the order of the angles inside a shape by choosing one angle to start on, then going either clockwise or anti-clockwise and noting down the next angle that you see.

Keep on going until all interior angles are noted.

If the order of the angles don't have properties that match, then the two shapes are definitely not congruent.

Be aware that if the order of the angles don't seem to match initially, try again but in the opposite direction.

So clockwise rather than anti-clockwise or vice versa.

So for example with this object-image pair, let's start on 104 degrees on the object.

Going clockwise, we then have 76 degrees, 104 degrees and then 76 degrees again.

Whilst on the image, let's also start on the 104 degrees to help make a better comparison with our object.

We then have 104 degrees again, then 76 degrees and then 76 degrees again.

The image has two 104 degrees and two 76 degrees in a row.

Whilst on the object we have an alternation between 104 and 76.

The orders are therefore definitely not the same because the properties differ slightly.

This is still true even if we try to note the order of the angles in the image again, but anticlockwise rather than clockwise.

For this check, let's start with the 135 degree angle and read clockwise.

Pause here to write down the order of the interior angles on both the object and the image.

For the object, starting at 135 degrees we have next 45 degrees, then 90 degrees, then 90 degrees.

However on the image starting at 135 degrees, we then have 90 degrees, then 45 degrees and then 90 degrees again.

Using these order of angles, pause here to choose the correct statement that explains why the object and image are not congruent to each other? The most simple explanation is that the object has two 90 degree angles in a row, but the image does not because it goes 90 degrees, then 45 degrees and then the second 90 degrees.

Great stuff, onto the practise.

For question one, pause here to explain in as much detail as you can how you know each pair of objects and images are not congruent with each other.

And for question two, pause here to identify as many images in each diagram as you can that are not congruent to the object at the top of that diagram.

Mark each non-congruent image with a cross.

Those are some brilliant observations on images that are not congruent to their object.

Onto the answers.

For question 1A, both object A and image A are different shapes.

Object A is a triangle, but image A is a hexagon.

Different number of sides definitely means that they are not congruent.

For 1B, both object B and image B have different angles.

For example, image B has two 90 degree angles whilst object B definitely has none.

All the angles in object and image need to be the same for them to be congruent.

And for 1C, object C has a side of four units whilst image C does not have any sides that measure to four units.

The corresponding length on image C is five units instead.

And for question two, pause here to check your answers with the ones on screen.

Now that we can identify when pairs of shapes are definitely not congruent, how can we justify when pairs of shapes are congruent? Well, let's have a look.

If two shapes are congruent, then all of the corresponding pairs of interior angles must be the same and the interior angles in each shape must be in the same order.

Furthermore, each pair of corresponding sides must be the same length, however they can be in any orientation or may have been reflected.

Remember, if two shapes do not have corresponding angles of the same size or if corresponding sides have different lengths, then they are definitely not congruent.

The lengths of sides of two shapes and the size of the angles of two shapes can be easily checked using rulers and protractors.

However, checking that the order of the angles in two shapes are the same is a little bit more tricky.

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

So for this first shape, we can choose either direction.

I'll choose clockwise.

We have the angles, 100 degrees, 100 degrees, 80 degrees, and then 80 degrees.

For the second shape, we need to start in a way where we want to find possible relationships between the two shapes.

The best way to do this is to identify if there are two adjacent angles that match the same two adjacent angles on the previous shape A.

In this case we have a 100 degrees and an 80 degrees, albeit only if we look at these angles when going anti-clockwise.

When we continue going anti-clockwise, we have 100 degrees and then 80 degrees.

Since we are at the end of the list, we can jump back to the very beginning to place that last 80 degree angle.

The angles are the same but they are not in the same order.

And so the two shapes, A and B, are definitely not congruent.

We can also see this non-congruence by looking at the two 100 degrees in a row in shape A, but no 100 degrees in a row in shape B.

Let's have a look at a different shape, shape C.

We already have the angles and their order for shape A, let's again identify 100 degrees and 80 degrees.

We can go clockwise to match the order of 100 degrees and 80 degrees, but this does mean that we need to go from right to left in the table.

We have 100 degrees and then 80 degrees if we wrap around to the end of the table for shape C.

We can see here that the angles are the same and also in the same order.

This means that they could be congruent, but we still need a little bit more information to know for sure.

So let's say we've already checked and two shapes have angles that are the same and in the same order.

Then the two shapes are congruent if and only if the pairs of corresponding sides are also equal in length.

We can show this through extra rows on our angles table.

This time that show the lengths of the sides on the shapes.

Notice how the length row of the table are shifted to the right compared to the angles row.

This is to show the side is adjacent to a pair of angles directly above it.

For example, this six metre long side is adjacent to both these two 100 degree angles.

Let's look at pairs of angles adjacent to each of these other sides.

Starting with the 11.

5 metre side, it is adjacent to 100 degrees and 80 degrees and so on.

And notice the 11.

5 metres is adjacent to the 80 degrees above it and the 100 degrees right at the very beginning of the table as we wrap around that end result.

Let's now compare these lengths to the possibly congruent shape C.

Pause here to try this for yourself.

Which side lengths go by which pairs of adjacent angles? And here are the results.

As Izzy says, "The lengths of each side of both shapes match exactly to their adjacent angles." This means shapes A and C are definitely congruent to each other.

Here we have a partially completed table for this quadrilateral.

Pause here to find the value of X.

X is adjacent to the 87 and 74 degree angles.

The side adjacent to these two angles on the actual shape has a length of 27 metres.

Here we have two tables showing the angles and their adjacent sides.

Pause here to finish the sentence, the shapes A and B are.

Either congruent, non-congruent, or not enough information.

The two shapes definitely are congruent as the angles and sides match up completely.

For this check, angles of these two shapes are both the same and in the same order.

From the two shapes, pause now to write down the values of A to H if possible, and not all of them will be possible.

B has a value of 21 metres whilst both C and F have a value of 38 metres.

Furthermore, pause here to identify the correct statements.

Even though both shapes have the same angles in the same order, they are not congruent.

This is because for them to be congruent, all pairs of corresponding sides, sides with the same pair of adjacent angles, must be the same length.

If even one pair of corresponding sides are not equal in size, then nope, they are definitely not congruent.

Brilliant onto some independent practise.

For question one, pause here to complete the table and explain whether shapes A and B are congruent or not or whether you just can't tell with the information given.

And for question two, pause here to practise the same again, but this time for these two pentagons.

And for question three, pause here to explain whether from the information given, you can tell if E and F are congruent or not.

Amazing, onto the answers.

For question one, these two shapes are definitely not congruent.

The pair of corresponding sides adjacent to the angles of 24 degrees and 131 degrees are not equal in length.

For question two, it is simply not possible to tell if these two shapes are congruent or not.

As no pair of corresponding sides have their lengths given.

We simply do not know enough information to disprove congruence and we need the lengths of all of the sides in order to prove it.

So at the moment we just cannot tell.

And finally, question three, these two triangles are congruent.

Both have the same angles in the same order and the pair of corresponding sides adjacent to the angles 62 degrees and 35 degrees are equal in length.

As this is a triangle, this is enough to confirm congruence by ASA.

We now have a formal method for identifying congruence after a translation, rotation or reflection.

But what about identifying similarity after an enlargement? Let's have a look.

If two shapes are similar, then all the corresponding pairs of interior angles are the same and the interior angles in each shape must be in the same order and they can be in any orientation or they may have been reflected.

And notably, each pair of corresponding sides must follow the same multiplicative relationship.

This is an example of a congruent image and this a similar image.

But what does each pair of corresponding sides follows the same multiplicative relationship actually mean? Well on the object we've got a length of four units, and on the image 12 units, the multiplicative relationship from four to 12 is a multiplied by three.

On the object, we've got a two unit length and on the image a six unit length, the multiplicative relationship between two and six is also a times by three.

These two rectangles are similar as the multiplicative relationship between each pair of corresponding sides is a multiply by three.

A similar table to the one that we used to show congruence can be used to check that the shape has the same angles in the same order and that the multiplicative relationship between each pair of corresponding sides is the same.

So for this, shape A going clockwise, we have 50 degrees, 135 degrees, 85 degrees and 90 degrees, and that these are the sides adjacent to the pair of angles above it on the table.

For shape B, going anticlockwise this time, we have the same angles in the same order, but this time we have these side lengths adjacent to the pair of angles above it on the table for shape B.

Let's compare corresponding side lengths using the adjacent angles to identify which pairs are corresponding.

This pair is related by a scale factor of three, as is this pair and this pair and this pair.

All the angles are the same and in the same order even if B is reflected.

And all pairs of corresponding sides have the same multiplicative relationship of a multiplied by three.

Therefore we have shown for certain that A and B are similar.

In fact, shape B has lengths that are three times the size of the lengths of shape A.

On the other hand, if even one pair of corresponding sides does not share the same multiplicative relationship as the others, then the two shapes are definitely not similar to each other.

Notice how here in this incorrect example we have two different multipliers.

From 14 to 49 the multiplier is 3.

5 and from 20 to 60 the multiplier is 3.

Since these two multipliers are different, there is no consistent scale factor.

These two shapes are not similar.

And furthermore, in the special case where the multiplicative relationships between all pairs of corresponding sides is a multiplied by one, then the two shapes are congruent.

Congruence is just a specific form of similarity.

For this check, pause here to identify how do you know that these two shapes are not similar to each other? Quite simply, the angles are not the same.

However, for these two shapes, the angles are the same and in the same order.

Pause here to find the values of A to D in this table.

To complete this table, we look at the side lengths for shape B.

We have the answers of 40, 8, 4 and 48 adjacent to the two angles above it on the table.

Using this table, pause here to confirm whether these two shapes are similar or not.

If they are similar, state the multiplicative relationship between each pair of corresponding sides.

They are similar to each other.

Each pair of corresponding sides has the multiplicative relationship of a multiply by four to get from shape A to shape B.

Brilliant, onto the final practise task.

For question one, complete as much of the table as possible and use the information gained from the table to identify if the two shapes are similar, congruent, or neither.

Pause now for question one.

And finally for question two, get ready to deal with a lot of information by creating and completing a table of angles and adjacent sides for each of these four shapes.

State where the pairs of shapes are congruent, similar, or neither.

Pause now to do this.

Great effort on this practise task.

For the answers to question one, pause here to check the information on screen and know that these two shapes are similar with a multiplicative relationship of a multiply by five.

But because they have a multiplicative relationship of a multiplied by five, they are not congruent as this multiplicative relationship is not a multiplied by one.

And for question two, pause here to compare your completed tables of information to the ones on screen.

And know that C and D are not similar to each other.

D and E are congruent.

And C and F are similar but not congruent.

Amazing effort, everyone, in analysing all of those shapes in a lesson where we have identified if two shapes are congruent by comparing angles, the order of the angles and checking that corresponding sides are equal in length.

Sometimes it is easier to spot non-congruence by identifying a congruence property, a pair of shapes do not follow.

We've also looked at similarity where we have compared angles, the order of angles, where all pairs of corresponding sides must also share the same multiplicative relationship.

That is all from me, Mr. Gratton, for today.

Thank you all so much for joining me and until next time, take care and good bye.