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 learning outcome today is to be able to understand and use the criteria for two triangles to be congruent, which is SAS.

There are some key words and some diagrams to explain some keywords on this slide.

They are words that you've known and you've learned before, but you may wish to pause the video and just refamiliarize yourself so that you feel confident before we make a start with the lesson.

So the lesson on checking and securing understanding of congruent triangles by SAS is gonna be split into two learning cycles.

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

So let's make a start at identifying congruence.

So to be congruent, two shapes need to be able to fit exactly on top of each other using rotation, reflection or translation if necessary.

So their angles and edges will be the same size and in the same relative position within the shape.

So here we have got three quadrilaterals and there are some similarities between them.

So the square and the rectangle have the same angles.

We know that from the property of a rectangle and a property of a square, but they are different shapes, so they are not congruent to each other.

The rectangle and the parallelogram have got the same edge length and in the same order.

However, they are not congruent because the angles are different.

So none of these shapes are congruent to each other.

For scale drawings one way to check if two shapes are congruent is to use a piece of tracing paper.

So just by without using a piece of tracing paper by eye, would you say that these two shapes are congruent? Would they fit exactly on top of each other? Are their edge lengths the same? Are their angles the same in the relative positions? Well now let's use a piece of tracing paper.

So we put our tracing paper and we trace over it.

We didn't have to use this one, we could have done the right hand side one, but I've chosen to use the left.

So now by moving my piece of tracing paper, this is gonna act as my reflection or my rotation or my translation.

Can I make this trace fits exactly on top of the second copy? Well, I need to rotate it and line it up and actually it doesn't.

So these two shapes are not congruent to each other.

So by eye you might be misled, it's important that you are identifying congruent shapes accurately.

And so using a piece of tracing paper on scaled drawings is one way of doing it.

Another method is to measure.

So measure the edges, measure the angles, user ruler, user protractor.

And if they are the same and in the same relative position, then you can say that they are congruent to each other.

Being sure that they are equal and they are the corresponding edges.

So here we've two quadrilaterals once again, and the four edges are the same.

So if you was to make a list of the edges that both quadrilaterals have, you would say five centimetres and 8.

1 centimetres, five centimetres and 8.

1 centimetres.

So they do both have a pair of five centimetre edges and a pair of 8.

1 centimetre edges.

But that does not tell you that they are congruent.

One reason is that the order is different.

And secondly, because the four angles are not the same.

So a quick check.

True or false? Are these two triangles congruent? Pause the video and then when you've made your decision whether they are or they're not, so it's true if you think they are congruent and it's false if you think they're not, then we'll move on to the justification.

So press play when you're ready.

They are congruent triangles, those two diagrams. So the triangles have the same angles and lengths or they are congruent because the triangles have the same angles.

Once again, pause the video and when you're ready to check your justification, press play.

The reason we can say that these are congruent is because the angles and lengths are the same and in the relative position.

So if two isosceles triangles, let's focus ourselves just on isosceles for the moment.

If two isosceles triangles both have two edges of eight centimetres, are they guaranteed to be congruent? The answer there is no.

And these two diagrams sort of highlight that.

So both of these are isosceles triangles.

We can use the hash marks to know that.

And the eight centimetres is the equal edges, but quite clearly they are not congruent triangles.

If we got a piece of tracing paper and sketched one on top of the tracing paper, move the tracing paper, you would never with any reflection, rotation or translation be able to make it line up exactly.

So what is different about these isosceles triangles? We know what's the same, they have two equal edges of eight centimetres, but why are they not congruent? What is different? Well, the third edge length, so two of their edges are the same, but the third edge is different, and that is because the angle between the equal edges is also different.

So if the two isosceles triangles have the same angle between the two eight centimetre edges, will they now be congruent? So two isosceles triangles with two equal edges of eight centimetres and the angle between those equal edges is the same.

Does that guarantee congruence? What do you think? So this is now, yes.

So here if we've got two isosceles triangles, both with eight centimetre edges for their equal edges and the angle between the equal edges 37 degrees, then you would have congruent triangles.

That angle at the top is fixing the base length.

And this actually is true for all triangles, not just isosceles.

So if two edge lengths and the angle between them are known, then only one triangle can be constructed.

The third edge is fixed and that third edge is fixed because of the angle.

So here I've got three copies of the same triangle.

A triangle that has got a 10 centimetre edge, an 18 centimetre edge, and a 47 degree angle in between them.

So these three triangles are congruent because there's only one that can be constructed.

If the third edge was to change.

So if we try to change it so that the third edge was no longer 13.

4 centimetres, then can it continue to have a 10 centimetre edge, an 18 centimetre edge, and a 47 degree angle between them? Well, no, there's two ways that you can change that third edge.

You could increase the angle.

So we can see that with the 60 degree that the angle is now changed from 47 to 60 degrees in order to accommodate a 15.

6 centimetre edge.

Or the angle could stay the same, but the other edge would need to change.

And so I've increased to 20.

6 centimetres in order to accommodate an edge of 15.

6.

And so this highlights that when you have two edges and the angle between them, there is only one triangle that can be constructed.

So this is a way that we can guarantee congruence between two triangles.

And so this is a criteria for congruence and this condition is known as side-angle-side or SAS for short.

So you must have two edges and an angle between them.

And that's the same on both triangles.

You may be thinking what Sam's thinking, "Does the angle have to be between the two known edges?" So we keep saying side-angle-side, could it be side-side-angle? Jacob says, "Yes." And he's gonna show Sam why.

So it does have to be the angle between.

So here is a trapezium.

We know it's a trapezium because it's quadrilateral with one set of parallel sides.

Jacob says that this is an isosceles trapezium.

And so in this isosceles trapezium, the two triangles created by a diagonal have two equal edges and we can see those marked.

So the two equal edges is the diagonal, because that's a shared edge, and because it's an isosceles trapezium, two edges are equal on an isosceles.

It also has an angle in common.

Those two triangles both have the same angle in common and that's because of alternate angles, but they're not congruent.

So here we have two sides and an angle, but that angle is not between the two known sides.

And you can quite clearly, as Sam said, they are obviously not congruent.

You can see that one of the triangles has an obtuse angle.

The other triangle has three acute angles.

They're quite clearly not congruent triangles despite the fact that they do have two edges of the same length and an angle.

So a quick check.

True or false? Two triangles are definitely congruent if they have two edges and an angle that are the same.

And justify it by saying, to confirm congruence by SAS, the angle must be between the known sides or this is the criteria side-angle-side.

So pause the video, read through that once again, true or false and your justification, press play when you're ready to check.

So that's false and it's because we do need the angle between the two known sides.

So here's another check for you.

Are these triangles congruent to each other? So look at the information you have, are they congruent to each other? Press pause and then when you're ready to check press play.

They are congruent.

So sometimes we have to do a little bit of extra calculation to be able to prove that they are congruent to make sure we have sufficient information.

So on the left hand triangle it had side-angle-side, it had two edge lengths and the angle between them.

On the right hand triangle we had the two same sides, a 7.

1 centimetre and a 7.

8 centimetre.

But we didn't have marked the angle between them.

And if that angle was not 61 degrees, then they wouldn't be congruent.

However, we can use the angles in a side of a triangle summing to 180 and calculate that that angle is actually 61 degrees and therefore they are congruent by SAS.

So onto the first task of this lesson on identifying congruence.

On question one, I want you to write down all the pairs of congruent shapes.

So remember to be congruent, their length and angles must be the same in the corresponding places.

You can use a piece of tracing paper here to be able to trace over and check whether with a reflection or a rotation or a translation, you can make your traced shape fit.

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

So on question two, there's two pairs of triangles.

So part A is the first column, and part B is the second column.

And given that each pair of triangles are congruent, fill in the blanks.

So using the definition of congruency, fill in the blanks.

Press pause, and when you come back there'll be question three.

So question three, which is the last question on task A.

Explain how you know that neither of the triangles here are congruent to triangle A.

Triangle A is the one that's in the box.

So press pause, think about how you know for certain that they are not congruent.

When you press play, we're gonna go through the answers to this task.

So on question one, you needed to write down all the pairs of congruent shapes.

So if the shapes were different, for example, you weren't gonna say a pentagon was congruent to a hexagon, so you were only really comparing the ones that were the same shape, then you were comparing whether their size was the same and tracing paper would've made that easy.

So B and H are the same size, pentagon.

And D and G are the same size, parallelogram.

Question two, we were given that they were congruent.

So if they are congruent, then their edges and their angles are the same.

And that was what you were making use of here in order to complete the gaps.

So on a, you should have written 9.

3 centimetres as an edge on the top triangle and 5.

3 centimetres as an edge on the bottom triangle.

On b, you should have written 8.

7 centimetres as the edge on the top triangle and the missing angle would be 90 degrees.

And on the bottom triangle it was 14 centimetres for the hypotenuse, because that is opposite the 90 degree angle.

It's always gonna be the longest as well 'cause that's the largest angle.

And then it was a 38 degrees that you should have written in the position of the angle.

It was had to be 38 rather than the 52 because the 38 degrees is between the 14 centimetre edge and the 11 centimetre edge.

So it does matter the position of the angle.

And then lastly, question three, you need to explain how you knew that neither of them were congruent.

Well the bottom one where we've got an edge of 127 millimetres and a edge of 12.

7 centimetres and an angle of 133 degrees, well 12.

7 centimetres is an edge on the triangle A, 127 millimetres is not 133 millimetres, but that's not the issue.

The issue here is that you cannot have two obtuse angles in a triangle because that would sum to over 180 degrees.

So you know that 133 degrees as an angle cannot be another angle within triangle A.

And if their angles are different, then they're definitely not congruent.

On the second one, the longest edge of the triangle cannot be 133 millimetres.

So on triangle A, 133 millimetres was an edge length, the other side of the 127 degree angle, and if the marked 133 on this triangle was true, then you'd have two edges of 133 millimetres, which would then mean that this is an is isosceles triangle.

And if it is an is isosceles triangle, the obtuse angle would need to be between the two equal edges.

So it cannot be 133 millimetres.

So we're now up to the second learning cycle, which is justifying congruence using SAS.

So in a proof of congruence, which we're gonna be looking at mostly in this learning cycle, the justifications are as important as the statement or the claim that you are making.

And the most common justifications that you'll be using is making use of defined properties of a shape, a shared or common edge or angle or any stated or provided information in the question or diagram.

So if we look at this diagram here, we're gonna try to use it and given with this information.

So given the triangle ABC is equilateral and angle BAD equals angle DAC, prove that triangle ABD is congruent to triangle ADC.

So here we've got our diagram and it's all of that information is about this particular diagram.

So we're told that triangle ABC is equilateral.

So what do we remember? What can we recall about the properties of an equilateral triangle? Well, I'm sure we can all recall that that means all edges are equal.

So it infers that AB equals BC equals AC.

So I can add hash marks to my diagram to show that those there's a quality between those line segments and that's a property of an equilateral triangle.

Continuing on, we are also told that angle BAD is equal to angle DAC.

And so therefore we can show that they two angles are the same and we can assume it to be true.

So any information that is told to you in the question, you can assume to be true.

You don't have to prove that yourself.

And then AD is a shared edge on both triangle ABD and triangle ADC.

So if it's an edge on both of them, whatever the length of it will be the same for both triangles.

And so we can justify the triangle ABD, and triangle ADC are congruent by SAS.

And this can be written succinctly as the following proof.

So we're gonna start by stating that AB is equal to AC and our justification is because triangle ABC is equilateral.

Then we can say that angle BAD is equal to angle DAC as we were given this information, AD is a shared edge and therefore triangle ABD is congruent to triangle ADC by SAS.

So here is a check where you need to complete the justifications in this proof.

So ABCDEF is a regular hexagon.

Prove that triangle BAF and triangle BCD are congruent.

And so complete the gaps, the justifications.

Press pause and then when you're ready to check press play.

So the first line of the proof is AF equals AB equals BC equals CD as and that is because it is a regular hexagon.

Then angle BAF is equal to angle BCD as it is a regular hexagon.

What you do not need to do is prove what regular means.

So regular is a mathematical term that is understood to mean that the edges are equal and the angles are equal.

Therefore triangle BAF and triangle BCD are congruent by SAS.

We've managed to say that there are two sides that are the same in each triangle and the angle between them is the same.

So here we've got a diagram with a pair of parallel lines.

We know that from the arrows and some transversal lines as well.

Given that DE equals GH.

So given that two line segments are equal and F is the midpoint of DH, prove that EF equals FG.

So this time we're not proving congruence, we're proving an equality of two line segments.

Laura says, "If triangle DEH and triangle FHG are congruent, then a consequence of this is that EF does equal FG." And here we are gonna make use of what it means for two things to be congruent.

So if two triangles are congruent, then we know that their edges are equal.

And so this is a way of proving that EF, which is an edge on triangle DEF, is equal to FG, which is an edge on the triangle FHG.

So Laura says, "I can prove that these two triangles are congruent by SAS." Can you see what she's about to say? So she continues to say, "As DE equals GH as stated." So that's a side.

Angle EDF is equal to angle FHG, which is equal to 57 degrees." We can see them labelled on the diagram.

So that's our angle.

"And then DF equals FH as F is the midpoint of DH." We were told that F is the midpoint and midpoint means that it's exactly halfway.

So that would break the lines DH into two equal parts of DF and FH.

So she's shown side-angle-side.

So she's managed to prove that those two triangles are congruent and by being congruent, that then means that EF equals FG, as a consequence.

Izzy said, "The angles will be the same regardless as they are equal alternate angles." So actually if this diagram didn't tell us that they were 57 degrees, it didn't give us the actual size of those angles, we could still state that those two angles were equal.

And they are equal because they are equal alternate angles because of the parallel lines.

So a true or false check for you, without knowing the size of any angles, we cannot prove that triangle DEF and FGH are congruent by SAS.

So using that diagram there and then there's some justifications.

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

So this is false and your reason is because the size of the angles is not relevant, just like Izzy said, it's only that they are equal.

And we can prove they are equal using equal alternate angles.

So we're now to the last task of the lesson.

And on question one you're given that ABCD is a kite, I'd like you to prove the triangle ABC and triangle ACD are congruent by SAS.

So pause the video and then when you're ready for the next question, press play.

Question two is the next question.

And it's given that ABCDE is a symmetrical pentagon.

Prove that triangle ABC and triangle ADE are congruence by SAS.

Once again, write down your proof.

You're trying to do it for SAS.

So you need to find a way of showing that two sides are the same, two angles are the same, and two other sides are the same between the two triangles.

Press play when you are ready for question three.

So this is the last question of the task, question three.

So given ABC is a triangle and P is a midpoint of AB, Q is a midpoint of BC and PBQR is a parallelogram.

Prove that triangle APR and triangle RQC are congruent by SAS.

I'd really encourage you to add notation to the diagram that's gonna help you look to see if you have got two sides and an angle that are the same in each of the triangles.

Press pause and then when you're ready to go through the answers to task B, press play.

So on question one, we were proving two triangles were congruent by SAS knowing that this was a kite.

So all of the justifications was based on the properties of a kite.

So AD equals AB as they are adjacent edges of a kite.

Angle ADC is equal to angle ABC as they are opposite angles in a kite.

BC equals CD as they are adjacent edges of a kite.

And you can see all the notation on the diagram and therefore triangle ABC and triangle ACD are congruent by SAS.

The order in which you stated those properties doesn't matter as long as the all three of are there that you've got two sides and an angle.

Question two, given that ABCDE is a symmetrical pentagon.

Prove that triangle ABC and triangle ADE are congruent by SAS.

So this one here is about the symmetry of the pentagon.

So AB is equal to AE because the pentagon is symmetrical.

Angle ABC is equal to angle AED as the pentagon is symmetrical.

And BC equals DE as the pentagon is symmetrical.

So we have shown that we've got two sides and an angle between them that are the same in both triangles, therefore we can say they are congruent by SAS.

Lastly, we've got this question here where we were told it's a triangle.

We've got some midpoint and a parallelogram.

So AP equals PB as P is the midpoint of AB.

So our justification of how we know that they are equal is because P is a midpoint.

We were told it's a midpoint, we didn't assume it's a midpoint.

BQ equals QC as Q is a midpoint of BC.

Then given that PBQR is a parallelogram, we're now gonna make use of the properties of a parallelogram.

BQ equals PR and PB equals RQ as opposite edges are equal in a parallelogram.

Then we need to think about the angles 'cause so far we've shown that we've got two sides that are the same on both triangles.

So angle ABC is equal to angle APR, which is equal to angle RQC as they are equal corresponding angles.

They are equal corresponding angles or we know that they are equal corresponding angles because we have parallel lines.

We have parallel lines because it's a parallelogram.

The reason ABC angle is in brackets is because actually that's not an angle within the triangles we're trying to prove are congruent, but it is a way of linking the other two angles as equal.

And therefore we can conclude to say that triangle APR and triangle RQC are congruent by SAS because we've shown there are two sides and the angle between them that are the same that match in both triangles.

So to summarise today's lesson, two triangles can be proved to be congruent if the two side lengths and the angle between them are the same if they match in the two triangles.

And you can see that on the diagrams. If two side lengths and an angle that's not between them are known, this does not prove congruence.

So remind yourself about that isosceles trapezium.

There can be two triangles with these details.

So if you were to try and construct a triangle knowing two sides and any angle that's not between them, then you could construct two different triangles.

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