Lesson video

Hi, everyone, it's Ms. Jones here, and today's lesson is the second part about congruence and triangles.

Before we can begin, please make sure that you have a pen and some paper, you've got a nice quiet space to work if possible, and you've removed any distractions like phones or tablets or anything that's going to distract you from what we're doing today.

Once you've got that all sorted, please pause the video here, and we can begin.

Okay, hopefully you are ready now to start.

So thinking back to the previous lesson, all about congruence, and the rules for congruence, I want you to look at these two triangles here.

We've got three identical angles in both of the triangles, but we don't know how big the triangles are, we don't know the size of them.

We know the size of the angles, but we don't know the size of the triangle altogether.

If they each have a side length of eight centimetres, are they congruent? I would like you, using your equipment to construct an example and a non-example.

Pause the video here to have a go at that.

So hopefully everyone managed to find an example.

As long as the side lengths that are eight centimetres are in the same place, we now have angle side angle, angle side angle, which is one of our conditions for congruence.

So they will be congruent if we have the angle side angle in the same place.

However, if we do not, if we have our side between 70 and 50 degrees is eight centimetres, or between 70 and 60, or between 60 and 50, we can see we've got three different triangles.

They are similar, but not congruent, they are not identical.

So the order of the sides affects whether the triangles are congruent or similar.

If the eight centimetre side is in the same position, then the triangles will be congruent as they follow the angle side angle rule.

So we need to make sure we really consider the order of our angles and of our sides to ensure that they follow the rules for congruence that we've learned before.

So, as a reminder, we could have three sides, and obviously with that, the order doesn't really matter.

We could have a side angle and side, so it would have to be for example, this side and this side, and then the angle in between it, angle side angle, as we've just discussed.

And the final one is a right angle, the hypotenuse and the side.

What I'd like you to do now is pause the video to complete your independent task.

This is this task here where you are asked to identify the triangles are congruent.

So thinking about those rules that we've just discussed, and number two, again, thinking about the rules, but this time we've got a kite as our shape, and we've not been given any specific angles or side lengths except for one of them.

So pause the video to have a go at this.

But hopefully you manage to find that all of these triangles here are congruent except for the middle one.

When I was working this problem out, the first thing I would be doing is filling in those extra angles, because that is something that we can do for some of these images.

So when we have two angles, we can always find the missing angle, the third angle for a triangle.

So this one, I'm adding them together and subtracting from 180 degrees because that's what angles in a triangle sum to, and this one's going to be 63.

This one is going to be oops, 63 degrees.

This one is going to be 77 degrees.

Now remember, the angles being the same, it doesn't make them necessarily congruent, so we also need to check the order of the side lengths that we have.

So we've got here side angle side, which is the same here as side angle side.

So we only have one angle here, so we needed the two adjacent sides to be the same, that were either side of that angle.

Not to be the same, sorry, to be the same as another one, which halves those two sides there.

So these two are congruent here, sorry.

In this one we've got three angles, in this one we've got three angles, so we can have a look at this for an angle side angle.

So you've got 77 degrees, eight centimetres and 63 degrees.

And again, 77, eight, 63.

So these two are also congruent under the angle side, angle congruency rule.

Then we can have a look at these two here or even better, probably these two here, because we've got a 12 centimetres in both and we've also got two angles to there 'cause we've got 63, 12, 40, 63, 12, 40, so we've got another angle side angle here.

This triangle here, we've only got one angle and we don't have the sides between the 40 degree angle.

We have this side, but we do not have this side, and therefore we don't have a side angle side that we can use.

So this one, we don't actually know if it's gone, when it might be, but we don't know it, so we cannot say that that one is congruent to the other triangles, really well done if you managed to get that correct.

For the kite, we know you've got a 90 degree angle here and here and therefore here and here.

So we also know we've got some common sides to some of these triangles.

So first of all, I've noticed that triangle ABC, this one here, is congruent to ACD, this one on the opposite side there, because they have three sides are the same, this side is the same as this side, this side is the same as this side, and they've got a common side, which is equal.

So those two triangles are common, they're congruent.

We also have two other triangles, BCE, these little two triangles on the top and CED.

Again, we've got a side that's common, the same, and we've got a side that's common to both of them, this one here, and we've got an angle here.

Those angles are equal 'cause we've bisected that kite straight through there, and there's other reasons you might have found as well.

You could have thought about Pythagoras' theorem, about how if this two sides are equal, we'd have the same square root.

So these two sides are going to be equal as well, lots of things you could have thought about.

So those two are congruent.

Then we've got ABE, so this triangle larger triangle here, and AED, so the two bottom triangles.

And again, lots of different things we could have discussed and decided, but again, this line here goes through it, bisects.

This angle here, the angle BAD, and so we've got two equal angles there, we've got equal sides here, and we've got an equal side here.

That's common to both, so that's why that one is congruent as well.

But again, lots of different reasons you could have found, well done if you managed to get any of those, brilliant job.

So now is the explore task.

I want to know how many different triangles you can draw with the following features.

So question one, an angle of 30 degrees, a side of five centimetres, and a side of three centimetres in that order.

So I want to know what would happen if you've got an angle of 30 degrees and then a side of five, and then a side of three in that order.

Number two, an angle of 30 degrees, then a side of five and a side of two in that order.

And then finally an angle of 30 degrees, a side of five and a side of seven in that order.

So for example, as a quick sketch, the triangle I'm thinking of for number one, your angle of 30 degrees could be here, and then it needs to be the side next to it, of five centimetres, and then another side next to that one, of three centimetres.

It's not this side of five and this side of three, 'cause otherwise we would have said a side of five centimetres, an angle of 30 degrees and a side of three centimetres, it needs to be in the order that it says there.

So once you've had a go at drawing as many triangles as you can for each of those three conditions, try some other lengths and see what you notice.

And hopefully you may be able to make a conjecture about this, and notice some patterns.

Pause the video now to have a go at that.

So for the first example, there were actually two different triangles that you could make.

I'm sure most of you managed to get this triangle here, but actually this triangle here also works, but quite often we imagine the triangles as the top one, that's what we imagine a triangle to look like.

But in fact, it could have also looked like the bottom one as well, so we wonder if you've got both of those.

For number two, it was actually impossible to draw that triangle, there were no solutions because this side was just too short, we could not much up and connect it with the five centimetre side and keep this angle as 30 degrees.

The third one just had one solution, there was no way we could have got that seven centimetres down here, so there was only one solution for number three.

And hopefully if you tried a few different lengths and angles, you would have noticed that generally this works for the following rules.

If the second side, so five was the first side, three was the second side, if that second side is smaller than the first, but also larger than half, there's going to be two solutions to it because it can fit here, but it also can fit here.

If the second side is less than half of the first side, so two is less than half of five, there are no solutions, it just cannot meet up.

When you think about your circle that you'd make, if you were to move that smaller side length around, it's never going to be able to meet up with this one, wherever it goes.

And for the third one, if your second side is larger than the first side, you're going to have one solution, as almost the opposite to the first one.

Really well done if you managed to get those answers, and if you managed to come up with some more conjectures, I'm sure there are loads that I haven't even thought of, so you've done absolutely brilliantly today, well done.

There was just some quite tricky things, there was lots of practise of construction, really, really amazing job, good.

If you would like to share your work with Oak National, then please ask your parent or carer to share your work on Twitter, tagging OakNational and #LearnwithOak.

Amazing job today, and well done again.