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 today's lesson is about being able to identify when triangles are congruent and prove them using the different criteria.

The definition of two shapes being congruent is on the screen.

You may wish to pause and read that before we make a start.

Today's lesson has got three learning cycles.

The first learning cycle is about choosing the correct criteria.

The second learning cycle is about being able to tell if they are congruent or not.

And the last one is to look at hidden congruence.

So problems where maybe it's not so clear that there are triangles that could be congruent.

So we're gonna make a start at choosing the correct criteria.

We're gonna make a start with just reminding ourselves about the different criteria that we could choose.

So two triangles can be proved to be congruent if certain criteria can be applied, and there are four different cases.

So here is an example of the first one, which is called SSS.

So if you know, the edges of two triangles are the same.

So in this case, we've got a 3.

6 centimetre edge, a 6.

5 centimetre edge, and a 4.

9 centimetre edge, and so does the other triangle, then they will definitely be congruent.

It doesn't matter if they're rotations or reflections.

The order will be the same in both.

You just might need to go anti-clockwise rather than clockwise.

And so these three edges will fix the triangle.

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

The next criteria is SAS, so side-angle-side.

The most important part here is that the angle is between your two corresponding sides.

So you need to know that two corresponding sides are equal and the angle between those edges is also the same.

If you can prove that, if you can see that, if you know that using properties, then you can say that two triangles are congruent by SAS.

The next two slides show the same criteria.

We just put the letters in a slightly different order depending on the scenario, but ultimately, they are equivalent to each other.

So this first one is ASA.

If you know two corresponding angles and the side edge between them, then you would say they are congruent by ASA, whereas the equivalent one is AAs, and this is where you have two corresponding angles and another edge.

But the other edge is not between the angles.

The reason these are equivalent is because you have two angles, you could calculate the third angle of the triangle, and this would allow you to say ASA instead or AAS if you were given with an ASA style question.

So two corresponding angles and a corresponding edge is sufficient information to be able to say that two triangles are congruent.

The last criteria is a special criteria for right-angled triangles only.

So this one is RHS.

It can only be applied if you know that both triangles are right angled.

So there is a right angle.

If their corresponding hypotenuse where the hypotenuse is the longest edge opposite, the right angle is the same.

And if you know one of the other two sides to be corresponding and equal, then you can use RHS.

So a check, which of these are not a criteria for proving congruence of triangles? So you've got seven options there.

So pause the video, read through them, and then when you're ready to check, press play.

So AAA, angle-angle-angle is not a criteria for proving congruence.

It does prove similarity but it doesn't prove congruence.

And then the other, one side-side-angle is not sufficient to prove congruence.

And the reason for that is because at times, two different triangles can be formed with that information.

So it doesn't guarantee or fix the triangle.

So if we think about choosing the correct criteria, so here are two triangles that are congruent, and which of the criteria can we use to prove this? So down the side, we've got SSS, SAS, ASA, but do remember that ASA is the same as AAS and RHS.

So when we find ourselves looking at two triangles, then we need to sort of go through a process to eliminate and to figure out which of our criteria we can use.

So firstly, do we know all three edge lengths? No.

So we're not gonna to use SSS to prove that these two triangles are congruent.

For SAS, we need to ask ourselves, do we know two corresponding edges and the angle between them? Well, we know two corresponding edges, but we don't know the angle between them, so no.

So we won't be using SAS as our criteria for this proof.

Do we know two corresponding angles and one corresponding side? Remember, it doesn't need to be the side in between the angles, it just needs to be corresponding side.

And the answer to that question for these triangles is also no.

If it has a right angle, do we know the hypotenuse and another corresponding side? Well, it does have a right angle.

We do know the hypotenuse on both of them are the same.

And do we have another corresponding side? Yes.

So for this one, we can prove that these two triangles are congruent by RHS, right angle, hypotenuse, and side.

Let's go through another example of that.

So here, we, again, we've got triangle ABC and triangle DEF.

We know they are congruent, but which criteria proves it? So do we know all three edge lengths? No.

So it's not SSS.

Do we know two corresponding edges and the angle between them? We only know one edge length, so no.

Do we know two corresponding angles and one corresponding side? Yes.

Check the location of your two angles.

99 degrees and 13 degrees are at either end of the 4 centimetre edge on ABC, and they're on either end of the 4 centimetre edge on triangle DEF.

So they are corresponding.

If it has a right angle, do we know the hypotenuse and another corresponding side? No.

It's not a right angle triangle.

We know it cannot be a right angle triangle even without the third angle, and that's because there is a 99 degree angle.

In a right angle triangle, the 90 degree angle, the right angle is the largest angle.

So these two triangles can be said to be congruent by ASA or AAS.

So let's look at these two triangles.

We've got triangle ABC and we've got triangle MON.

Do we know all three edge lengths? No.

Do we know two corresponding edges and the angle between them? Yes.

Do we know two corresponding angles and one corresponding side? Yes, we know that too.

The 12 centimetres is on both with the 27 degree angle and the right angle on either end of it.

And so they are corresponding, two corresponding angles and a corresponding side.

If it has a right angle, do we know the hypotenuse and another corresponding side? Yes.

So these two triangles have sufficient information to show that these are congruent by SAS or ASA or RHS.

So there are gonna be times where actually, you have enough information to choose which of your criteria to use.

So here's a check.

Izzy says, "These two triangles are congruent by AAS." Is Izzy correct? Pause the video, make your decision, think about why she is or she isn't.

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

So, no.

Although there are two angles and a side known on both of the triangles, the sides are not corresponding.

So if you look at the position of the 7.

5 centimetres, it's opposite the 52 degree angle on one of the triangles, and it's opposite the 91 degree angle on the other.

So they're not corresponding sides.

So it's really important that we've got two corresponding angles and a corresponding side.

So let's continue with thinking about choosing the correct criteria.

But this time, these two triangles, although are congruent are, it's less obvious that they are congruent.

So which of the criteria can we use to prove this using only the given information on the diagrams. So do we have all three edges? No.

Do we know two corresponding edges and the angle between them? No.

Do we know two corresponding angles and one corresponding side? No.

If it has a right angle, do we know the hypotenuse and another corresponding side? No.

So from the information that is just given to us on the diagrams, none of our four criteria hold for the given information, and this is why it's less obvious that they are congruent.

So from that information, what else can we calculate? Well, we can calculate the third angle in both triangles.

So because angles in triangles sum to 180 degrees, we can add them, add the two we know, subtract it from 180 to find the third angle.

So now, using all the information, including this third angle that we just calculated, let's run through the questions again.

Do we know all three edge lengths? We still don't know all three edge lengths, so that's a no.

So we're not gonna use SSS.

Do we know two corresponding edges and the angle between them? No, we don't have two corresponding edges.

We've got the 39 centimetres on both triangles, but we have a 36 centimetres on one and a 15 centimetres on the other.

So we don't have two corresponding edges.

Do we know two corresponding angles and one corresponding side? Yes.

If it has a right angle, do we know the hypotenuse and another corresponding side? No.

So it is, it does have a right angle.

We do have the hypotenuse, but once again, the fact that the 36 centimetres is on one and the 15 is on the other, we do not have another corresponding side.

But we can prove that these are congruent by ASA.

Using the 39 centimetre side and the two angles of 63 degrees and 27 degrees, we can prove that these are congruent.

Here's a check for you.

These two triangles are congruent.

Which of these criteria can we use to prove it? So pause the video and then when you are ready to check your answer, press play.

Well, we can use B, C, or D.

We do not have three edge lengths, so we can't use SSS.

We can use SAS because we can calculate the third angle in both triangles.

And then we know that there's a 38 degrees between the 9 centimetres and the 7.

1 centimetres.

We can use AAS because we, again, can calculate the third angle and we can use RHS because if we calculate the third angle on the left hand triangle, we see that it is a right angle to triangle.

So task A, which criteria proves each pair to be congruent to each other? So there are four pairs, A, B, C, and D.

You need to just decide which of the four criteria hold for each pair.

Press pause, and then when you're ready for the answers, press play.

So A, you could use AAS or ASA, depending on if you did calculate at the third angle so that you had the angles either end of the 8 centimetres, you would've wrote ASA.

If you just went with the given information, you probably went for AAS.

On B, RHS would be a criteria that holds because it's a right angle triangle.

So if it has a right angle, do you know the hypotenuse? Yes.

And do you have another corresponding side? Yes.

So RHS.

On C, it was SAS.

There was a side-angle-side, they're corresponding, two corresponding sides and the angle between them.

And lastly for D, you did need to calculate the third angle in both triangles to see that they did have three angles that were the same.

And then because of the 14 centimetres being between the 62 degree and the 57 degree angle on both triangles, then they are congruent.

Learning cycle two is about being able to see if they are congruent or not.

So on the screen, there are two triangles.

Are they congruent to each other? Well, we cannot be sure that they are congruent.

There isn't enough information to prove congruence, and that's because although we have two corresponding edges, we do not have the angle between the two corresponding edges on both triangles.

Are these triangles congruent to each other? At first glance, you may not have been sure, but the first triangle ABC, indicates that this is an isosceles because both edges are 3 centimetres.

So knowing that it's an isosceles means that you can calculate the base angles, which you know will be equal, and they will both be 40 degrees.

So therefore, we have enough information to say that the two triangles are congruent by AAS.

On this check, I want you to say which of the triangles is not congruent to the other two.

So pause the video, and then when you decide which is the odd one out, press play.

So B was the odd one out.

So on task B, question one, you need to decide which of the triangles are congruent to X, Y, Z.

X, Y, Z is the one in the green box, and also giving the criteria that proves it.

All the edges are in centimetres.

So press pause whilst you decide if those five triangles are congruent to X, Y, Z.

And if they are, how do you know it? Which criteria would prove it? Press pause whilst you do that.

And then when you're ready for question two, press play.

Question two is a very similar question to question one.

You need to decide which of the triangles are congruent to ABC, and given the criteria that proves it, once again, all edges that are marked are in centimetres.

Press pause whilst you go through those six triangles, deciding if they are congruent.

And if so, why are they? Press play when you're ready for the answers to both question one and question two.

Question one, A, you could say was congruent by ASA.

You had two corresponding angles and a corresponding side.

B was incongruent.

If you look, the 7.

5 edge should have been opposite the 52 degree angle.

In this case, it is opposite than 91 degree.

C is congruent, and we can say it's congruent by SSS.

D is congruent by SAS.

We've got two corresponding edges and the angle between them.

Lastly, E, we can't guarantee that it's congruent with that information.

It might be congruent but we just do not have enough information to tell at this point.

Question two, question two involves a right angle triangle.

So you probably have used an RHS proof at some point.

So part A were congruent because of SSS.

The three edges were the same.

B, congruent by RHS because they're both got a right angle, their corresponding hypotenuse are the same and a corresponding side.

C, also congruent by RHS.

D is incongruent.

The 13 should be the hypotenuse of the triangle.

So the angles work, the angles make it a right angle triangle, but the 13 is not the hypotenuse.

So this is incongruent.

It would be a completely different size.

E, you can prove they are congruent by using ASA or AAS.

And lastly, F, you can prove they're congruent by ASA or AAS, but ASA was sort of given to you with angle-side-angle.

So we're up to the last learning cycle of the lesson, which is called hidden congruence because sometimes, firstly, you need to identify the triangles and then go through the proof of congruence.

And so here, we have exactly that.

We've got a regular pentagon, and at first glance, there are no triangles.

So how could we possibly be looking to prove congruent triangles? So by drawing on some diagonals, we can make triangles.

We can go further to show that the triangle ABC is congruent to triangle AED.

And this is how we would do it.

We would know that AB is equal to BC is equal to AE or is equal to ED because they are all edges of a regular pentagon.

We could also say angle ABC and angle AED are equal because they are the interior angles of a regular pentagon.

And hence, we can see and show that triangle ABC is congruence to triangle AED by SAS.

We have two corresponding sides and a corresponding angle between them.

Then we might have a problem like this, ABCD and DEFG or squares prove triangle CDG and ADE are congruent.

Jun has a problem with this.

"So there are no triangles in this diagram.

How am I proving triangles that do not exist?" Sofia suggests that we add the line segments to form them.

So that would be triangle ADE, and that would be triangle CDG, Jun's okay with that.

"Okay, I can see in the triangles now, but still I'm unsure where to start." And you might feel like Jun sometimes, that you've got a problem.

"There's no numbers, how do I start this question?" So Sofia suggests that we add any properties that we know.

So what properties do we know? Well, we've been told that they are both squares.

If they're squares, then the edges are all the same.

So on square ABCD, we can give them each edge one hash mark to indicate that they are all the same length.

And we'll have to use two hash marks for square DEFG because it's quite clearly a different length.

We also know by being square that the angles are all right angles.

I could mark all of the right angles on there.

But because there is the line segment CG that passes through a right angle and the line segment ae that passes through a right angle, I decided it could get quite complicated.

So I've just put right angles there to remind me that there are right angles on this diagram.

So Jun says, "I can see now that both triangles have two edges of the same length." So have a look what is Jun talking about.

Look at the triangle CDG, look at the triangle ADE.

Can you see that they have two equal corresponding edges? Sofia says, "Plus, the angle between them is the same." So the angle between those two equal corresponding edges is the same, and that's angle CGD and angle ADE.

Did you notice that they would be the same? Sofia has told us why.

And the reason they are the same is 'cause they both include the 90 degree from the two squares.

The angle is made of the 90 degree from the squares, plus the angle ADG.

We don't know the size of angle ADG, but it is part of the two angles.

And therefore, we can say that these two triangles are congruent by SAS.

AD = DC, the edges of the same square.

GD = DE, they are edges of the same square.

And angle CGD is equal to angle ADE because they are both made of a 90 degree plus angle ADG.

And so yes, they are congruent by SAS.

Here is the same diagram that Jun and Sofia have just been working on, but it's a check for you.

So triangle CGD is congruent to triangle ADE by SAS.

What angle corresponds to angle DAE? Pause the video, find the angle, think about which one it would correspond to in the other triangle knowing that they are congruent.

Press play when you're ready to check.

Okay, it's GCD.

Just to note here, GCD, angle GCD is the same as angle DCG.

It's just the order of the vertices that you went in.

Let's look at this problem then.

So again, we're thinking about hidden congruence.

So given that PQRS is a parallelogram and PT = RU, prove that TQ = SU.

So in this problem, we haven't been asked to prove that two triangles are congruent.

We've been asked to show that two edges or two lengths are equal, and it's at this point that we need to recognise, "Well, congruence would really help us do that." Because if you have shown that two triangles are congruent, then you are saying that all of the properties of that triangle are the same.

So the edges are equal, the interior angles are equal.

And on this diagram, I'm hoping you can see there are two triangles.

There's triangle PTQ, and there is triangle SUR.

So if we can show that those two triangles are congruent, then this will imply that corresponding edges are the same length, and hence, TQ, which is an edge of one of the triangles, is equal to SU, which is an edge on the other triangle.

So TP = RU.

That was given in the question in the statement.

So we know that those two corresponding edges are equal.

PQ = SR, because they are opposite sides of a parallelogram, we were also told that this quadrilateral was a parallelogram.

So we can use any facts about a parallelogram in our proof.

An angle TPQ will be equal to angle SRU because they are opposite angles in a parallelogram.

Every parallelogram, the opposite angles are equal.

So triangle PTQ is congruent to triangle SUR by SAS.

We've got two corresponding sides and the angle between them.

So we know that they are congruent.

And therefore, if they're congruent, TQ will equal SU.

Here's a check.

So complete the statements, pause the video whilst you read through it and figure out what they should have in those blank spaces.

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

So you can see from the markings on the diagram that this is SAS.

We've got two corresponding sides and the angle between them as equal.

And therefore, if those two triangles are congruent, then BE corresponds with GD and would be equal, and angle AEB is angle AGD.

Once again, the three letters could have been written as DGA rather than AGD.

And likewise, for the line segment, you could have written DG.

So the last task of this lesson, question one, ABCDEFGH is a regular octagon.

So I've not just spammed the keyboard there.

They are the eight vertices of this regular octagon in anti-clockwise direction.

Part A, prove triangles ABC and triangle HGF are congruent.

And part B, what other triangles are congruent to ABC? Pause the video whilst you are writing down your proof in part A and then finding the other triangles that are also congruent to ABC.

Press play when you're ready for question two.

Here, we have question two.

You are told that ABDF is a square, BCD and DEF are equilateral triangles.

Prove that triangles CDE and FDC are congruent.

So once again, you're probably going to want to add some lines to the diagram to find the triangles you're trying to prove are congruent.

Secondly, you're gonna want to add any properties that you know from the fact that this is squares and equilateral triangles before you start trying to rewrite your proof.

Press pause whilst you do that.

And then when you press play, we're gonna go through question one and question two.

So here, we've got the answer to question one.

So firstly, you needed to prove ABC and HTF were congruent triangles.

So AB = BC = HG = GF because the octagon is regular.

Angle ABC and angle HGF are equal because they are interior angles of a regular octagon, and therefore triangle ABC is congruent triangle HGF by SAS.

What other triangles are congruent to triangle ABC? Well, BCD, CDE, DEF, EFG, GHA, and HAB are congruent triangle ABC because them same features apply.

They all include two edges on the octagon so they would be equal in length because it is regular and they all include an interior angle of an octagon that is regular.

Question two, so I'm hoping you sort of took my advice and you did draw on the line segments to create the triangles, and then you added properties from a square and equilateral triangles.

So firstly, CD is a common edge to both triangles.

You may not have written this in exactly the same order, but you need to have had these parts to it.

So CD is a common edge to both triangles.

DF = DE as they are edges of the same equilateral triangle.

Angle CDF would be equal to 150 degrees, and we can calculate that because we know it's got the 90 degree of the square and the 60 degree angle of the equilateral triangle.

Angle CDE can be calculated using angles around a point summing to 360.

And you know that the other angles around that point is two 60 degree angles coming from equilateral triangles and a 90 degree angle coming from the square, and that calculates to 150 degrees.

So therefore, you can say that angle CDF is equal to angle CDE.

You've calculated that they both equal 150 degrees, and therefore they're congruent by SAS.

We're at the end of the lesson.

We're gonna summarise what we've been doing.

So we've been applying the criteria for congruence.

So if one of the criteria, SSS, SAS, ASA, or RHS apply, then the two triangles can be said to be congruent.

In some cases, more than one can be used to prove congruence.

And if two triangles are proven to be congruent, then you can also prove equal lengths.

Really well done today.

I look forward to working with you again in the future.