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 to prove that two triangles are congruent by ASA.

On the screen there are some key words that I'll be using during the lesson.

They're not new words, they're words that you will have learned previously in your studies, but you may wish to pause the video here so that you can reread the definitions, make sure you're feeling confident before we make a start.

So the lesson is to check and secure understanding of congruent triangles using the criteria ASA.

And we're gonna look at this by breaking it into two learning cycles, the first of which is to be able to identify congruence, the second is to justify congruence.

Let's make a start at identifying congruence.

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

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

So here on the screen there are three quadrilaterals and there are differences as well as similarities between them.

So if we focus on the square and the rectangle, you know that they both have four right angled interior angles and therefore they have the same angle, but they are different shapes, and the different shape means that their lengths are not the same and therefore they're not congruent.

If we then focus on the rectangle and the parallelogram, their edges are the same length and in the same order.

So they're in the relative position that they should be, but the angles are different.

So none of these are congruent to each other.

One way that you can check if two shapes are congruent is to use a piece of tracing paper.

This only works if you know that these are drawn to the same scale.

So if we grab a piece of tracing paper and trace over one of the shapes, we can then use that tracing paper to see if it matches up, if it is the same shape, so I can translate the tracing paper, I can move it on top.

Clearly that's not orientated correctly.

So I can rotate the sheet of paper and try and line it up.

Having lined it up, it doesn't look to be congruent.

The sketch on my tracing paper is not on top of exactly the one below, but maybe I need to reflect the sheet as well.

So if I turn that over and line it back up, we can see that actually this is even worse.

It's nowhere near close.

So these two shapes are not congruent.

So with piece of trade and paper, is a really quick and easy way of identifying if two things are congruent.

Here is a check, two shapes will always be congruent if they're reflections of each other, if they're rotations each other or if they're enlargements of each other.

So pause the video.

And when you've decided on what the correct solution to this is, press play.

Well, it's actually A and B.

So hopefully you've got at least one of those, but did you get both? So when it's a reflection or a rotation, the object and the image will always be congruent.

On an enlargement, it's not true that it will always be congruent.

There are some cases, some scale factors that make the object and image congruent, but in the most part they will be similar.

So two triangles are similar if they have two corresponding angles that are the same.

So similar means that they are enlargements of each other, they're scaled up or down.

Andeep says, does it not matter about the third angle of the triangle? So here I've told you that two corresponding angles are the same, then you know that the two triangles are similar.

Sofia says the third angles will be the same.

Why does Sofia know that the third angle will be the same? Andeep's recognised, "Oh, yes, because of the interior angles, that sum to 180 degrees." So by knowing two of the angles, you actually can calculate the third angle.

So we don't need to have all three angles known to be able to say that these two triangles are similar.

You just need to know that two of them are corresponding angles and are the same.

So Sofia says, "Exactly, we only need two." So two triangles are guaranteed to be congruent.

So remember that congruent means exactly the same size of the angles and the lengths, and one way we can guarantee they're congruent is if they have the same edge lengths.

And this is the criteria SSS, side-side-side.

So here we have a pair of congruent triangles.

We can see that they've both got a 6.

5 centimetre edge.

They've both got a 4.

9 centimetre edge, and they've both got a 3.

6 centimetre edge.

Remember, it doesn't matter if it is a rotation or a reflection, they will still be congruent.

Another way that we can guarantee congruence in triangles is if we have two edges and the angle between them that match or are the same.

And this is the criteria SAS.

So in this pair of triangles, both of them have a 2.

4 centimetre edge, then an 89 degree angle, and then a 5.

3 centimetre edge.

So side-angle-side, remember the angle must be between those two edges.

So if we know that two triangles are similar when they have two angles that are the same, how would we know that they are congruent or when would they be congruent? So just have a moment to think about that.

If you've got two angles that are the same, which actually means all three angles are the same, that makes them similar.

But when would they be congruent? If the size of the two triangles were the same as well, so if the edge lengths were the same as well as the angles, then they would be congruent.

So if we have an edge on the triangle, we would then be able to guarantee the congruence because that one edge length is going to fix the size of the triangle.

The edge length does not have to be between the two angles because if you think about it, if the third angle was marked here instead of the 99 degree angle, it's still the same triangle because I could calculate the 99 degree angle.

So if you do have the angle's either end of a line segment, then we tend to call it ASA.

If you had two angles in the triangle and a side that is not between them, then we might call it AAS.

But in essence, they're exactly the same thing because we can work out the third angle.

So this is another criteria for proving two triangles a congruent, and this is when you've got two angles and a side.

So how do we prove that triangle QPR is congruent to triangle TUV by either ASA or AAS? Have a moment to think.

So we can look at what we've got given here, both of these triangles have a 97 degree angle, and opposite that 97 degree angle is a side of 12.

3 centimetres.

So they are corresponding edges, they're in the same relative position.

We've got the 97 degree, so that's a corresponding angle.

However, the difference here is that we've got a 38 degree angle and a 45 degree angle.

So currently we do not have two corresponding angles that are the same and a corresponding edge, but we can calculate using angles inside of a triangle summing to 180.

We can calculate that the third angle on this right hand one, I could have done the third angle on the left hand triangle.

So the angle TVU is actually 38 degrees.

And now I do have two corresponding angles that are the same and a corresponding edge.

So because the edge is not between the 97 degrees and the 38 degrees, we would tend to say AAS, because if we sort of look at it, it's an angle, then an angle, and then a side.

However, if I had worked out angle QRP instead, that would be 45 degrees, and then I could make use of the 38 degrees and the 45 degrees and say ASA.

And that's because they ultimately are the same.

So here's a check for you.

Why are these two triangles not congruent by ASA? Pause the video, look at the diagrams, and then when you're ready to check whether you've come up with the right answer, press play.

The reason is because the two 6.

2 centimetre edges are not corresponding.

So if you look on triangle A, B, C, the 6.

2 centimetres is opposite the 74 degrees.

Whereas if you look on triangle DEF, it is opposite 47 degrees.

So we haven't got corresponding edge, we have corresponding angles, but we do not have a corresponding edge.

So on the first task of the lesson, I would like you for question one to sort these into congruent, similar or neither.

So some of them are congruent to each other, some of them are similar to each other, and some of them are neither congruent nor similar to any of the others.

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

Here's question two.

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

So that's using the definition of being congruent to be able to fill in the missing information in each diagram.

So part A and part B.

So pause the video and once you've completed question two, you've completed task A.

So when you press play, we'll go through our answers.

Okay, so question one, you were sorting these out.

So A, C and D are congruent to each other.

If you had grabbed a piece of tracing paper, traced over the parallelogram A, then by rotating or reflecting the tracing paper, it would fit exactly on top of C and D.

B and E, the parallelograms B and E, are similar to each other.

If you measure the angles within them, they are the same, but clearly one is larger than the other.

So they're not congruent.

However, they are similar.

And F is not congruent to, nor similar to any of the others.

So if you were to measure the angles in F, they are different to the angles in A, D, C, B and E.

And if you were to measure the edge lengths, they are also different.

Question two, given the fact that these were congruent triangles fill in the missing information.

So on the left hand triangle on part A, you needed to fill in the 13.

3 centimetre edge.

On the right hand triangle, we needed to fit in the 11 centimetre and the seven centimetre edge, and it does matter where you put them.

So on the left hand triangle, we can see that the seven centimetre edge is opposite the 32 degree angle, and so therefore the seven centimetres needs to be opposite the 32 degree angle on the other triangle.

The other information that you needed to fill in here was the 93 degree angle, which you needed to calculate using interior angles in a triangle summing to 180 degrees.

It's gonna be opposite the longest edge, which is 13.

3 centimetres because it's the largest angle.

Moving on to part B, on the left hand side triangle, you had to add the 5.

9 centimetre edge length and also the angle of 56 degrees.

The 56 degrees was not in the other triangle, but once again, you can calculate it because if you have two angles in a triangle, you can calculate the missing angle because of the fact that they summed to 180 degrees.

On the second triangle in that pair, you needed to add the 4.

5 centimetres and the 7.

1 centimetre edge.

Once again, we can see from the first triangle that 7.

1 centimetre is opposite the 85 degrees and therefore it needs to be in the same relative position in that other congruent triangle.

Really well done if you are successful through task A.

So the second learning cycle is to look at justifying congruence, especially looking at the criteria of ASA.

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

And common justifications that we'll use are defined properties of a shape, a shared or common edge or angle, and any stated or provided information in the question.

So given that A, B, C, D is a parallelogram in this diagram and E is the point of intersection of the diagonals, prove that triangle AED, and triangle BEC are congruent.

So that's what we're trying to do.

We're trying to prove congruence between those two triangles.

Laura says, the properties of a parallelogram give the information for proving the congruence.

And remember that's one of the main justifications is making use of any defined properties of a shape.

So let's have a look at how Laura does this.

So she says AD is equal to BC as they are opposite edges of the parallelogram.

So she's put hash marks to indicate their equality.

AD and BC are also parallel to each other, and AC and DB are transversal.

So we've added the feather marks to indicate parallel lines.

Laura says angle DAC is equal to angle ACB because they are equal alternate angles, we know they are equal because of the parallel lines.

Similarly, angle ADB is equal to angle DBC because they are equal alternate angles.

So we've got our arcs there to indicate the equality of the angles and therefore triangle, AED and triangle BEC are congruent by ASA.

Laura has managed to justify and prove that those two triangles are congruent despite the fact there are no numerical values on our diagram.

So which of the following would be valid justifications in a proof of congruence? So pause the video and read through the options.

Once you've decided on which ones are valid, then press play to check how you got on.

So A, B, and D are all valid.

Justification by the way the diagram looks is not a valid justification.

Unless you have been told explicitly that the diagram is drawn accurately, then you can't just measure things and say, "Oh, they are equal" because it may not be drawn to scale.

So make sure you are using either its stated given relationship, such as two line segments are equal or properties of shape which are defined for that shape or a shared edge or angle.

So we're onto the last task for you.

And question one is complete the proof.

So the proof has already been partly written and there are some blanks for you to fill in.

So triangle ACE is congruent triangle BFD is what you are trying to prove.

You were told that within that diagram and ACDF is a parallelogram and also the angle ACE is equal to the angle BFD.

So pause the video and complete the proof.

When you press play, we'll move on to the next question within this task.

So question two, here is a diagram that we are told angle MNO is equal to angle PON and also the NQ is equal to QO.

You are trying to prove that triangle MNO is congruent to triangle PNO.

So pause the video, make sure you're adding some information to the diagram, the information that you've been told, the information that can be inferred, and write down your proof including the justifications.

So press play when you're ready for question three.

Question three has six parts.

It's gonna take you a little while to get through this one.

So for each quadrilateral, so part A is a square, part B is a rectangle, part C is a kite, part D is an isosceles trapezium, part E is a rhombus, and part F is a parallelogram.

I'd like you to prove that the diagonal bisects the shape into two congruent triangles by ASA.

So using the criteria of ASA or AAS if possible.

So pause video, work through that question, and then when you press play, we'll go through our answers.

On question one, we've got the completion of a proof, so there were some missing parts.

So angle ACE equals angle, BFD, as, and I've written given in question, you may have said stated in question or something along those lines.

So that was told to us and so therefore we can assume that to be true.

AC is equal to DF.

If you have written FD, that is the same line segment, so you are also correct.

And our justification was that they are opposite edges on a parallelogram.

So we're using a property of a shape.

Then angle CAF is equal to angle CDF because they are opposite angles in a parallelogram.

So using the property of a shape again, we can state that two angles are equal, therefore triangle ACE is congruent triangle BFD by ASA.

We have proved and justified that two corresponding angles match and a side matches in the two triangles that we're trying to say are congruent.

Question two, there were lots of triangles within this diagram, so it's important that you actually had identified which ones you were trying to prove to be congruent.

So as NQ equals QO, that was stated and given in the question.

Then triangle NQO is an isosceles triangle.

So we can, because we know those two edges are equal, then it is an isosceles triangle.

Hence angle QNO is equal to angle QON.

The base angles are equal in an is isosceles.

So we're now using a property of a shape.

We've been able to say explicitly that is an is isosceles triangle so we can use the properties.

NO is a shared edge to both the triangles we're trying to prove to be congruent.

And angle MNO is equal to angle PON as given.

So we've now shown that both of these triangles have two angles and an edge that they share.

So therefore triangle MNO is congruent to triangle PNO by ASA, angle-side-angle.

Lastly, on question three, remember there are six parts to this.

So on part A it's a square.

So opposite edges of a square are parallel.

So angle BAC is equal to angle ACD as they are alternate angles in parallel lines.

So using the properties of a square, AC is a shared edge.

So that's a justification.

It's shared on both triangles so it therefore is the same length.

Angle DAC is equal to angle ACB as they are alternate angles in parallel lines.

We've already stated the opposite edges of a square are parallel, so we can make use of angles in parallel lines.

Therefore, triangle ADC and triangle ABC are congruent by ASA.

This is not the only way that you could have proved that these triangles are congruent by ASA.

So you may have done a proof similar to one of the ones I'm gonna go through in a moment for the square, as long as what you've stated is true mathematically with a correct justification, then your proof will be valid.

On B, the rectangle I've said opposite edges of a rectangle are both parallel and equal.

Hence FG equals EH and angle FGE is equal to angle GEH as they are alternate angles in parallel lines.

So the properties of the shapes I've then made use of them.

Angle GFE equals angle GHE and we know that to be 90 degrees as they are interior angles of a rectangle.

So once again, the properties of a rectangle are being used.

So I can say that triangle GFE and triangle EHG are congruent by ASA.

Both triangles have got a right angle, the opposite edges are equal, so that edge is the same and the angles are equal because of alternate angles being equal in parallel lines.

On the C, the kite, it's not actually possible to use the criteria ASA to prove those two triangles are congruent.

However, it is possible to use SSS and SAS to prove they're congruent instead.

So those two triangles are congruent.

The diagonal, that particular diagonal does bisect the kite into two congruent triangles.

But ASA is not a criteria that we could use.

On D, the isosceles trapezium.

triangle MPO and triangle MNO are not congruent.

So no criteria.

SSS, SAS or ASA will prove them to be congruent because they are not congruent.

E is a rhombus.

So you may have proved this the same way as the square proof, or you may have done it like this or you may have done a bit like the rectangle.

So here opposite edges of a rhombus are parallel.

So angle QRT is equal to angle RTS as they are alternate angles in parallel lines.

So you can see the arcs there to indicate two angles are the same.

QR equals ST as all edges are the same length on a rhombus.

So that's the property of a rhombus.

Angle TQR equals angle RST as opposite angles are equal in a rhombus.

Again, look for the arcs there.

And therefore triangle TQR and triangle TRS are congruent by ASA.

We've shown that there are two angles and a side that are corresponding angles and edges that are the same in both triangles.

Lastly, we have F, the parallelogram.

So opposite edges of a parallelogram are parallel.

So angle UXV equals angle XVW as they are alternate angles in parallel lines.

UX equals VW as opposite edges are equal in length on a parallelogram and angle XUV equals angle VWX as opposite angles are equal in a parallelogram.

Therefore, the two triangles are congruent by ASA.

Once again for all of the parts here that did prove to be congruent by ASA, your proof may have been slightly different to the one on my screen.

So just make sure that mathematically what you've written down is true and your justification is clear and well written.

So to summarise today's lesson about congruent triangles and ASA, two triangles can be proved to be congruent if two corresponding angles and a corresponding side are known to be the same.

It's really important that they are corresponding.

If the corresponding side is between the two known angles, then we say they are congruent by ASA, angle-side-angle.

However, if the corresponding side is not between the two known angles, we say they're congruent by AAS.

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