Lesson video

Hi, I'm Mr. Bond, and in this lesson, we're going to learn how to prove that two triangles are congruent.

In this lesson, we'll be proving that triangles are congruent.

So first, let's recap the conditions under which two triangles are congruent.

They're congruent if they satisfy one of the following conditions.

SAS, you might remember this stood for side-angle-side.

And this is where two sides and the included angle are the same in two triangles.

ASA, or angle-side-angle, and this is where two angles and the included side are the same in two triangles.

SSS, or side-side-side, and this is where all three sides are the same length in each triangle.

And then, finally, RHS, right angle-hypotenuse-side, and this is where two right-angled triangles have the same hypotenuse and one shorter side the same length.

Here's a question for you to try so that you can recap the conditions of congruency.

Pause the video to complete the task and resume the video when you're finished.

Here's the answer.

For this question, we can use the fact that we know two of the angles to find the third angle by considering that angles in a triangle sum to 180 degrees.

Once we've done this, we can, again, compare the angles and the included side.

And once we notice that they're the same, we can say that that congruent by angle-side-angle.

Here's something for you to have a think about.

Which condition should you use to show that two right-angled triangles are congruent? Pause the video to have a think, and resume the video when you're finished.

We can use any condition, not just right angle-hypotenuse-side.

Here's another question for you to try.

Again, pause the video to complete your task and resume the video when you're finished.

Here's the answer.

For both triangles, we know an angle because we know the right-angled triangles.

And we know two sides.

But can we say their congruent by side-angle-side? Well, no, because the right angle isn't the included angle for either triangle.

But because we have a right-angled triangle, we can use Pythagoras to find the length of the missing side in each triangle.

But then, we can see that they're congruent because all three sides are the same length.

But that wasn't the only way we could do it.

We could also say they're congruent by right angle-hypotenuse-side or by side-angle-side.

Here's another example.

CDEF is a parallelogram.

We want to prove that triangles CDF and DEF are congruent.

Since we know that this is a parallelogram, we know that opposite angles in a parallelogram are equal.

So that means that these two angles are equal.

So we can write that angle DCF is equal to angle FED.

We also know that opposite sides in a parallelogram are equal in length.

So we can write that CD is equal to EF and DE is equal to CF.

So this means, yes, the triangles are congruent.

And the condition we used was side-angle-side.

I'd like you to have a think.

Are there any other ways to prove the triangles are congruent? Here's another question for you to try.

Pause the video to complete your task and resume the video when you've finished.

Here's the answer.

BD, the diagonal of the rectangle, splits the rectangle into two right angle triangles.

Because the shape is a rectangle, we know that opposite sides are the same length.

We also know that BD, the shared side in both triangles, is the same length.

So we can use right angle-hypotenuse-side because we know they're both right-angled, the hypotenuse is of the same length, both BD, and AD is equal to the length BC.

We could have done this in other ways.

And here's the final question for this lesson.

Again, pause the video to complete your task and resume the video when you're finished.

Here's the answer.

You can see that I've added quite a lot of notation from the information that was given in the question.

I've got the arrows to show that AB and CD are parallel, which is given in the question.

M is the midpoint of BD.

And this means that BM is the same length as MD, so I've marked that on with a Hatch mark.

And I've also marked on that the angles ABM and MDC are equal.

And I know that because alternate angles are equal.

Now that we know that, which condition have I used? Well, I've used angle-side-angle because I know that the angles AMB and DMC are equal because vertically opposite angles are equal.

Like I mentioned before, the angles ABM and MDC are equal because they're alternate to each other.

And also, I know the included sides, BM and MD are also equal in length because M's the midpoint.

That's all for this lesson.

Thanks for watching.