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.

Today's learning outcome is to be able to understand and use the criteria by which triangles are congruent and in particular RHS.

So on the slide, there are some keywords that I'll be using during the lesson that you need to be confident with.

You will have learned them before, so it's not a new keyword, but you may wish to pause the video and read the definitions just to ensure you are feeling confident as we start the lesson.

So this lesson about check and securing understanding of congruent triangles by RHS.

And we're gonna do that by breaking it into two learning cycles.

The first of which is about identifying congruence and the other is justifying congruence.

So let's make a start on identifying congruence.

So if we have a right-angled triangle with a hypotenuse of five centimetres, so we've got a diagram indicating this here.

Remember hypotenuse is the longest edge opposite the 90 degree angle.

We have a 90 degree angle because it is a right-angled triangle.

Then Aisha says A and B can be any value.

So A and B are labelled to be the other two edges of this right-angled triangle.

Is Aisha correct? Do you agree with her that they can be any value? So she's not quite correct.

There is many values that A and B can take, but they need to be values that make Pythagoras theorem hold.

So all of these right-angled triangles have a hypotenuse of five centimetres and so A and B are taken different values.

But the important thing is that A and B and the five make Pythagoras theorem hold.

So just a reminder on that.

If we do three squared plus four squared, it is equal to five squared.

Similarly, two squared plus root 21 squared is equal to five squared.

And lastly, one squared plus two root six all squared equals five squared.

So this is just three examples, but there are many more.

So are these triangles congruent to each other? Well, no.

The only thing they have in common is the right angle and the hypotenuse of five.

So to be congruent, they need to have all the same edge lengths and all the same angles.

A right-angled triangle has a hypotenuse of five centimetres and a shorter edge of three centimetres.

So now the A is the only unknown edge length.

Aisha says so A has to be a value that makes Pythagoras' theorem hold for it to be a right-angled triangle.

So how many values are there for A? There's just the one, the length of the third side is four.

The only value for A that will make Pythagoras' theorem hold with a shorter edge of three and a hypothesis of five is when A is equal to four.

So this implies that the third edge has to be four and this is a unique triangle regardless of the orientation.

These three triangles are congruent to each other.

If we were to cut them out and as little tiles, they would fit exactly on top of each other.

We might need to rotate them or to reflect them, but they are congruent.

So if a triangle has the three edges, three centimetres, four centimetres, and five centimetres, what type of triangle is it? Well, it's definitely scalene as the three edge lengths are different and the sum of the squares of the two shorter sides is equal to the square of the longest side so it's a right-angled triangle as well.

So one way to show that a triangle is right-angled is to see if Pythagoras' theorem holds.

And with these three edge lengths it does.

So it's a right-angled scalene triangle.

So to guarantee congruence between two right-angled triangles, knowing only the hypotenuse is not sufficient and we can see the two examples we had previously that just because it's a right-angled triangle with a hypotenuse of five does not guarantee congruence.

These two triangles are not congruent to each other.

Knowing the hypotenuse and a shorter edge is sufficient because of the special relationship between the sides, which is Pythagoras' theorem.

So it is therefore not necessary to know all three edges if you know that the triangle is right-angled.

And this is a special case of proving congruence by knowing two sides and an angle that holds for right-angled triangles.

So we know it as RHS and that stands for right-angled hypotenuse side.

So if we have two right-angled triangles, you know them to be a right-angled triangle and their hypotenuse is the same length and a shorter edge is the same length, then you are guaranteed that they are congruent because of the special relationship between the sides, which is Pythagoras' theorem.

So with these two triangles here, the special relationship Pythagoras' theorem does hold and we can prove that they are congruent because they are both right-angled, they both have a hypotenuse of 17 metres and they both have another edge of 15 metres.

We could use Pythagoras' theorem to work out what that third edge would be.

We don't need to to prove they are congruent because by Pythagoras' theorem, because this is a right-angled triangle, we have the right-angled, the hypotenuse and a side, they are congruent.

So are these two triangles congruent? Pause the video and when you've made your decision on that press play to check.

So yeah, we can say that these are congruent by RHS.

They are labelled to both be right-angled triangles.

We can use that notation.

We know that means 90 degrees of right-angled.

The hypotenuse, which is the longest edge, both of which are 17 centimetres, and a shorter edge is eight centimetres.

So because we have the right-angled, the hypotenuse and the side to be the same in both triangles, then we can say that these are congruent by the criteria RHS.

Task A has the three questions on the screen here.

So question one is missing some words in that statement.

Question two, there is also a word missing.

And question three is given the triangles are congruent, fill the blanks.

So pause the video whilst you work through those three questions.

When you finish those three questions and you wanna check your answers, press play and we will go through them.

So all answers are on the screen here.

Question one was congruent triangles have the same angles and the same edges.

If you had put same edges and the same angles, then that's fine and is also correct.

So that's the definition of being congruent for all shapes.

But here I've written for congruent triangles.

Question two, to prove two right-angled triangles are congruent.

They must have the same hypotenuse and short edge.

So this is actually a special case of SAS where the angle is not between the two sides and it only works for right-angled triangles.

And question three, given that triangle ABC, and triangle DEF are congruent, fill the blanks.

So if you know they are congruent, then the edges will be the same.

So the 25 centimetres is the hypotenuse and the shorter edge on triangle ABC will be 24 centimetres.

You could use Pythagoras' theorem to work out the length of the third edge, but you do not need to have the length of the third edge if you have the hypotenuse and a short edge in a right-angled triangle.

So the second learning cycle is justifying congruence, and we are focusing on the criteria of RHS.

So in a proof of congruence, regardless of which criteria you use, the justifications are as important as your statement or your claim.

And the common justifications that we'll be using are defined properties of a shape.

So if you are told that it is a parallelogram, thinking about all of the properties of a parallelogram, if you know it's as an equilateral triangle, what are the properties of an equilateral triangle that you can make use of? Another common justification is a shared or common edge or maybe a shared or common angle.

So if there is a shared edge, that length will be the same for both triangles.

And the lastly, any stated or provided information.

So sometimes you might be told a relationship between two line segments that they're equal, for example, or given a ratio of two line segments and you can make use of that within your proof.

So if it's been given to you, it's been stated and provided, then you can assume it to be true and make use of it.

So given that this triangle ABC is equilateral and AD is perpendicular to BC, prove that triangle ABD and triangle ADC are congruent.

So this is our diagram that we have and we've got some information provided.

So from that information we can say that AB equals AC, angle ADB equals angle ADC, which happens to be 90 degrees and therefore triangle ABD and triangle ADC are congruent.

So this is not a proof of congruence and the reason it isn't a proof of congruence is because it has no justifications to why those mathematical statements are true.

So they are true, but there isn't any justification.

So in your proof, you must make sure you justify why you are claiming or stating a mathematical statement.

So this one would be better.

As triangle ABC is equilateral, AB equals AC.

So our justification is using the given fact that that is an equilateral triangle and the property of an equilateral triangle is that its edge lengths are equal.

As AD is perpendicular to BC then angle ADB is equal to angled ADC, which is 90 degrees.

So we're using the definition we were told they're perpendicular.

And what does perpendicular mean? Well perpendicular means that they meet at a right-angle.

So we can then say that those two angles are equal because of this idea that they are perpendicular, they're both 90 degrees.

AD is a shared edge.

So that line segment from A to D is an edge on both triangle ABD and triangle ADC.

So it's a shared edge and therefore triangle ABD and triangle ADC are congruent by RHS.

We have shown that there is a right-angled triangle for both, they're both right-angled triangles.

We have shown that the hypotenuse of both of these right-angled triangles is the same and we've also shown that one of the shorter sides is the same.

And so we have to conclude the proof and state the criteria that has been used.

But by proving that these two triangles are congruent, then other relationships or connections can also be stated.

So it might be sometimes that you're not actually asked to prove that triangles are congruent, but instead you're asked to prove a relationship.

And the way that you can prove the relationship is by proving that triangles are congruent.

So if those two triangles are congruent, which we now know they are, then we can also say that BD equals DC because if they are congruent triangles, then all the edges and all the angles are the same.

So BD is an edge of one of the triangles, and DC is the corresponding edge on the other triangle.

So they must be equal in length.

We can also say that angle BAD and angle DAC are equal.

Once again because the triangles are proved to be congruent, then the interior angles will be the same.

So here's a quick check for you.

ABCD is a kite.

Prove that triangle DAC, and triangle ABC are congruent by RHS.

So the proof is there, you just need to fill in the blanks.

So pause the video, read through it a couple of times I would suggest think about the properties of a kite and how that's gonna help you.

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

So angle ADC is equal to angle ABC, and they are both 90 degrees, they're given in the diagram.

So we are not assuming that they are 90 degrees because they look like a right-angle.

They are marked to be a right-angle.

So we can make use of that.

AC is the hypotenuse of both triangles.

So AC is a shared edge, it's the diagonal across the kite, and because of that right-angle, that edge is opposite the right-angle and therefore the hypotenuse.

DC is equal to BC as they are adjacent sides on a kite.

So you may have written CB instead of BC, it's the same line segment so that is exactly the same.

So it's a property of a kite, is that we have pairs of equal edges and they're adjacent to each other.

Hence triangle DAC and triangle ABC are congruent by RHS.

So we're now on the last task of the lesson, which is justifying congruence and making use of the criteria RHS.

So given that ABCD is a rectangle, prove the triangle ABC and triangle ADC are congruent by RHS.

So pause the video as you work through this question.

Make sure you're not only writing mathematical statements that are true, you are justifying them, concluding with a statement that tells us that you have proved that the two triangles are congruent by RHS.

Press play when you're ready to move to the next question.

So this question, given that ABC is a triangle, DF equals EF, DF is perpendicular to AB, and EF is perpendicular to AC.

Prove that angle BAF is equal to angle FAC.

So I'm not gonna say too much on this one.

I want you to rethink about how you are going to prove those angles are equal.

Pause the video, work through that, and when you have finished with question two, you've finished with the task, press play and we will go through our answers.

Question one, you were proving congruence of the two triangles by RHS.

So angle ABC and angle ADC are both 90 degrees as it is a rectangle.

So that justification is the property of a shape.

We were told it is a rectangle so we can use the properties.

AC is a shared edge and is the hypotenuse of each triangle.

It's opposite the right-angle in both cases.

AB equals CD as they're opposites sides of a rectangle.

So again, the property of the shape, hence triangle ABC and triangle DAC are congruent by RHS.

A couple of things just to mention here.

The order in which I've stated it so I did the right angle, then I did the hypotenuse, and then I did the side.

I've done that because I tried to do it in the order of the acronym RHS, just to sort of help myself.

The side I chose was AB and CD.

You may have chose to do AD and BC, they would also be equal because they are opposite edges on a rectangle.

So if you chose to use a different shorter edge, that's not a problem as long as you are justifying it correctly.

Now moving on to question two.

So question two was not actually about proving congruence.

That's not what the question asked you to do.

The question was asking you to prove that there was an equality of two angles.

How we were gonna do that was by using the congruence.

And then a consequence of that congruence is that the angles would be equal.

So prove that triangle ADF and triangle AEF are congruent.

And then that will imply that the angles are equal because if they are congruent, the edges and the angles are the same.

So angle ADF, and angle AEF are both 90 degrees due to the perpendicularity.

You were told in the question that DF is perpendicular to AB and EF is perpendicular to AC.

So we can make use of that.

And that tells us that there will be a 90 degree angle or a right-angle in both triangles.

DF equals EF, we were told that, that was a given fact so we can assume that to be true.

And AF is the hypotenuse of both triangle ADF and triangle AEF.

It's the hypotenuse because it is opposite those right-angles.

It's a shared edge between the two triangles so therefore we know it is the same length, it is exactly the same edge.

So hence the triangle ADF and the triangle AEF are congruent by RHS.

So as we know, the two triangles are congruent, then that means all of their edges will match.

They have the same edges and they have the same interior angles.

So angle BAF and angle FAC are equal.

We have managed to prove they are equal by proving that the triangles are congruent.

Really well done if you managed to get through that by yourself before this feedback part.

So in summary of today's lesson, for a right-angled triangle only, so this criteria only is true for right-angled triangles.

You can prove congruence by using the hypotenuse and one other side, and this is the criteria that we know as RHS where R stands for right-angle.

hypotenuse is the H, and side is the other side.

So right-angled hypotenuse and side, it's a criteria for proving congruence of two triangles, but only if they are right-angled triangles.

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