Lesson video

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

I really hope you're ready to try your best and ready to learn.

So our outcome today is to be able to understand and use the criteria for congruence, specifically SSS.

On the screen, there's some keywords I'll be using during the lesson.

You have learnt them before, but you may wish to pause the video so that you can read those definitions and make sure you feel confident before we make a stop.

(object clicks) So the lesson is about checking and securing your understanding of congruence SSS.

We're gonna do this by splitting the lesson into two learning cycles.

The first learning cycle is identifying congruence and the second learning cycle is justifying congruence.

So let's have a look at identifying that two triangles are congruent.

So shapes which have the same lengths and angles are said to be congruent.

So here are some examples of pairs of congruent shapes.

So two rectangles where a translation has taken place, the object and the image would be congruent.

Two rectangles where a reflection has taken place where one has just flipped over, then the object and the image would be congruent.

And lastly, if a rotation has taken place.

The object and image are still the same size, their angles and their lengths haven't changed.

So here are some pairs of incongruent shapes, shapes that are not congruent.

So firstly, although the shapes are the same, they're both rectangles, they're different sizes.

So it isn't true that the lengths are the same.

It's true that the angles are the same but not the lengths.

And then lastly, this might seem a little bit obvious, but a rectangle and a triangle are not the same shape, so they don't have the same amount of edges, nor do they have the same edge lengths or the same angles.

So to be congruent, two shapes need to be able to fit exactly on top of each other.

And we can use rotation, reflection, or translation to make this happen.

The angles and edges will be the same size, so we just saw that previously, and in the same relative position within the shape.

So here we've got a rectangle, a square, and a parallelogram.

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

So just because two shapes have the same angles and in the same positions doesn't mean they will be congruent.

The rectangle and the parallelogram have the same edge lengths and they are in the same relative positions.

They're in the same order, but they have different angles, so it's not sufficient to have the same lengths but not the same angles.

We need to have both lengths and angles the same to be able to say they are congruent.

So none of these shapes are congruent to each other.

(Lashley gulps) One way to check if two shapes are congruent is to use a piece of tracing paper.

So here I've got two shapes.

Do you think they're congruent to each other from looking at them? Well, let's use a piece of tracing paper.

We'd trace over one of them.

It doesn't matter which one we choose, but we trace over one, and we're now gonna see if that shape that I've traced on the tracing paper can overlay onto the other one.

So if I just translate it to the right, clearly it doesn't overlay.

But remember we can rotate and reflect as well as translate to get our congruent shapes to fit.

So if I now rotate it? It's not lined up properly, so let's move it slightly, or actually it doesn't fit on top exactly.

So it may have been that you thought, yeah, they look congruent, they look like they're the same size and the same shape, and the angles are the same, but actually there's a slight change to the angles which has changed the shape.

So these two are not congruent to each other.

So you can use a piece of tracing paper to quickly check if two shapes are congruent.

So give a reason why these shapes are not congruent.

So pause the video, think about what's the reason that you know these are not congruent to each other.

Press play when you wanna check.

Well, they're not the same shape.

So it was a bit like the rectangle and the triangle.

If the shape is different, then they are not congruent.

One's a rhombus, one's a trapezium.

Okay, another check.

Give a reason for why these rectangles are not congruent.

So this time the shape is the same, but why are these not congruent? Pause the video.

and when you're ready to check your answer, press play.

They're not the same size.

So remember, it doesn't matter about the orientation, it can be rotated and still be congruent, but actually these are slightly different-sized rectangles.

If you did have a piece of tracing paper, you'd be able to see that quite easily.

And another check, give a reason for why these rhombi are not congruent.

Rhombi is the plural of rhombus.

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

They have different angles.

So a rhombus, the shape is the same, the size is the same, in terms of the edge lengths are both seven centimetres, and a rhombus, all edges are equal, but their angles are different.

You can see there's a slightly different tilt to the right-hand rhombus to the left-hand one.

Again, a piece of tracing paper would clearly show that as well, that there wouldn't have been any way of rotating, reflecting, or translating to make it fit exactly.

So another method we could use rather than tracing paper is to measure, to use a ruler and to use a protractor.

So being sure that corresponding angles and edges are equal is a way of checking if they are congruent.

So although the four edges are the same in both of these shapes here, the order of them is different.

These two quadrilaterals are not congruent.

So when you are measuring with a ruler, with a protractor, it's important that you don't just think, ooh, okay, there's a five centimetres and there's an 8.

1 centimetres on both of them, that makes them congruent, because the order is different.

We can see that with the hash marks.

It's also quite clear here to see that these are different shapes.

One we would say is a kite, and the other we would say is a parallelogram.

So a quick check on that, true or false, the two triangles here are congruent.

Press pause, make a decision, and then press play to move on to the justification.

Okay, so it's true.

These two triangles are congruent.

Now justify how you knew that.

Is it because the triangles have the same angles? Or is it because the triangles have the same angles and lengths? Once again, you might wanna pause the video, but when you're ready (cat meows) to check, press play.

So the justification is they have the same angles and lengths.

The reason we need both, it's just because having the same angles does not guarantee congruence.

There's a couple of ways we can remind ourselves about that.

A square and a rectangle, they are different shapes, but they both have four 90-degree angles, or two rectangles can both have four 90-degree angles but not be congruent.

They could be different sizes.

So it's important that it's not just the angles, it's the angles and the lengths.

(object faintly scrapes) So all equilateral triangles are similar to each other.

So here we've got three examples of equilateral triangles.

They are similar to each other.

Why is this? So you may wanna pause the video, you might need to go and look back at what the definition of similar means.

But why are all equilateral triangles similar to each other? Well, that's because they all have three 60-degree angles.

So if you know that all of the angles are the same between two shapes in the relative same positions, corresponding positions, then we can say they are similar.

When would two equilateral triangles be congruent to each other? So all equilateral triangles are similar to each other, but when would two equilateral triangles be congruent? Well, they would be congruent if the edge lengths were the same as well.

So here, my three examples, none of them are congruent to each other.

They're different sizes.

The angles are the same, but their edges are different.

So if the edges were the same as well as the angles, then those equilateral triangles would be congruent.

So the only triangle that can be constructed with three edge lengths of eight centimetres is an equilateral triangle.

Hopefully you agree with me on that one.

So by definition of equilateral, all edges are equal, so if you were constructing a triangle with three eight-centimeter edges, you would have constructed an equilateral triangle.

So if you were to construct another triangle with three edge lengths of eight centimetres, it would be congruent to the first one that you constructed because there is only one particular triangle that would've been constructed.

And this is true for all triangles.

If the three edge lengths are known, so here we've got a 12-centimeter edge, a 14-centimeter edge, and a 19-centimeter edge.

There is only one triangle that you can construct.

The angles are fixed because of those three edges.

And so if I was to construct a second one, it would be identical.

But if an angle was to change, so I'm trying to show here that the angles are fixed.

We know on an equilateral triangle the angles are fixed to 60 degrees, but is that a special case for equilateral triangles? No.

If you have the three edges, it fixes the angles because if one of the angles was to change, then you can see that our 14-centimeter edge and our 19-centimeter edge no longer intersect at a vertex.

So one of the edges, I've changed the 14-centimeter edge, would have to change in order to construct and close the triangle.

So Izzy and Aisha are just reminding us a little bit about construction.

So if three edges are known, only one triangle can be constructed and this fixes the angles.

So Izzy says, "Hang on, I remember when we did constructions for SSS.

There are two points where the arcs could cross, so there are two triangles that can be constructed." So she's questioning, "You're saying there's only one that can be constructed?" But she can remember from the constructions unit that she did that there are two.

So there is a link there to a GeoGebra file.

I'm gonna talk through it now and then we'll come back to the slideshow.

So this is the GeoGebra file which will allow us to look at the construction of a triangle given SSS.

So the stages, if you were doing this with pencil and a pair of compasses would, first of all, you'd draw one of the edges.

So here I've got a 14 edge drawn and we can move it anywhere on the screen.

We can rotate it so it doesn't have to be going in the direction that it currently is.

And by moving, I can place it in a different location.

But on a sheet of paper, normally you sort of draw it quite horizontal probably.

And so that is our first edge of the triangle that we're trying to construct.

Next, we would use our pair of compasses to draw a circle that shows all the possible positions of one of the other sides, and that would be bisecting it to the correct radius.

So if we want to draw a radius, a circle with radius 12 centimetres centred on A, if I tick it here, this is what you would have on your sheet of paper, having drawn a circle with a radius of 12 from the centre A.

Then any point along that circumference is 12 centimetres from the vertex A.

And so that is our second edge of our triangle.

When we draw our third edge, which is 19 centimetres long centred at B, we would draw another circle.

And this is where we end up with two points that are intersecting on those two circles that we've drawn.

One are sort of above the first edge and one below the first edge, so we can then draw our triangles.

So if I go to this intersection point, the distance from vertex A to that intersection point is 12 because it is a point on the circumference with a radius of 12.

(object faintly clicking) If I then connect that point of intersection to the other end of the line segment, then our construction is complete.

So we've constructed a triangle SSS with 12, 14 and 19-centimeter edges.

However, I could have chosen to draw the triangle above.

I could've drawn a radius from the vertex A, the centre A to the point of intersection there and joined that to B, and I would construct the same triangle.

These two triangles are congruent.

The edge lengths, the three edges are the same.

So this is sort of a end product that we ended up with in that GeoGebra file.

Slightly different orientation to where I ended mine, but we have the two triangles that Izzy is discussing.

Aisha said, "I see what you mean in terms of there are two, but you could draw that triangle in loads of different positions And it's still the same triangle.

They are congruent." So mathematically, they're the same triangle but in a different position.

So it's just a copy of the other one.

(object faintly clicks) So two triangles are guaranteed.

Remember, we're trying to identify congruence.

We can identify that two triangles are congruent to each other if they have the same three edge lengths.

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

So this is only true for triangles.

If you have three sides that you know the three side lengths and they match to another triangle that has the same three side lengths, then those two triangles are definitely congruent.

So using that fact, these two triangles are congruent, complete the missing edge lengths.

Pause the video, and when you're ready to check your answers, press play.

So we're gonna use the right-hand triangle to fill the missing edge on the left-hand triangle because you know these two triangles are congruent.

It means that this one will also need to have a 67-meter edge.

And likewise, the right-hand side can be finished by putting a 96-meter edge because the two triangles are congruent.

So onto the first task of the lesson for you, which of these shapes are congruent to each other? So a group of them are actually congruent within here.

You probably want to pick up a piece of tracing paper to make your life easy.

Remember, you can rotate your tracing paper.

You can reflect your tracing paper to check if they are congruent.

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

On question two, I don't want you to use tracing paper.

Instead, I want you to grab a ruler.

So by measuring the sides, decide if the pairs of triangles are congruent to each other.

So the pairs are vertical, so it's like three columns.

Pause the video.

When you're ready to move on to question three, press play.

Question three, given that the pairs of triangles are congruent to each other, fill in the missing information.

So this is a little bit like the check we just did.

So pause the video, and then when you press play, we've got one more question left of the task.

So question four, you need to decide if these pairs of triangles are congruent to each other.

So is ABC congruent to DEF.

Pause the video, and then when you press play, we're gonna go through our answers to Task A.

(object faintly tapping) So question one, you needed to identify which shapes were congruent.

And using tracing paper would've been the quickest and easiest.

It's not sufficient enough to just do it by eye because as we can see, c looks quite congruent by eye but actually isn't.

So a, b, e and h are all congruent to each other.

Question two, by measuring the edges, then are they congruent? Well, a was a pair of congruent triangles and c was a pair of congruent triangles.

b was not.

Question three, fill in the missing information given that they are congruent pairs.

So 63 metres and 97 metres needed to be added onto the first pair in part a.

And then 12 centimetres on the left triangle for part b.

And then on the right-hand triangle, nine centimetres and six centimetres.

The order and where their location was was important because the six centimetres needs to be opposite the 29-degree angle.

You can see that on the first triangle.

And then you needed to use angles in a triangle adding up to 180 to calculate the value, the size of the obtuse angle.

Question four, were they congruent? Well, part a, yes, we can easily match that both triangles have the same three edge lengths.

On part b, it was also yes by SSS, but you needed to convert some of the units.

So they were given a mixed unit, so it was harder to compare, but once you change them to the same unit, maybe that was centimetres, then you can compare them to see that they have got the same equivalent three edge lengths.

So our learning cycle two, it goes on to justifying congruence, focusing on SSS.

So here is a kite with one of the diagonals marked.

Are the two triangles, ABC and ADC congruent? So Aisha says, "Well, we could measure the three edges with a ruler." So we saw that in the first learning cycle, that as long as you can show that the three edges of the triangles are the same in both, that's one way of showing they're congruent by SSS.

Jun says, "I don't have a ruler, and it's probably not that accurate anyway." And that's because a ruler will be accurate to the nearest millimetres if you've got millimetres marked on it but it's not overly accurate.

However, he thinks he can show that they are congruent.

So as it is a kite, AD equals AB.

So adjacent edges of a kite are equal, that's a property, And BC equals DC also because it's a kite.

AC is an edge on both of the triangles, so that is the diagonal that was marked.

And if you look at the two triangles that we're trying to prove are congruent, it is an edge on both of them.

Therefore, triangle ABC and triangle ADC are congruent by SSS because they have the same three edges.

And Aisha notices, "Okay, you didn't use a ruler, you haven't measured anything, you haven't used tracing paper, but instead you've used the properties of the shape." So for your check, can you prove that triangle ABC and triangle ACD are congruent in this rectangle? So pause the video, and then when you're ready to go through the answer to this check, press play.

So as it is a rectangle, so using the properties of the shape, AD equals BC, the opposite edges are equal.

It's also true that AB equals DC.

Opposite edges are equal because it is a rectangle.

Just a note on this, you may have written yours the other way around.

So you may have spoke about the shorter edges first before the longer edges, that doesn't matter.

And also the letters, if you wrote DA, that's the same line segment as AD.

Going on.

AC is a shared edge on both triangles, so that is the diagonal that is marked, and therefore triangle ABC and triangle ACD have the same three edges so they are congruent by SSS.

We can justify and prove that those two triangles are congruent.

So in a proof of congruence, the justifications are as important as your statement or your claim.

That statement or claim might be AB equals DC.

The common justifications will be defined properties of a shape.

So like we just saw with the rectangle, opposite edges being equal.

It might be a shared or common edge or angle.

So again, the diagonal was common to both triangles, so it's a fixed length, whatever it is, and that's an edge on both triangles.

And lastly, a stated or provided piece of information.

So you might be told something specifically about a diagram and then you can assume that to be true in your proof.

So looking at this diagram here, if we are told that triangle ABC is equilateral and the line segment BD is equal to the line segment DC, we need to prove that triangle ABD is congruent to triangle ADC.

So firstly, we're told that the triangle is equilateral.

Well, we know some properties of equilateral triangles and that infers that all three edges are equal.

So I've put hash marks onto my diagram to indicate that they are all the same.

So AB equals AC equals BC.

They are the same edge length, and that's a property of being an equilateral triangle.

Then we were told, we were given this information that BD is equal to DC, so we can assume that to be true because we were told that in the question.

I've used double hash marks now because it's a different length to the edges of the equilateral triangle, but we do have a.

There's equality within the diagram.

AD is a edge on both of the triangles that we're trying to prove to be congruent.

So it's shared, it's common, it's the same length for both of them.

And so we can justify that this triangle, ABD, is congruent to triangle ADC by the criteria SSS because we have just shown that both triangles have the same three edges.

This can be written succinctly as the following proof.

So what we just went through, we can write it slightly shorter.

So we can say AB equals AC as triangle ABC is equilateral.

It's true that BC is also equal to those two, but it's not relevant for what we're trying to prove.

BD equals DC as we were given the information, so that was a given information.

And AD is a shared edge, therefore triangle ABD is congruent to triangle ADC by SSS.

So we've got a conclusion, which is what we were trying to prove, and we've given the criteria, which is SSS, our congruence criteria.

So for your check, I would like you to complete the justifications in this proof.

So pause the video, read through it at least once before you try to fill in the blanks.

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

(Lashley gulps) So the blank on the first row would be it is a regular hexagon.

You do not need to go and define that regular means all edges are equal.

That's absolutely fine that that is a mathematical terminology that we understand to mean that.

But that is the reason we can justify that FE equals ED, that they are edges of a regular hexagon.

BF equals BD as it was stated in the question, so if you said it was given, then that's fine.

We were told that.

We can assume it to be true.

We can follow on with the truth.

BE is a shared edge, and therefore triangle BEF and triangle BDE are congruent by SSS.

So the last task of this lesson is to look at justifying congruence, not measuring using ruler or protractor or using a tracing paper, we are justifying that they are congruent by using the criteria SSS, and making sure we know it for certain by properties of shape, given information, or shared edges or angles.

So given that ABCD is a parallelogram, BM equals ND, where M is the midpoint of AD and N is the midpoint of BC, complete the proof that triangle ABM is congruent to triangle CDM.

So you need to fill in the blanks for this proof to show that the two triangles are congruent by SSS.

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

So question two, given the ABCD is a parallelogram, AN equals MC, M is 2/3 along AD and BN to NC is in the ratio one to two, prove that triangle ABN is congruent to triangle MCD.

So you need to write a full proof here.

You want to add things to the diagrams to support you.

That's absolutely fine.

Make sure you've got that concluding statement at the end, given the criteria of congruence that you have used.

Press pause, and then when you're ready to move to question three, press play.

So question three, there are six parts to it.

This one's gonna take you a little bit of time to do.

So for each quadrilateral, prove that the diagonal bisects the shape into two congruent triangles by SSS if possible.

So they are all quadrilaterals, the square, the rectangle, the kite, the isosceles trapezium, rhombus, and a parallelogram.

And you are trying to prove that the two triangles you can see on each of those quadrilaterals are congruent to each other, but only if possible.

So pause the video, and then when you're ready to go through all the answers to Task B, press play.

So here is question one where you needed to complete the proof.

So on the first line of the proof, AB equals, you were saying it was equal to DC.

If you wrote CD? That's absolutely fine.

It means the same thing, it's the same line segment, and the justification is because the opposite edges on a parallelogram.

Then on the second sort of part of the proof, AD equals BC as they are opposite edges of the parallelogram, therefore AM equals NC as M and N are both midpoints.

So we know they are midpoints 'cause that was given in the question.

If they are midpoints, it means exactly halfway along, and as they are opposite edges of a parallelogram, they will be equal in length.

Lastly, BM is equal to ND, given in the question so you're not assuming that, you don't need to measure it.

You were told that those two line segments are equal so you can use it.

And to finish the proof in that sentence, triangle ABM is congruent triangle CDN by SSS.

So our congruence criteria is side-side-side.

So here is question two.

You need to write a foolproof to show that triangle ABN was congruent to triangle MCD.

Just to mention, the order in which you gave your statements and justifications doesn't matter, so as long as they are there.

So I've started by saying that AB is equal to DC as they're opposite edges on a parallelogram, as long as at some point within your proof, you have said that, then that's perfectly fine.

The second thing I've said is that AD equals BC as they're opposite edges of a parallelogram, so my justification is the properties of the shape, I'm now gonna use that fact.

Therefore, MD equals BN as A to M to M to D is the ratio two to one.

So this is now used in the fact that you were told some information about M and N, those points and they actually were equivalent to each other.

Because the opposite edges were equal, then 2/3 was equivalent to 2/3 on the other side, and 1/3 was equivalent to 1/3 of the other edge, so they are equal.

And then AN equals MC was given in the question.

So therefore triangle ABN is congruent to triangle MCD by SSS.

(Lashley gulps) (object faintly clicks) Question three, there were six parts, we're gonna have each part on a different slide.

So here is part A, which was for the square.

We're trying to prove by SSS that the diagonal bisects into two congruent triangles.

So because it's a square, we know that all edges are equal.

So AB equals BC equals CD equals AD as they are edges of a square.

AC is a shared edge, therefore triangle ABC is congruent to triangle ACD by SSS.

What's that also show? It also shows us that they are isosceles triangles.

On the rectangle, again using the properties of the given shape.

FG equals EH as they are opposite edges of a rectangle.

FE equals GH as they're opposite edges of a rectangle.

So there's our properties.

Again, the order that you wrote those two doesn't matter.

EG, which is the diagonal, is a shared edge, and therefore we can say that those two triangles are congruent by SSS.

Moving on to the kites, where you looked at the kite earlier in the section of the lesson, IJ equals JK because they are adjacent edges on a kite.

That is our justification.

IL equals LK as they are adjacent edges on a kite.

You can see the hash marks to indicate the equality.

JL is a shared edge, and therefore triangle JIL is congruent to triangle JKL by SSS.

Moving on to d, the isosceles trapezium.

So triangle MNO and triangle MPO are not congruent.

So this is one that it was not possible to prove that the diagonal bisects into two congruent triangles.

What we can say is that MN is equal to PO because it is an isosceles trapezium, and MO is a shared edge.

So we can identify that two of the three edges are the same.

What we can't prove and justify is that the third edge is the same.

So MP does not equal NO, and if it did, it would be a parallelogram.

So isosceles trapezium does not split into two congruent triangles.

Moving on to e, the rhombus, this is very similar to the square because of the properties.

So QR equals RS equals ST equals QT as they are edges of a rhombus.

So all edges are equal on a rhombus.

RT, the diagonal, is shared on both triangles, and therefore triangle QTR is congruent to triangle TRS by SSS.

And the last one, f, parallelogram.

This one's similar to the rectangle, not surprisingly.

So we can say that the opposite edges are equal on a parallelogram.

So we've got UV equals XW, and likewise UX equals VW.

VX is a shared edge, and therefore we have shown that all three edges are the same on both triangles, so it's congruent by SSS.

So to summarise today's lesson where we've been checking and securing understanding of congruent triangles using the criteria SSS.

Two triangles can be proved to be congruent if you know the three side lengths are the same.

So you don't need to worry about the angles because they will be fixed by the three edges.

But if you know the three edges are the same on both triangles, then you can guarantee congruence.

The corresponding angle pairs will therefore be the same incongruent shapes.

If you can prove that two shapes, two triangles are congruent, then the corresponding angle pairs will also be equal.

Really well done today, and I look forward to working with you again in the future.