Lesson video

Hello, I'm Mrs. Lashley and I'm gonna be working with you as we go through the lesson today.

I really hope you're ready to try your best as we go through the lesson.

So in today's lesson, we're going to be looking at congruent triangles and being able to apply the criteria of SSS to prove that two triangles are congruent.

Congruent is a word that you'll have been familiar with from previous learning, such as transformations.

You may wish to read the definition for congruent to make sure you're familiar with that before we get going.

Today's lesson has got two learning cycles, the first part is guaranteeing congruence.

So how much information do you need to know in order to be able to say that two shapes are congruent? And the second learning cycle is justifying congruence of two triangles by using SSS.

So let's make a start on the first learning cycle.

So Jacob has got four sticks of different lengths, and he's labelled them A, B, C, and D.

And that's gonna help us to identify which one's which.

By placing them end to end, he can create and form a quadrilateral.

And you can see the quadrilateral that he's formed.

So you can see stick A, and then stick B, and then stick C, and then stick D going anti-clockwise, or stick A, then D, then C, then B, going clockwise.

Andeep has the same four sticks as Jacob, so they're exactly the same, they're the same length.

And Jacob says, "Because your sticks are the same lengths as mine, whatever you do, you'll create the same quadrilateral.

So they will be congruent quadrilaterals." Do you agree with Jacob? Do you think that's correct that when you've got exactly the same copies, that when we place them end to end, the only quadrilateral will be the one that Jacob created? Well, Andeep doesn't agree.

Andeep says, "I'm not sure that the edges being the same is enough to guarantee congruence.

So let's try making some." So we are trying to work out how much information guarantees congruence.

So Andrew says, "I've made this quadrilateral.

It isn't congruent to Jacob's." So he said to Jacob, "It isn't congruent to yours.

Having the same length is not enough to guarantee congruence." So have a look at the two quadrilaterals, the one that Jacob made originally, the one that Andeep has made with the same four sticks.

How do we know they're not congruent? But Jacob has recognised what's gone on, so he says, "The four side lamps are the same because they're using the same sticks, but they're not in the same order." Jacob is going anticlockwise around his quadrilateral, he's got stick A, then stick B, then stick C, then stick D.

Whereas Andeep is A, then C, then B, then D.

So they are the same four sticks, but they've been placed together in a different order and therefore creating a different quadrilateral.

So Andeep says we need the same sides, and for them to be in the same order.

So we're trying to work out how much information is necessary to be able to guarantee that two shapes are congruent, where they would fit exactly on top of each other, whether they're rotated or reflected or just translated.

So Andeep says, "If the order is different, then they're definitely not congruent." So that's a really quick way of being able to say two quadrilaterals are not congruent if their edges are not in the same order.

So here's a quick check on that, what other order can the sticks be placed to form a quadrilateral that is guaranteed to not be congruent to the other two? So pause the video whilst you think about that.

And then when you're ready to check, press play.

So we needed to come up with a different order for those four sticks from what we already had, we've had A, B, C, D, we've had A, C, B, D.

So if you did it A, B, D, then C, then that is a different order.

Jacob is gonna continue with this idea.

Okay, so we're trying to make a congruent quadrilateral to his.

So we now know that the four sticks be in the same length is not sufficient.

For them to have a chance of being congruent, they need to be in the same order.

So he says, "Make yours congruent by joining A to B, then join C to B, then use D to join it up." 'Cause then the order will be the same.

So A to B, B to C, and then C to D.

So Andeep asked, "Well, I've made this.

Is this congruent?" "It might be." Andeep join them up clockwise rather than anti-clockwise, but they are still the same lengths in the same order.

So being the same lengths in the same order, does that guarantee congruence, Andeep thinks he's found an example to show that, that isn't enough to guarantee the congruence between these two quadrilaterals.

What changed? Well, the order didn't change, A, B, C, D.

Clockwise, it reads A, B, C, D, which is the same order as Jacob's, A, B, C, D in anti-clockwise, but they're quite clearly not congruent.

I think you'd agree with Jacob.

So they have the same lengths in the same order.

But what makes these different to each other? It's the angles that are different.

That angle made by B and C is a reflex angle in Andeep's, whereas on Jacobs it's an obtuse angle.

So it's quite clear there that these are very different.

So a check, from what we've just seen using Andeep and Jacob's examples, which of these properties are necessary when proving two quadrilaterals are congruent? So it's not sufficient, but they must be there for them to have any chance of being congruent.

Pause the video whilst you decide, and then when you're ready to check, press play.

So the order of the edges needs to be the same if they have any chance of being congruent.

So that's necessary.

The edge lengths need to be the same, the corresponding edge lengths need to be the same if there is a chance of them being congruent.

Again that's necessary to be able to prove that they are congruent, and their interior angles need to be the same 'cause we just saw when you didn't have interior angles the same, then they couldn't be congruent.

So for there to be any chance of two quadrilaterals being congruent, then those three things must exist.

The orientation doesn't matter.

So for two things to be congruent, they need to be exactly the same size and shape in the sense of if you were to cut them out, flip them over, rotate them, could you stack them neatly on top of each other? So the orientation doesn't matter.

So as I was just saying, to be congruent, two shapes need to be able to fit exactly on top of each other.

You can use rotation, reflection or translation in order to make that happen.

So their angles and their edges will need to be the same size and in the same relative position within the shape, and that's to do with the order.

So on the screen we've got three quadrilaterals.

The square and the rectangle have got the same angles, but they are different shapes.

So you can see them there, by definition of being a square, by definition of being a rectangle, they've got four 90 degree angles, but their edge lengths are different.

So we can see quite clearly that just having the same angles is not sufficient to say that they are congruent.

The rectangle and the parallelogram have the same lengths in the same order, but they have different angles.

So once again, we've got the same edge lengths, they're in the same order, we go from a three centimetre edge to a six centimetre edge to a three centimetre edge back to a six centimetre edge.

We do that both on the rectangle and the parallelogram, but that's still not enough to say that these are congruent because the angles are different.

So none of these three quadrilaterals are congruent to each other.

They've got properties that overlap, like the square and rectangle, all of their angles being right angles, and the rectangle and the parallelogram or their edges be in the same, in the same order, but there isn't enough of their properties that are the same to make them congruent.

This is another example of two quadrilaterals.

So both of these quadrilaterals, they've got two edges of five centimetres and 8.

1 centimetres.

They both have two angles of 97.

1 degrees.

Are they congruent? Have a look at them.

Would you say that if you was to cut them out, they would fit exactly on top of each other? I think it's very clear that they are not congruent.

The left hand side quadrilateral is a kite, and the right hand side quadrilateral is a parallelogram.

But there are features again that are the same, so no, the other two angles are not the same, so all four interior angles, the corresponding angles do not match, and the edges are also in different positions, so the order is different.

So how do we be sure that two quadrilaterals are congruent? So how do we guarantee that they are congruent? Well, the corresponding edges need to be the same.

The order of the edges need to be the same.

So we just saw the kite and the parallelogram, they might have the same edges, but they weren't in the same order.

So both of those things need to apply.

The interior angles of the quadrilaterals need to be the same, and the angles need to be in the same location.

So if you only know one or two of these, that is not sufficient to guarantee or to prove that two quadrilaterals are congruent.

Jacob wants to know, "But is this necessary for triangles?" So do you need to know all four of the bullet points above for triangles or is this just for quadrilaterals? Andeep says, "Well, the order of the edges is always the same." So that second bullet point says, for a quadrilateral, you need to know that the order of the edges were the same and we saw with the sticks that there were different orders that you could put them in.

But Andeep suggesting with a triangle there is only one order that edges can be in.

Is that true? Andeep is sure of himself.

He says "Yes, if you've got three edges, A, B, and C, then swap two of the edges.

And this order is still the same.

So here you've got a triangle made out of three sticks, A, B, and C.

So if we swap B and C, I can still go around the triangle edges in the order A, B, C.

I'm just going anti-clockwise now rather than clockwise.

So on the original triangle, clockwise you went A, B, C.

On this new one where we've swapped the edges B and C, we are going around anti-clockwise, but we still get A, B, and C.

Okay, if we swap A and C, this is the result without rotating or twisting the edges, just swapping them over, then we can still read A, B, C.

It's anti-clockwise, if you start on the stick A, you can then go to B and then go to C.

And finally, if you swap A and B around, then you still can read A, B, C going anti-clockwise, and Jacob has recognised that they are always next to the other two sides.

So because there is only three of them, wherever you place one of the sides, it has to be joined to the other two.

So are these two triangles congruent? We know that it's not about the order of the edges because there is only one order when you've got three edges.

For a quadrilateral, we need it to have the same edge lengths, the same order of edges, the same angles in the same places.

So we've already recognised that the order of the edges is not necessary for proving congruence between two triangles.

So let's check about angles.

So here, 60 degree, using the protractor, 60 degree, we could calculate what this should be, but we can measure it as well, another 60 degree.

So that left hand triangle has got three 60 degree angles.

The right hand triangle, it's got a 60 degree, another 60 degree, and another 60 degree.

So what we can see are that both of these are equilateral triangles.

They've got three equal interior angles of 60 degrees.

So does that mean these two triangles are congruent? If they're both equilateral, if they've both got three 60 degree angles, does that mean that they are congruent to each other? It means they are similar because they've got the same interior angles than they are similar to each other.

So we need to check whether their edges are the same length in order to see if they are congruent.

So this one has got an edge of 6.

5 centimetres using the ruler, and this one has got an edge of 6.

3 centimetres.

So their edge lengths are different, so they are not congruent because if we were to cut these out, they wouldn't fit exactly on top of each other.

The 6.

5 centimetre edge equilateral triangle would be overlapping the 6.

3 centimetre.

It's a slightly smaller sized equilateral triangle.

So it's true to say that all equilateral triangles are similar to each other because they will all have 60 degree angles, but they're not necessarily congruent to each other.

Here we've got two triangles that look to be scaling from, just looking at them.

So we need to check that their angles are the same and their edges are the same in order to be sure that they are congruent.

So using our ruler again, we can measure each of the three edges.

So A, B, C, the triangle ABC has those three edge lengths.

And PQR has the same three edge lengths.

So, so far we are on track to be able to say that they are congruent.

If one of those edges were different, then automatically we'd say that they're not congruent.

But so far they have got three edges that match up the corresponding edges.

So AC equals PR, they're both the 7.

2 centimetre edge.

AB equals PQ, they're the 3.

9 centimetre edges, and BC equals QR, they're the 6.

5 centimetre edges.

So we've now got to check that their angles are the same.

So this one reads 85, and this one reads 33.

This one reads 85, and this one reads 33.

I don't need to measure the third angle because we can calculate what the third angle is Using angles in a triangle add up to 180 degrees.

So the angles between the corresponding equal sides are the same.

So if you look on triangle ABC, the angle ABC is 85 degrees and that angle is created by the 3.

9 centimetre edge length and the 6.

5 centimetres.

And angle PQR is in the same relative position on that triangle, so they are corresponding angles.

So the answer is yes, these are congruent, their edge lengths and their interior angles are the same.

We're not worrying about the order of the edges because there is only one order.

So, check the angles are the same, therefore the triangles are congruent.

Is Lucas correct? Pause the video whilst you make your choice on that, and then when you're ready to check, press play.

So unfortunately Lucas has made a mistake here.

So when the angles are the same, then we can say that the triangles are similar.

We need to know the edge lengths to be able to tell if they are congruent.

And think back to that equilateral triangle example, they both had 60 degree angles, but their edge lengths were different.

Izzy says if two triangles have the same angles, that means they are similar.

So that's what we just saw on that last check.

And if the corresponding sides and angles are the same, then the two triangles are congruent.

So if we only know angles, we can only suggest that they're similar, and if we know angles and sides and the corresponding ones are the same, then we can say they are congruent.

We are now gonna think, "Well, is that necessary to know all that information to be sure to guarantee the two triangles are congruent?" So we're gonna take a look.

So with two side lengths connected at a vertex, that creates an angle, and you can see three examples of three different angles created with the same two side lengths.

By connecting the ends of those line segments, we actually have drawn three different triangles.

So the triangles have got a common feature of having an edge of 3.

1 and an edge of 4.

8.

But that's all that they have in common because the third edge is a different length and the angle is different.

So can two triangles with the same three sides have different interior angles? So here are three circles, and the radius of each is the fixed length of an edge.

So when you construct a triangle using three edge lengths, you actually use your pair of compasses to be able to draw the arcs that have the radius of each edge length.

So that's what's here, we've got the three circles that when we put them together in a particular way will construct a triangle.

The triangle it will construct here is the triangle with an edge of 3.

2, 2.

1, and 3.

9.

So the lengths of the radii.

So if we make two of the circles centred at the same point, then an angle is formed with those two edges, and that's like what we saw previously where we have three different triangles created from the same two edge lengths.

And as the angle changes, the third length will also change.

So if I just rotate my radius, I've created a larger angle between those two edges.

Here and here.

So to create a triangle with the third edge of that last circle, it needs to be on one of the ends of the radius, and the other radius needs to be on the circumference of the circle.

That sounds quite complicated, but basically here we've got, by having only two radii centred at that same point, we create a corner, if you like, of the triangle.

What we now need to do is find how we get a third edge, which is connecting the two radii together that's a specific length.

And how we do that is we centre that circle on one of the radii, and we need the other radius to be on the circumference.

So if we move that circle so that the centre is on the end of a radius, have chosen that smallest one, 2.

1, and at the current angle that is there does not form the triangle we are looking for because the radius, the 3.

2 centimetre radius is not on the circumference of the 3.

9 centimetre circle.

But where would that be? Well, if we remove that circle, just to clear our diagram a little bit.

So we've got our edge fixed at 2.

1 centimetres in that orientation, and the radius of 3.

2 centimetres can still sort of move around to create any angle.

But what we want to do is make sure that when we connect the end of that radius 3.

2 to the radius one of 2.

1, it will be the length 3.

9.

And this point here is a point where it is on the circumference of the 3.

9 centimetre.

So when joined to the centre it will be 3.

9 centimetres.

And it's also a point that's on the circumference of the 3.

2 centimetre circle.

So when connected to the centre of that circle, the length will be 3.

2.

And so this angle here is open enough that it joins those two points.

And when we draw the radius on, we can see this triangle.

So it's got the three edges we want, and that angle is there between 2.

1 and 3.

2 centimetres.

But hopefully there was somewhere else on the circles that you thought it could have joined up.

And that's on the other side, so there's actually two points of intersection.

These two triangles are congruent to each other, they are just reflections of each other.

On the screen there is a link to a GeoGebra file, which you could pause the video and click on that link and explore what we've just gone through, being able to change the different side lengths and to see what we've just seen through animation.

So can two triangles with the same three sides have different interior angles? That was the question we were trying to investigate.

And so no, the side lengths fix the size of the angles.

When we only had two of the side lengths, then the angle could be any size whatsoever and the third length changed dependent on it.

But if that third edge needed to be a specific length, we saw there was only two places it could go and they were actually congruent because they were reflections.

So only one triangle can be formed from three edges.

So the two triangles with the same three edges will be congruent is what Izzy wants to just check that she's following along.

Yes, if we know the edges are the same, then we know they are congruent.

Do we need to know that all the interior angles are the same and all the corresponding edges are the same? The answer is no.

If you know that corresponding edges are the same in two different triangles, then you are guaranteed that those two triangles are congruent because once you have three edge lengths, it fixes the shape of that triangle, it fixes the interior angles.

And this condition for congruence is called SSS for short, which stands for side, side, side.

So if you have two triangles where you know that the corresponding edges are the same, then you can say that they are definitely congruent by SSS.

Here's a check for you then, so which of these is not congruent to the other two? Pause the video whilst you have a go, it may be that you want to sketch them out because there are no diagrams of triangles here.

So you may wanna sketch the triangles to try and think about which one is different to the other two.

Press play when you're ready to check.

It was C.

And that was because if you look at A and B, it had the same three edge lengths.

It didn't matter which one was where because remember the order is the same, it just may be going anticlockwise or clockwise, it might be a reflection or a rotation, but their edges are the same.

Whereas if you look on C, it's an isosceles with a 10 centimetre or three centimetre and a 10 centimetre.

So you've got a task now on guaranteeing congruence.

So question one, "These two quadrilaterals are congruent.

Fill in the blanks." So if you know that these two copies are exactly the same, then remember all the things you need for congruence to fill in the blank.

Pause the video whilst you have a go at question one, and then when you're ready for the next question, press play.

Okay.

Question two has got four parts to it, you need to give an example of quadrilaterals that show that these conditions are not enough to prove congruence.

So you are looking for a counter example, an example that is gonna disprove that just knowing, for example A, just knowing that the angles are the same in two quadrilaterals does not prove congruence.

Pause the video whilst you're thinking about that, and then when you're ready for question three, press play.

So question three, there are three constructed triangles.

So the construction has been done for you.

You can see the arc marks there, they would've come from using pairs of compasses.

Part A is using a protractor, you need to measure the three angles in each of the triangles.

And part B, what do you notice about what you've just measured? Press pause, and then when you press play, we will be going through the answers to questions one, two, and three.

Question one was filling in the blanks.

So this was about looking for corresponding edges and corresponding angles because if they are congruent, which we know they are 'cause you were told that they are, they have the same edges, as in the lengths will be the same on both of the quadrilaterals, they need to be in the same order.

The angles, the interior angles will be the same, and again, they will be in the same relative position.

So if the 83 degree is between the 4.

5 centimetre and the 3.

6 centimetres on one of them, then it's in the same position on the other.

Question two, remember there was four parts and you were looking for examples or counter examples to show why the condition angles are the same is not sufficient, so is not enough information to prove congruence.

So an example, you may have come up with your own examples, but an example for showing this is that isosceles, trapezium and parallelograms can have the same angles but are not congruent.

So an isosceles trapezium would have two 60 degree angles and two 120 degree angles, or this particular one does, and that parallelogram also has two 60 degree and two 120 degree angles.

So they have the same interior angles, but they're clearly not congruent.

For B, you needed to show that the angles were the same and in the same order was still not enough information.

So squares and rectangles is a good one to show that because they both have four 90 degree angles, therefore they're in the same order, but they're different.

Or you may have done parallelograms and rhombi, they can have the same angles in the same order, but they're not congruent.

Part C was to show that edges being the same is not sufficient.

And again, squares and rhombi was a good example because they are both shapes or quadrilaterals, which have this, they're both quadrilaterals where their edges are all equal.

So they could could have exactly the same edge length, but their angles are different, they are different shapes and not congruent.

Similarly, you could have used kites and parallelograms, they both have two pairs of equal edges.

They're not congruent, they're not the same shape.

And lastly, we needed to show a counter example or some quadrilaterals, which show that just having edges of the same length in the same order is not enough information to prove congruence in quadrilaterals.

And a parallelogram, and a rectangle works for this because opposite edges are equal in both of those and it's goes short length, long length, short length, long length, all the way round and clockwise or clockwise, but they're not the same shape.

Question three, you needed to measure the three interior angles of the constructed triangles.

You've used a protractor, so you're not going to be able to measure to the same degree of accuracy that's on the screen to one decimal place.

You'll be measuring to the nearest degree, but it should be about 41 degrees, 61 degrees, and 78, 79 degrees.

What do you notice? Well, they are the same in each construction.

So this really highlights that if the three triangles have just been constructed using a different base each time, their lengths are the same, their angles are the same, that rotation doesn't affect congruence.

So now we're gonna look at justifying congruence by using SSS, so side, side, side.

So we can be sure that two triangles are congruent if we know that the three edges are the same on both triangles.

The orientation and the sense does not need to be the same.

So if one is reflected, if one is rotated, that doesn't mean they are not congruent, as long as the corresponding edges are equal.

So a check using the fact that these two triangles are congruent, fill in the blanks.

Press pause whilst you have a go at that, and then when you're ready to check, press play.

So you know that these are congruent, which means that the three edges will be the same.

Between the two diagrams, we can work out what those three edges are because on the left hand triangle, you've got an 8.

1 centimetre and a 3.

6 centimetre.

And on the right hand triangle you've got a 3.

6 centimetre and a seven centimetre, which means that the three edges are 8.

1, 3.

6, and seven.

So the missing edge on the left hand one is seven centimetres, and the missing edge on the right hand triangle is 8.

1 centimetres.

So we've got a rectangle with one of its diagonals marked.

Are the two triangles, ABC, and ADC congruent? Andeep suggests that we could measure the three edges with a ruler.

So if we know that the three edges on both of the triangles match that they're equal, then we can say that they are congruent by SSS.

So Andeep suggests we can do that by measuring them.

Laura says, "Well, I haven't actually got a ruler" and it wouldn't be completely accurate if she did, A ruler might measure to the nearest millimetre and they might not be an exact millimetre, but she thinks she can see a way to show that they are congruent even without a ruler.

Have you got any thoughts on how she's about to do this? Using the fact that it's a rectangle, we know that AD is equal to BC, opposite edges in a rectangle are equal.

And similarly AB is equal to DC.

There are different length, but they are equal to each other.

So we're using hash mark, one hash mark for one set and two hash marks for the other set.

AC, so that's the diagonal is an edge on both of the triangles, it's a common edge or a shared edge to both triangles.

So therefore ADC and ABC have the same three edges.

So they are congruent by SSS.

Remember, SSS is the acronym that means side, side, side.

Andeep realises that Laura has managed to do that without the ruler because she's just used properties of the shape.

So here's a check, prove the triangle ABC and triangle ACD are congruent in this kite.

So if you're not sure about the properties of a kite, ask someone close to you, pause the video, search for it because they are the properties that are gonna help you do this check.

When you think you finished the check, press play and we'll go through it.

As it's a kite, AB equals AD adjacent edges are equal in a kite.

And that is also true for BC and DC.

And we've used the hash marks again to indicate that.

AC is an edge on both of the triangles, so it's a common edge, it's a shared edge, therefore triangle ABC, and triangle ADC have the same three edges, so they're congruent by SSS.

And so once are your task for this part about justifying congruence by SSS.

Question one needs you to measure the sides of the pairs of triangles to decide if they are congruent or not.

So you will need a ruler to measure those sides.

Pause the video watch you're doing in question one, and then when you're ready for question two, press play.

So here's question two, by using properties of shapes, identify the triangles that are congruent by SSS.

So in part A, it's a parallelogram with the diagonal drawn, on part B, it's a symmetrical Pentagon with two diagonals drawn.

And for part C, it's a symmetrical hexagon with three diagonals drawn on.

So you need to decide which triangles within each of those shapes are congruent to each other.

Pause the video whilst you're doing that, and then when you're ready for the last question of the task, press play.

So question three, these triangles are congruent.

Using this fact, add to the edge length marked X.

So which edge would be X? Pause the video whilst you decide on that edge, when you're ready for the answers, press play.

So question one, A and B, the first pair were congruent and the second pair were not.

So you needed to measure the edges, and as soon as you found that the three edges on one were not equal to the three edges on the other means they cannot be congruent.

Question two, you needed to identify the congruent triangles within each of these shapes by using the properties of shapes.

Some of the properties were labelled using the hash marks.

So on the parallelogram, triangle ABC and triangle ADC are congruent.

The triangle symbol means triangle, and the symbol between them means congruent two.

So it's a shorthand way of saying that triangle ABC is congruent to triangle ADC.

On part B, we can see with the symmetry here that TP equals TS.

We've got the hash marks, and PQ equals SR, we've got the double hash marks.

TQ and TR will be equal in length because of the symmetry.

And so triangle TPQ is congruent to triangle TSR.

And lastly, there were two pairs of congruent triangles in part C, IJK was congruent to MLK, and IMK was congruent to MNK.

Question three, 6.

1 centimetres is the value of X, and you should have been able to get to that by knowing the angles in a triangle add up to 180.

Therefore, on the right hand triangle, the missing angle was 80 degrees, and therefore the 80 degrees is opposite the 6.

1 centimetre.

So the edge length needs to be in the correct position.

So to summarise today's lesson, two triangles can be proved to be congruent if you know the three side lengths are the same, the corresponding angle pairs are the same in congruent shapes.

Well done today and I look forward to working with you again in the future.