Lesson video

Hello, my name is Dr.

Rowlandson and I'm delighted that you'll be joining me in today's lesson.

Let's get started.

Welcome to today's lesson from the unit of Geometrical Properties with Polygons.

This lesson is called Interior Angles of a Polygon, and by the end of today's lesson, we'll be able to find the interior angle of any polygon.

Here are some previous keywords that you may remember and we'll be reusing again in today's lesson.

So you might want to pause the video at this point if you want to refamiliarise yourself these words before press and play to continue.

This lesson contains two learn cycles.

In the first learn cycle, we'll be putting triangles together to make polygons, and that's so we can learn and understand the sum of interior angles of a polygon 'cause then in the second learn cycle, we'll be using the sum of interior angles of a polygon.

But let's start off with making polygons with triangles.

Here are two triangles, and at this point we understand that the sum of interior angles in a triangle is 180 degrees.

So with a triangle on the left, if I labelled those angles, a, b, and c, we know that a plus b plus c is 180, and with the triangle on the right, if I label those angles, d, e, and f, we know that d plus e plus f must also be 180.

Now, in this case, the longest edge in each triangle is equal in length.

That means we can put these two triangles together to make a quadrilateral, like this.

Now, if we compare the quadrilateral to those two triangles focusing particularly on the angles, we can see that all of the interior angles of the quadrilateral are made up of angles from those triangles, and all of the interior angles from those triangles can be found somewhere at the vertices of the quadrilateral.

For example, if you look at angles a, b, and c in that left hand triangle, we can see those here in the quadrilateral.

And if you look at angles, d, e, and f from the right hand triangle, we can see those here in the quadrilateral.

So that means all of the angles in that quadrilateral are made up of angles from those triangles, and all the angles from those triangles are in that quadrilateral.

Therefore, we know that the sum of interior angles in each triangle is 100 degrees, and we have two of those together to make the quadrilateral.

That means the sum of the interior angles of the quadrilateral must be equal to the sum of the interior angles of those two triangles, which is two blocks of 180.

So that's one quadrilateral.

But quadrilaterals in general can be split into two triangles by drawing a line segment from one vertex to its opposite vertex.

For example, in this quadrilateral, we could split into two triangles by drawing a line segments going from left to right, and it will look like this.

Or we can draw the line segment going from top to bottom, and it'll split into two triangles that look like this.

Each interior angle in the quadrilateral can be compared with angles in these two triangles.

So if you look at this particular example, if we label the angles in the triangle a to f.

Let's compare w, x, y, and z with angles a to f, we can see that angle z in the quadrilateral is equal to angle c in the triangle, and angle x from the quadrilateral is equal to angle e from the triangle.

And then let's look at angle w.

What's that equal to? Well, angle w is equal to the sum of a and d from the triangles.

And then what about angle y? Well, that is equal to the sum of angles b and f from the triangles.

So what can we learn from that? Well, if we look at the sum of w, x, y, and z, and think about what that must be, if we substitute w for a plus d and x for e and so on, we can see that w plus x plus y plus z must be equal to a plus b plus c plus d plus e plus f.

And that must mean that w plus x plus y plus z must be equal to 180 degrees, which is the sum of a, b, and c plus another 180 degrees, which is the sum of d, e, and f.

Therefore, the sum of w, x, y and z must be equal to 360.

This same method doesn't just work for quadrilaterals.

It can be used to find a sum of interior angles in polygons with any number of sides.

Let's take a pentagon.

A pentagon has five sides, and it can be split into three triangles, like this.

The interior angles of each triangle sum to 180 degrees.

Therefore, the interior angles of the pentagon must sum to three lots of 180 degrees, which is 540.

So the interior angles of a pentagon sum to 540, that's three triangles.

Now, there are multiple ways you can split a pentagon into triangles.

It's not just one way we can do it.

For example, we could split like this like we've seen already, or we can split it into three triangles, this way.

We can split it into three triangles this way, this way, and also this way.

Now, with however we do it in all of these examples, the angles for each triangle can be found at the vertices of the pentagon.

Let's take one vertex at a time from the pentagon and see where those angles are in the triangles.

Let's start off with the top left vertex of the pentagon and see where that angle is in each of the examples.

In two of the examples, we can see that angle is in its complete form, and that's those two examples on the right, you can see that angle's not broken up, but in the other three examples, we can see where the angle has been broken up into two or three parts, but it's all there in those triangles.

We then look at the top vertex.

We can see that angle can be found in the examples in various ways.

Here's where the right hand vertex is in each triangle.

The bottom right vertex can be found here in each triangle, and the bottom left vertex can be found here in each triangle.

So all of the angles in the pentagon are accounted for in the triangles that it's split into, and there are no angles in those triangles that are not present somewhere in the vertices of the pentagon.

This means that the sum of the interior angles of the pentagon is therefore equal to the sum of the interior angles of three triangles and three lots of 180 degrees is 540 degrees.

Here we have Alex.

Alex splits the pentagon into triangles in a different way.

He does it like this.

He says, "By crossing the line segments, I'll split the pentagon into five triangles." Now five times 180 is 900.

Therefore, he deduces that the interior angles of a pentagon must sum to 900 degrees.

What's wrong with Alex's reasoning? Pause the video while you think about this and then press play to continue.

The problem here is that we have line segments that intersect with each other inside the pentagon.

When we draw line segments inside the pentagon that intersect with each other, it does create more triangles.

But the problem with those extra triangles is they contain angles, additional angles that are not at any of the vertices of the pentagon.

Therefore, the sum of interior angles of the pentagon is not equal to the sum of the angles in five triangles.

So how can we avoid this situation? Well, splitting a polygon by drawing line segments from a single vertex ensures that no extra vertices are created inside the shape.

So in other words, you can choose a vertex and draw line segments from that vertex to the other vertices, and they will not cross or intersect anywhere inside the shape.

We could do it here, could do it here, here, here, or here.

It also means that the polygon is split into its minimum number of triangles.

In this case, it's split into three triangles, and it can't split a pentagon into fewer triangles than that.

So let's check what we've learned so far.

In which diagram has a hexagon then split into its minimum number of triangles? Is it A, B, or c? Pause the video while you think about this and then press play when you're ready for an answer.

The answer is C.

In that example, we can see we've started at one vertex and drawn line segments to the others.

That's created four triangles, which is the minimum number, whereas with A and B, you can see that the line segments intersect somewhere inside the shapes, so they create additional angles.

Which calculation finds a sum of interior angles of a hexagon? You've got four choices, A, B, C, and D.

Pause the video, make a choice, and press play when you're ready to continue.

The answer is B, 4 times 180 will find the sum of the interior angles of that hexagon.

That's because there are four triangles and each triangle has angles summing to 180 degrees.

A heptagon again has seven sides.

What is the minimum number of triangles that it can be split into? Is it 5, 6, 7, or 8? Pause the video while you make a choice and then press play when you're ready for an answer.

The answer is five, and you'll look like this.

Let's think about some tricky situations you might come across along the way.

For example, polygons that contain reflex angles can also be split into triangles.

However, you may need to think carefully about which vertex you want to draw line segments from.

For example, if you choose this vertex and draw line segments from here to every other vertex in that shape, you'll get something like this.

Here we can see that we cannot draw line segments to all of the vertices without leaving the shape.

Therefore, you may want to think about choosing a different vertex, for example, here.

From this point, we can draw line segments to all the vertices without leaving the shape.

Now, as we've been working through these examples, you may have been wondering whether or not there's a relationship between the number of sides a polygon has and the minimum number of triangles that it can be split into.

Well, the answer is yes, there is a relationship between the number of sides that a polygon has and the minimum number of triangles it can be split into.

Let's take a look at the examples we've seen so far altogether on the same slide.

We had a quadrilateral that has four sides and can be split into two triangles.

We saw a pentagon that has five sides and can be split into three triangles.

We saw a hexagon that has six sides and can be split into four triangles, and we've seen a heptagon that has seven sides and can be split into five triangles.

Hmm.

The minimum number of triangles that a polygon can be split into is always, hmm, less than the number of sides.

Can you spot what should go in that blank on that sentence? Pause the video while you think about this, and then press play when you're ready to continue.

If we compare the number of triangles in each polygon with the number of sides it has, we can see that the minimum number of triangles that a polygon can be split into is always two less than the number of sides, four sides, two triangles, five sides, three triangles, and so on.

So let's check what we've learned there.

For which vertex, is it possible to draw a line segment to every other vertex without going outside the shape? Is it A, B, or C? Pause the video while you make a choice and press play When you're ready to continue.

The answer is B.

From that vertex, you can draw line segments to every other vertex without leaving the shape.

Here we have a decagon.

Again, a decagon again has 10 sides.

What is the minimum number of triangles that a decagon, again can be split into? Pause the video while you have a go at this and press play when you're ready for an answer.

The answer is eight triangles.

The number of triangles is always too less than the number of sides.

Right, it's over to you now for task A.

This task contains two questions, and here is question one.

In part A, you need to split each shape into its minimum number of triangles, and in part B, you need to fill in the gaps of the table, which is looking at the minimum number of triangles and the sum of interior angles.

And then part C, you need to fill in the blank on that sentence.

Pause the video while you have a go at these and press play when you're ready for question two.

And here is question two.

The table gives some information about some polygons, and you can see that there are gaps in various places around that table.

You need to fill in the gaps in the table.

Pause the video when I have a go at this and press play when you're ready for some answers.

Well done with that.

Let's now go through some answers.

Question one in part A, you can split the shapes in multiple different ways, but however you split that heptagon you should get five triangles.

And here's an example of how you might do that.

With the octagon, you should get six triangles, and here's an example of how you might do that.

Yours might look different, and with the nonagon, you should have seven triangles, and here's an example of how you might do that, and that's probably a good vertex to choose because you can reach all your vertices easily without leaving the shape.

In part B, if we fill the gaps, well, when there's six sides, there are four triangles, and the sum of interior angles is 720 degrees.

For seven sides, we have five triangles, which sum to 900 degrees.

For eight sides, we have six triangles, which sum to 180 degrees.

And for nine sides we have seven triangles, which sum to 1,260 degrees.

And the minimum number of triangles is always two less than the number of sides.

And then with question two, we need to fill in the gaps in this table.

Now, one column is completed for us already, that's 10 sides, has eight triangles and a sum of interior angles as 1,440 degrees.

For 20 sides, well, that would have 18 triangles, and the sum of interior angles would be 3,240 degrees.

Now the next column, we don't know the number of sides, but we do know there are 20 triangles, and the number of triangles is always two less than the number of sides, so must have 22 sides, and the sum of interior angles is 3,600 degrees.

For a 9,000 sided shape, there would be 8,998 triangles, and the sum of interior angles would be 1,619,640 degrees.

But if the minimum number of triangles was 9,000, well the number of sides will be 9,002, and the sum of interior angles would be 1,620,000 degrees.

But what if that 9,000 is in the interior angles? Well, if we first think about the number of triangles that make 9,000 degrees, how many 180 is going to 9,000, that'd be 50, and the number of sides must be two more, so that'd be 52.

How to get on with that? You've been doing great so far.

Now, we've been using triangles to understand the sum of interior angles in a polygon.

Let's now use the sum of interior angles in a polygon to solve some problems. We can use the sum of interior angles to find a missing angle in a polygon when the other angles are known.

For example, here we have a pentagon, because it's got five sides and five angles, we know the size of four of the angles, and there's an unknown angle, which is labelled X.

How might we go about finding the value of X? You might want to pause the video at this point while you think about what approach you might take to finding the value of X, and then press play when you are ready to work through this together.

We can find the value of X by first thinking about what the sum of the interior angles in this shape must be.

This shape has five sides, so if I was to split it into triangles, I'll split it into three triangles, and therefore the sum of the interior angles would be three lots of 180 degrees, and that's 540 degrees.

That means these five angles must add up to 540 degrees, and X is one of them.

So the sum of the known angles is 390 degrees.

So then let's find X by doing 540, subtract 390, and that gives 150, so the value X must be 150.

Okay, let's check what we've learned so far.

We'll go do a series of three problems that are all very similar, and the first couple of problems, some of the workings could be done for you and you just need to complete the working yourself, but the amount of work you need to do each time, it'll get more and more.

So in this example, we've got a nine sided shape, a nonagon, and we need to find the value of X.

And we can see that the first three steps of working have been done for you.

You need to do the final step to find the value of X.

Pause the video while you have a go at this, and then press play when you're ready to go through the answer.

Okay, so far we can see that the number of sides is nine, therefore, the sum of interior angles must be seven lots of 180, which is 1,260.

The sum of the known angles is 1,133.

So the final thing we need to do to find the value of X is do 1,260, subtract 1,133, and that is 127.

So here's a second problem, very similar again, but this time you have more work to do yourself this time.

We can see we have a seven sided shape, and the sum of the interior angles is 900 degrees.

You need to continue the working and find the value of x.

Pause the video while you have a go at this and press play when you're ready to go over an answer.

Okay, your next step to solve this problem is to find the sum of the known angles.

That is 648.

And then to find a value of x, do 900, which is the sum of interior angles, subtract 648 to get 252.

And the final one, you need to do it all yourself this time.

You've got a five-sided shape, a pentagon, and you need to find the value of X.

Pause the video while you have a go at this, and then press play when you're ready to work through it together.

Okay, your first step is to work out what is the sum of the interior angles? Well, there's five sides.

You've got three triangles there, which means a sum of interior angles is three lots of 180, which is 540 degrees.

Your next job is to find a sum of the known angles, which is 495 degrees, and then subtract that from the sum of interior angles to get 45 degrees for your value of X.

Sometimes angles have markers on them, and the markers indicate when two or more angles in a diagram are equal, so the angles of the same markers are the same size.

Here we have an example of a hexagon because it's got six sides and six angles.

Now we can see that for three of the angles we're given the size very explicitly, 207 degrees, 98 degrees, and 74 degrees, but we have two angles where no number has been given, but there are markers on these angles, which give us a bit of a clue.

So how might we go about finding the value of X in this situation? You might want to pause the video at this point and consider what strategy you might take and then press play to work through it together.

Well, let's start off in a very similar way.

We can see that there's six sides, therefore, the sum of the interior angles must be 4 times 180 degrees, which is 720 degrees.

The next thing we want to do is find the sum of the known angles.

Now, for three of the angles, we can see that they are labelled with numbers 207, 98 and 74, but for two angles, they're not labelled as numbers, but they do have markings on them.

We can see that these two angles, these must be equal to each other because they have the same markings, so they must both be 98 degrees and these two angles, these also have the same markings as each other, so they must be equal, they must both be 207 degrees.

So now we've got that information on the diagram.

It's much easier to find the sum of the known angles.

We've got two lots of 207, two lots of 98 and a 74.

And when we find the sum of our knowns, we get 684, and then we can subtract that from the sum of the interior angles to get 36 as our value of x.

Here is another problem that is similar but a little bit different to last one.

Maybe pause the video at this point and consider what steps you might take in this problem and then press play to continue solving it together.

Well, let's start in the same way we've done with the previous problems. Let's look at how many sides it has.

It has seven sides.

If it's got seven sides, it can be split into five triangles, which means the sum of interior angles is five lots of 180, which is 900 degrees.

Our next step is normally to find the sum of the known angles, but we have an an angle there that doesn't have a number on it.

It has got a marking on it though, and that marking is the same as the angle labelled X, which means that both of these angles must be equal to X.

That means we only have five known angles.

So let's find the sum of those.

We get 630, and then subtract that from the sum of the interior angles to get 270.

So does that mean that the value of X is 270? No, because we have two angles left, which sum to 270.

So the value of X must be 270 divided by 2, which is 135.

Okay, let's check what we've learned there.

Focusing particularly on angles of markings on.

Which angle is the same size as the angle marked X in this diagram? Is it A, B, C, or D? Pause the video while you have a go and press play when you're ready for an answer.

The answer is B, and we can see that because they both have the same markings as each other.

They both have two little dashes on the angle marker.

Here once again, we have a angle problem where some of the calculations are done for you and you need to complete the calculations to find the value of X.

Pause the video while you have a go at this, and then press play when you're ready for an answer.

Okay, we can see our next step is to subtract the sum of the known angles from the sum of interior angles, which is 720, subtract 339, which makes 381, but we then have to divide that by three because there are three angles which are equal to X, and that means we get 127.

Okay, it's over to you now for task B.

This task contains just one question, and here it is.

You need to find the value of the unknown in each question.

Find a value of x, then y and then z.

Pay extra attention to whether or not there are any angles with markings on which indicate that they are equal to other angles in the diagram.

Pause the video while you have a go at these and then press play when you're ready to go through some answers.

Well done with that.

Here are the answers that we should have have got.

x is equal to 45, y is equal to 108, and z is equal to 240.

If you've got anything different to those, you may want to go back for your working and see if you can spot your mistake or re-attempt the question again from the start and see if you can get the right answer a second time.

Fantastic work today.

Here's a summary of what we've learned during this lesson.

Any polygon can be split into triangles, and the minimum number of triangles that a polygon can be split into is always two less than the number of sides.

And we can use these triangles to demonstrate the rule for the interior angle sum for any polygon.

And knowing the sum of interior angles of a polygon can help us with finding a missing angle in there when all your angles are known.

Thank you very much.