Lesson video

Hi there.

Welcome to today's lesson.

My name is Mr. Peters, and in this lesson today we're gonna be thinking about extending our understanding with regards to angles in triangles and thinking about how we can use this knowledge to help us solve problems. If you're ready to get started, then let's get going.

So, but in this lesson today, you should be able to say that I know that the angles in a triangle sum to 180 degrees.

In this lesson today, we've got a few key words we're gonna be referring to throughout.

I'll have a go at saying them first and then you can repeat them after me.

The first word is quadrilateral.

Your turn.

The second word is triangle.

Your turn.

And the final word is sum.

Your turn.

Let's remind ourselves of the meaning of these words.

A quadrilateral is a polygon with four straight sides and four vertices.

A triangle is also a polygon with three straight sides though, and three vertices.

And the sum is the result of adding two or more numbers together.

In this lesson today, we've got two learning cycles.

The first cycle, we'll think about angles in the triangle and the second cycle we'll think about identifying missing angles.

If you're ready to get started, let's get going.

Throughout this lesson today, we've got both Lucas and Aisha.

As always, they'll be sharing their thinking as well as any questions that they have to support us in developing our thinking and our reasoning throughout the lesson.

Our lesson starts here with Lucas and Aisha, and at the moment they're cutting up some quadrilaterals.

Lucas has noticed that when he cuts up a quadrilateral from one vertex to another vertex, he always creates two triangles.

Let's have a look at an example.

That's really interesting, isn't it? Aisha's now wondering if that's the same for lots of other different quadrilaterals as well.

Here's a different quadrilateral.

Again, if we cuts across from one vertex to another vertex, we can see that we've also created two triangles again, haven't we? What about this example? This is a slightly different quadrilateral this time, isn't it? Let's cut across from one vertex to another.

Oh look, another two triangles as well.

Hmm.

That's interesting, isn't it? In the previous example, the two triangles actually look pretty similar, didn't they? Whereas these two triangles look completely different, didn't they? So, here are the three examples we had so far.

Here are the three quadrilaterals that we cut into two parts, and if that means each time we cut a quadrilateral from one vertex to another, we can create two triangles, so what does that mean for all of the angles in the quadrilateral then when the quadrilateral is cut into two parts? Okay, well, Aisha's reminded us from prior learning that we know that the sum of the angles in the quadrilaterals come to 360 degrees.

So, let's try and use that to help us a little bit more.

Hmm, let's start with a square then.

We know that a square has four lots of 90 degrees, which is equal to 360 degrees.

That's because a square has four right angles, doesn't it? And each right angle has a value of 90 degrees.

Okay, and now if we take one of the vertices to cut through to another vertex like this, we can now see that actually each of those angles have been cut into half, haven't they? So, each one of these angles would be equivalent to 45 degrees.

There we go.

We've now got two lots of 45 degrees at each one of these angles.

Therefore, when thinking about finding the sum of the angles in a triangle, we can now see that we've got two triangles here.

One of the angles is 90 degrees and the other two angles are 45 degrees.

So we can find the sum of the total of the angles here to be 90 degrees plus 45 degrees, plus another 45 degrees, which is equal to 180 degrees.

Yeah, and that's exactly right, Aisha.

That's the same as the size of the angles on a straight line, isn't it? Great connection to make there.

Well done you.

Yep, and that's a good idea, Lucas.

I think we should rip off the vertices here and place them on a straight line to see if that works, shall we? Well, here we go.

Now we've ripped them off.

We can place them on a straight line together like this, can't we? There we go.

Where all the points meet.

Yep, and now you can see that we've created the angle on a straight line here with 90, 45, and another 45 degree angle.

All together that's equivalent to 180 degrees.

Okay, well let's think about this from the point of view of a parallelogram.

We know again that the opposite angles in a parallelogram are equivalent to each other.

Now, when we cut a parallelogram into two parts, we can see here that the angles aren't exactly the same as they were in the previous example.

And that's because a parallelogram, if we were to fold it along this line as well, the angles would not exactly overlay on top of each other to be exactly the same, unlike the previous example where we could fold the shape into exactly half over the line that we used to cut it in half, which would then show that the angles were exactly the same size.

So we can say this angle here is roughly 40 degrees by estimating the size of it.

If we say that this angle here is 40 degrees, then the remaining part of the angle must be 35 degrees because the total size of the angle, at that vertex, it was in fact 75 degrees.

So here is our 35 degrees now then.

Having done that now we can work out the missing angle on the left-hand triangle, can't we? We know that the sum of the angles must be 180 degrees, and we can subtract the known angles.

So we can subtract 105 degrees, and then again subtract another 40 degrees.

And then by subtracting both of those known angles, that will leave us with an angle size of 35 degrees.

Once again, if we know this angle is 35 degrees, then because the total angle at that vertex was 75 degrees, then the missing angle on the right-hand triangle must also be 40 degrees.

Hmm.

So what do you notice here then? That's right, when the parallelogram is divided into two equal triangles, then both triangles have exactly the same angles, don't they? They're just placed in a slightly different position.

Let's have a look here at a trapezium now.

There are two sets of angles that are the same.

We know these two angles here are the same, and we know that these two angles here are the same.

Aisha's thinking back to what happened to the parallelogram, and she thinks the same is going to apply here to the trapezium as well.

Let's have a look, shall we? If we were to divide this shape into two triangles through these two vertices here, then we can estimate the size of this angle here.

We know that the total size of this angle at the vertices here was 120, and we can estimate that this angle here is roughly 80 degrees.

So if this one here is 80, then that means the remaining part of that angle must be 40 degrees.

Once again, we can now work out the missing angle in the bottom triangle here.

If we know that the sum of the angles in the triangle must add up to 180 degrees, then we can say that 80 degrees plus 60 degrees is 140 degrees.

Therefore the angle in the bottom triangle must be 40 degrees, and therefore the missing angle of the top triangle must be 20 degrees, because the sum of this angle was 60 degrees, wasn't it? There we go.

So, let's have a look at the examples we've looked at so far and remind ourselves, what can we notice about each of these examples are all together? That's right, Lucas, all of the triangles here that we've looked at, the sum of the angles of all of these triangles sum to 180 degrees, don't they? Yeah.

And great thinking, Aisha.

Aisha thinks that's because all quadrilaterals can be composed of two triangles, and therefore if the sum of the angles in one triangle is 180 degrees and then if you add the sum of the angles in another triangle to that, that would give us the 360 degrees altogether, wouldn't it, which is the sum of the angles in a quadrilateral.

Well done if you managed to recognise that for yourself too.

Okay, time for you to check your understanding now.

The angles in a triangle sum to a, b, c, or d? Take a moment to have a think.

That's right.

It's b, isn't it? The angles in a triangle sum to 180 degrees, don't they? And can you tick the expressions that could represent the angles in a triangle? Take a moment for yourself to have a think.

That's right.

It's a and b, isn't it? And why could these two expressions represent the angles in a triangle? That's right.

The angles in a triangle sum to 180 degrees, and if you add the numbers in each of the expressions up together, that'd be equal to 180 degrees each time.

Well done if you managed to get that too.

Okay, onto our first practise for today then.

Can you look at the angles that have been provided? Can you combine three of them to make a triangle? It would be good if you could go on to sketch your ideas out each time.

Here's an example that Lucas created earlier on.

There we go.

You might like to extend this onto Lucas's challenge here, where he says, can you create a triangle which has two or three of the same sized angles as well? Good luck with that task, and I'll see you back here shortly.

Okay, let's see how we could have tackled this then.

Here are three examples of triangles that we were able to create.

You can see in the first two examples, we've got different sized angles at the vertices in these triangles.

And then finally for the last one, we did manage to create one triangle with all the same size angles.

Of course, if the sum of the angles in a triangle add up to 180, for all of the angles to be the same size, then we need to divide it into three parts, and those parts would need to be equal.

So therefore we can say that three lots of 60 degrees would be equivalent to 180 degrees.

So a triangle which has all three angles at the same size, all of those angles must be 60 degrees each time.

Well done if you managed to come up with some of your own.

Okay, moving on to cycle two now, identifying missing angles.

So, let's have a look here now then.

We've got a triangle here and we've got two known angles, and we've got one unknown angle, which is named angle a.

Lucas is saying that now we know that the angles in a triangle sum to 180 degrees, we can find the size of the missing angle a, can't we? Aisha is reminding us that this here is a scalene triangle.

A scalene triangle is a triangle where all the sides have a different length, and Aisha's come up with a good idea here.

She's begun to estimate what she thinks the size of the missing angle will be.

Just by looking at it first of all, she's recognised that angle a is roughly the size of a right angle.

So she's expecting our answer to be around the 90 degree angle mark.

We can work out the size of the missing angle by writing a missing addend equation.

We know two of the angles, so these would represent two of the known addends.

And we also know that the sum, the sum would be equal to 180 degrees, so we can write an equation like this.

45 degrees plus 38 degrees plus angle a would be equal to 180 degrees.

Of course, we've got a missing addend problem here, so we can write this as a subtraction equation too.

It might be helpful now to sum the two angles that we know together.

So we can do this by adding the 45 and the 38 together.

So 45 degrees plus 38 degrees is equivalent to 83 degrees.

So we can say 83 degrees now, plus the missing angle is equal to 180 degrees.

Now of course, we've got a missing addend problem here, so we can use subtraction to help us solve this.

We can do 180 degrees minus 83 degrees, and that gives us a total of 97 degrees.

So the missing angle is in fact 97 degrees, and as Aisha estimated, she thought it was gonna be roughly around the 90 degree angle size.

So we can say that we think this would be a suitable correct answer for our problem.

Let's have a look at another example here then.

Here we've got a triangle.

And what do you notice about this triangle? That's right.

It's got two extra little lines on it, hasn't it? And Lucas is now asking, what do these two extra little lines mean? Well, Aisha is right.

It means that the two sides with the two little lines on are exactly the same in length.

If we have a triangle which has two sides that are exactly the same in length, then this is known as an isosceles triangle, which we should know from previous learning.

So let's crack on and work out the size of this missing angle here then.

We know we can write this as an equation where we have two known addends, and we know the sum of the angles, don't we? So each of the known angles would represent the addends.

So we've got 64 degrees plus 58 degrees, plus the missing angle, which would be b, and that would be all equal to 180 degrees.

Once again, we could sum the known angles together.

So 64 degrees plus 58 degrees is equal to 122 degrees.

So now our equation becomes 122 degrees plus the missing angle is equal to 180 degrees.

And once again, we can now use a subtraction equation here because we have a missing addend.

So we can say 180 degrees minus 122 degrees is equal to the missing angle, which in this case is 58 degrees.

There we go.

Take a quick moment.

What do you notice here? That's right.

So not only does an isosceles triangle have two sides that are the same length, we can see here that it also has two angles that are the same size.

So we can say that the angles opposite two sides of the same length in an isosceles triangle would be equal in size.

Okay, and what about this example here? What do you notice? That's right, Lucas.

There are three little lines this time, aren't there? So that must mean that all of these sides are the same in length.

And of course, a triangle which has three sides that are all the same in length would be known as an equilateral triangle.

Hmm, let's work out the missing angle then.

Ah, Aisha's not even calculated it.

She says it's 60 degrees.

Why is that, Aisha? Oh yeah, of course.

We know that an equilateral triangle has three sides that are all the same length and three angles that would also be all the same size, so therefore the missing angle would be 60 degrees, wouldn't it? Yeah, and you're exactly right there, Lucas.

180 degrees divided into three equal parts or three equal angles in this example would be 60 degrees, wouldn't it? Okay, and one more for us to have a think about here.

What do you notice this time? Well, we can see here that we have an equilateral triangle, don't we? And then we've got a scalene triangle attached to it on the right-hand side.

And that's interesting, Lucas.

You think you can work out the size of missing angle d, do you? Ah, you are saying that because we know two of the angles in the scalene triangle, we can sum these together and then subtract that from 180 to find the missing angle d.

Let's have a go at doing that first of all then.

We know we can write an equation which is 32 degrees plus 28 degrees plus the missing angle is equal to 180 degrees.

Now we can sum these two addends together.

That gives us 60 degrees.

So now we can say 60 degrees plus angle d would be equivalent to 180 degrees, and therefore we can now write this as a subtraction equation.

180 degrees minus 60 degrees is equal to 120 degrees.

But Aisha thinks there's a quicker away.

I wonder what your thinking was, Aisha? Well, Aisha's recognised that the angle on the left, of course was an equilateral triangle, and therefore all of the angles in the equilateral triangle must be 60 degrees.

Ah, and of course at the bottom here we can see that where these two angles meet, it creates the angles on a straight line, doesn't it? And we know that the angles on a straight line sum to 180 degrees, don't they? So we can just minus the 60 degrees from the equilateral triangle from 180 degrees, and that would give us the missing angle d as well.

So that would be 120 degrees.

Great thinking, Aisha.

Two very different strategies to solving this problem here.

It just comes down to which one you think was the most efficient.

Okay, time for you to check your understanding again now then.

Can you find the missing angle e? Take a moment to have a think.

Okay, in order to do this, we can add the three angles together, and we know that would be equal to 180 degrees.

We know two of the angles, so we can sum these two angles together now.

That would give us 145 degrees, and of course we can now subtract that 145 degrees from 180 degrees, which in total would leave us with 35 degrees.

So the missing angle e is 35 degrees.

Well done if you managed to get that.

And here's another example.

Can you find missing angle f here? Well, you might be thinking, I've only been given one angle here and I don't know the angle at the top here at the moment, so that's not gonna be easy to calculate.

That's right.

It's an isosceles triangle, isn't it? And we know that because two of the lines have exactly the same length, which is indicated by those two little lines.

And we know that because two of the sides are exactly the same length, which is indicated by those two little lines.

If it's an isosceles triangle, then we also know that the two angles opposite the equal sides are going to be also the same size and angle, which is exactly what Lucas has now just pointed out.

So we can say that f is also going to be 45 degrees.

Well done if you managed to spot that for yourself too.

Okay, onto our final tasks for today then.

What I'd like to do is have a go here at calculating the missing angles for a, b, and c.

And you're also gonna do the same here for d, e, and f.

And then once you've done that, I'd like to have a go at working out all of the internal angles in the shapes below.

Good luck with that, and I'll see you back here shortly.

Okay, welcome back.

Let's work through these then here.

So from the first example, we've got a right angle, which we know would be 90 degrees, and we've also got 48 degrees, so a right angle plus 48 degrees.

Therefore, adding the two known angles together and then subtracting that from 180 degrees would leave us with 42 degrees.

So angle a is 42 degrees.

For b, we've got two known angles, so again, we can sum these together and subtract that from 180 degrees.

That would give us the size of angle b, which in this case is 64 degrees.

And then for c, once again, we've got two known angles.

We can sum these together and subtract that from 180 degrees.

That would be equal to 95 degrees.

So c is equal to 95 degrees.

For d, e, and f, let's have a look.

Well, the first shape you may have noticed is an isosceles triangle.

Therefore, we know that the two angles opposite each other from the equal sides would be the same.

One of the angles is 70, so the other angle will also be 70 degrees.

So d is equal to 70 degrees.

Hmm, triangle e is in fact an equilateral triangle.

We know that all the angles in an equilateral triangle are equal to one another.

So angle e in this case would also be 60 degrees.

And then finally, for f here.

Hmm, we're trying to find actually an external angle here.

This angle is on the outside of the triangle.

I wonder how we could tackle this.

Well, we can see that angle f is a part of an angle on a straight line.

So if we can work out the interior angle, then we can work out the exterior angle of angle f.

So to work out the interior angle, we can sum the known angles together and subtract that from 180 degrees.

That means that angle there would be 67 degrees.

Now we know that 67 degrees and f would be equal to 180 degrees because that's the angle on a straight line.

So we can do 180 degrees minus 67 degrees, and that would leave us with the size of angle f, which in this case is 113 degrees.

Well done if you managed to get all of those too.

Okay, and now finally, working out all the internal angles in the shapes below.

Well, exactly that, Aisha.

You can use your knowledge of triangle types to help you with this, can't you? An equilateral triangle, which is the triangle at the top, we know has all the angles that are the same size.

So each of these angles must be 60 degrees.

We can see that the triangle below is in fact a right angled triangle.

So I'm gonna place my right angle in here.

If we now know this is a right angled triangle, and this has got 90 degrees and 27 degrees, we can now work out the missing angle in that triangle by summing these two together and subtracting that from 180.

That will leave us with 63 degrees.

Looking on the right-hand side, we can see we have another triangle here, which is an isosceles triangle because it has two equal sides.

We know that the angles opposite each other from the equal sides must also be the same size.

So in order to work that out, we've got one known angle so far, which is 136 degrees.

If we subtract that from 180 degrees, then the size of the angle that's remaining, we can divide that into two equal angles, can't we? So that would give us an angle of 22 degrees for each one of these angles here.

And now finally, the angles in the triangle at the bottom.

Aisha said she was able to use her knowledge of angles on a straight line to help her with this.

We can see at the top of this triangle here, it forms a part of an angle, which sums up to 180 degrees on a straight line.

We know two of those parts.

We've got 27 degrees and 22 degrees now.

So if we add these two together and then subtract that from 180, that will leave us with the size of the missing angle at the top of the triangle.

That, therefore, is 131 degrees.

And now that we've got that, we can find out the missing angle in the bottom right-hand corner of that triangle as well.

We can add 131 degrees with 23 degrees together, and then subtract that from 180 degrees, which gives us the total of 26 degrees.

There we go.

We've managed to work out the size of all of the internal angles in a range of these different triangles when they've been put together.

Well done if you were able to do that, and that's some fantastic reasoning along the way.

Okay, and that's the end of our lesson for today now then.

Let's have a think about how we can summarise what we've learned.

We know that the angles in a triangle sum to 180 degrees.

We know that all quadrilaterals can be cut into two triangles.

And you can also use your knowledge of types of triangles to help you identify missing angles.

That's the end of our learning for today.

Hopefully you've enjoyed that lesson and are feeling a lot more secure about the size of angles within a triangle and how we can calculate some of these.

Good luck.

Take care, and I'll see you again soon.