Lesson video

Hello, my name is Dr.

Rowlandson, and I'll be guiding you through today's lesson.

Let's get started.

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

This lesson is called Exterior Angles of Polygons.

And by end of today's lesson, we'll be able to find the sum of exterior angles of a polygon.

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

So you may wanna pause the video at this point while you remind yourself the meanings of these words, before pressing play to continue.

This lesson contains two learn cycles.

In the first learn cycle, we're gonna learn what exterior angles are, and work on understanding the meaning of exterior angles.

And the second learning cycle, we're gonna find the sum of exterior angles for a polygon and then use that fact to find some missing angles.

But let's start off with understanding the meaning of exterior angles.

Here we have Laura.

Laura is using an activity on GeoGebra that allows you to draw images by inputting commands to a turtle, and the activity looks something a bit like this.

We have a few buttons and a couple of sliders.

The forward button makes a turtle move forward in a straight line, and the distance slider determines how far the turtle moves.

So for example, the distance slider is currently set to four, so if we press the forward button, the turtle will move four units forward.

And then we have two buttons that say turn left and turn right.

These rotate the turtle left and right, and the angle slider determines the angle of turn.

So we have the angle slider set to 45 degrees at the moment.

If we press turn left, the turtle will turn left 45 degrees, and if we press turn right, the turtle will turn 45 degrees to the right.

Now, if you wanna have a play around on this activity yourself, on the slide deck on this slide, you will find a link to this application at the bottom of the slide.

So feel free to either have a go now and play around and see what you can do with it, or at the end of the lesson or after the lesson, feel free to have a go at it and see what shapes or images you can draw.

Laura inputs the following commands: forward five, left turn 90 degrees.

Forward five, left turn 90 degrees.

Forward five, left turn 90 degrees.

And forward five, left turn 90 degrees.

What shape do you think Laura draws when she inputs these commands? Maybe pause the video at this point and consider what shape you think Laura will draw, and also consider why you think it will draw that shape, and then press play when you're ready to continue.

Okay, well, let's see what it does draw.

So first the turtle moves forward five, and then it turns left 90 degrees, moves forward five, turn left 90 degrees, move forward five, turn left 90 degrees, move forward five, and turns left 90 degrees to return to its starting position.

It has drawn a square.

Laura wants a programme to draw an equilateral triangle.

Now the link at the bottom of this slide of the slide deck takes you to a similar GeoGebra application where it has a series of different shapes you can have a go at drawing if you wanna have a go at it yourself.

Either way, let's see what Laura does with this.

Laura says, "I'm going to set it to draw three lines of equal length." So she plans to ask the turtle to move forward three times, and each time the turtle moves forward, it moves forward the same amount, in this case four units.

And Laura says, "What angle should I use for each turn?" Maybe pause the video at this point and consider what angle you think Laura should use for the turtle's turn.

Pause the video and then press play when you're ready to continue.

Laura says, "All the angles inside an equilateral triangle are 60 degrees, so I'll set the turtle to turn 60 degrees each time." Do you think this is going to work? Maybe imagine what the turtle is gonna do as it goes through each of these commands.

Let's take a look.

The turtle first moves forward four, and then turns left 60 degrees, and then moves forward four, then turns left 60 degrees, then moves forward four and turns left 60 degrees.

Hmm.

It hasn't done what Laura was expecting it to do.

Had it done what you expected it to do? So, why did the turtle do that? Pause the video and consider why the turtle moved in the way it did based on these commands, before pressing play to continue.

Laura considers what happened at the first turn.

So the turtle has moved forward four at this point, and it's about to now do its left turn of 60 degrees.

Laura says, "At this point I wanted the turtle to face this way," like we can see on the screen here, so that it would draw the triangle.

"That would've made an angle inside the triangle of 60 degrees." But the turtle was initially pointing outside of the triangle, so the 60 degree angle was outside of the shape, and the turtle did not turn far enough.

So, what angle do you think Laura should use instead? Pause the video while you consider this, and then press play when you're ready to continue.

Okay, do you have an angle in mind? Let's see what Laura does.

Laura says, "The turtle will be turning on the outside of the triangle." Now the remaining angle outside the triangle is 300 degrees, the conjugate pair of 60 degrees.

So she sets the turtle to turn 300 degrees each time.

Do you think this will work? Maybe imagine what the turtle will do this time.

Let's take a look.

The turtle moves forward four, and then turns left 300 degrees.

Moves forward four, turns left 300.

Oh, hang on a minute, the turtle's nearly gone off the edge of the page.

It's clearly gone the wrong direction.

So, why did the turtle do that? Pause the video and consider what you think happened this time.

Why did the turtle start to draw this image? And why did 300 not work? Pause the video while you think about it, and then press play when you're ready to continue.

I wonder what we thought.

Laura considers what happened at the first turn this time.

So the turtle has moved forward four, and it's about to turn left 300 degrees.

Laura says, "At this point, I wanted a turtle to face this way, so it would've made an angle inside the triangle of 60 degrees.

But 300 degrees is nearly a full turn, so turtle turned too much.

It's the same as if I had programmed the turtle to turn 60 degrees to the right." So if that's not worked, what angle should Laura use instead to draw this equilateral triangle? If you haven't got it yet, pause the video now, have a think about it again, and press play when you're ready to continue.

Hmm, I wonder what we thought.

Let's see what Laura thinks.

Laura prepares to try drawing an equilateral triangle again, and she thinks really carefully about what the turtle does at this point after its moved forward four.

Laura says, "At this point, if the turtle kept moving in a straight line, then it would go in this direction.

I want it to turn so that there is a 60 degree angle here," inside the triangle.

So what angle does the turtle need to turn? The angle on the outside and the angle on the inside are adjacent on a straight line, so would sum to 180 degrees, because they are supplementary.

So if the inside angle is 60 degrees, the outside angle must be 120 degrees.

So Laura has a third attempt at drawing an equilateral triangle, and this time uses 120 degrees for her left turn each time.

Let's see how it goes.

The turtle moves forward four, turns left 120 degrees, moves forward four, turns left 120 degrees, moves forward four, and turns left 120 degrees to return to its starting position.

It's worked.

Fantastic.

So let's try and make sense of what's gone on here.

An interior angle is an angle formed inside a polygon by two of its edges.

For example, this one here is an interior angle.

An exterior angle is an angle on the outside of a polygon, but it's not the entire outside of that vertex.

An exterior angle is an angle on the outside of a polygon between an extension of an edge and it's adjacent edge.

So if we take that horizontal edge of a triangle and extend it, then the angle between that extension and the adjacent edge, which is the one the right side of the triangle, that is the exterior angle at this vertex.

And an exterior angle at a vertex could go in one of two directions, depend on which edge is extended.

In this example, we're using the vertex in the bottom right corner of this triangle.

Well, that vertex is made of two edges, the horizontal edge and the edge at the right side of the triangle.

In the example we can see on the screen here, it's the horizontal edge that's been extended, but instead we could extend the edge on the right of the triangle, and consider the exterior angle between that and its adjacent edge.

They'll both be the same in either direction.

The edge and its extension form a straight line segment, therefore adjacent interior and exterior angles sum to 180 degrees.

For example, if the interior angle is 60 degrees, like in the equilateral triangle, then the exterior angle would be 120 degrees.

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

In which diagrams is an exterior angle marked? A, B, C, or D, and it may be more than one.

Pause the video while you make some choices and then press play when you're ready to continue.

The answers are C and D.

So, why is that? Well in A, the angle that is marked is inside the polygon, that's an interior angle.

In B, yes the angle that is marked is on the outside of the vertex, but that is not an exterior angle 'cause the exterior angle is not the entire outside of a vertex.

It is between the extension of an edge and its adjacent edge, and that's what we can see in C and D.

And here's another question, an interior angle and it's adjacent exterior angle sum to? Write down what you think should go in that blank.

Pause the video while you have a go and press play when you're ready to continue.

An interior angle and it's adjacent exterior angle sum to 180 degrees.

True or false.

The diagram shows an example of an interior angle and it's adjacent exterior angle.

Is that true or is it false? And then there's two justifications to choose from at the bottom.

Pause the video while you make some choices and press play when you're ready for some answers.

The answer is false because the two angles do not sum to 180 degrees.

They should do if they are an interior angle and it's adjacent exterior angle, but they don't, so that cannot be an exterior angle.

Here we have Andeep.

Andeep wants to programme the turtle to draw the triangle below.

What exterior angles do you think you'll need to use at each turn? Now, once again on the slide deck at the bottom of this slide, there is a link to an activity with a series of different shapes to draw, and this is one of them, if you wanna have a go at it yourself.

Either way, pause the video now and consider what angles you think Andeep should use to draw this shape, and then press play when you're ready to continue.

Okay, let's draw this shape together now.

So the turtle first moves forward four units, and then we should consider the direction that the turtle is currently facing and how much we want it to turn in order to face the direction we want it to face.

Now, if we want the interior angle at this vertex to be 37 degrees, to get the exterior angle, we need to do 180 degrees subtract the interior angle to get the exterior angle.

So that is 180 subtract 37 to get 143.

So we'll set a total to turn 143 degrees to the left, and then move forward five, like so.

And then we think, right, okay, we want the interior angle to be 53 degrees.

So what should the exterior angle be at this point? Well, if we do 180 subtract the interior angle, we get the exterior angle, so that's 180 subtract 53 to get 127.

So we set the turtle to turn left or 127 degrees and move forward three to get here.

And then we just need the turtle to return to its starting position.

What angle would it need to turn? Well the interior angle is 90 degrees, so the exterior angle would be 180 subtract that, which would be 90 degrees.

Here we have Izzy.

Izzy has programed the turtle to draw the quadrilateral below, and we can see the commands that Izzy used.

We can see how far forward the turtle moved each time, and we can see how much the turtle turned to the left each time, and those are the exterior angles.

My question is what are the interior angles in this quadrilateral? Pause the video at this point to consider how could we work out these interior angles, and what are they at each vertex, before pressing play to work through this question together.

Okay, well, we need to consider that the angles that the turtle turns each time is the exterior angle, and the sum of the exterior angle and its adjacent interior angle is 180 degrees.

We'll go use those facts.

So at this first vertex here, well, the exterior angle is 97 degrees.

If we do 180 subtract that, we get the interior angle at A, B, C, which is 83 degrees.

And then at the next vertex, the exterior angle is 49 degrees.

So to get the interior angle, we'll do 180 subtract 49 to get DAB as being 131 degrees.

At the next vertex, the exterior angle is 104 degrees, so the interior angle will be 180 subtract that to get 76 degrees at angle CDA.

And finally, the exterior angle at this last vertex is 110 degrees, so to get the interior angle, we'll do 180 subtract that to get 70 degrees, and that will make the turtle return to its starting position.

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

Find the size of the exterior angle at vertex B.

Pause the video while you work this out and press play when you're ready for an answer.

Okay, well, we can see the interior angle is 160 degrees, so to get the exterior angle, we'll subtract that from 180 degrees to get 20 degrees.

And that could be 20 degrees here as the exterior angle, or it could be 20 degrees here as the exterior angle, depending on which edge you choose to extend.

Find the interior angle at B, C, D.

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

Well, we can see the exterior angle is 75 degrees, so if we subtract that from 180, we'll get the interior angle, which is 105 degrees.

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

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

Here we have a pentagon, and you need to draw an exterior angle at each of the five vertices on this pentagon, and then calculate the size of each exterior angle as well.

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

And here is question two.

Here we have Jun.

Jun programmes the turtle to draw a quadrilateral by using the commands below.

We can see the commands and we can see the quadrilateral.

In part A, you need to find the size of each of the four interior angles and label them in the correct positions on the diagram.

And then in part B, you need to justify why this quadrilateral is a trapezium, and write down a sentence or two to explain how you know it's a trapezium.

It looks like one, but how do we know for sure? Pause the video while you have a go at this and then press play when you're ready for some answers.

Well done with that.

Let's look at question one together.

Here's one way that your answer might look.

You could extend each edge with your exterior angles going in an anti-clockwise direction around the outside of the shape, and then work out each exterior angles to get these numbers on the screen here.

Now, while that is one way you might look, you might have extended all of your edges in the other direction and it would look like this.

Note though that the numbers are all the same.

Whichever way you extend the edge at a vertex, you should get the same exterior angle either time, because you are subtracting the interior angle from 180 degrees.

Now, at each vertex you can extend either of the edges to create an exterior angle in either direction.

However, it can be quite helpful for yourself to choose a direction for a particular shape.

So either always create your angles going clockwise round the shape or anti-clockwise round the shape, just so you can keep track of how many exterior angles you have at each vertex.

And this is particularly important when we start working with the sum of exterior angles, which we're gonna move on to in the second part of today's lesson.

And in question two, we need to find the size of each of the four interior angles and label on the diagram.

Well, what's in Jun's commands there are the exterior angles, so we just need to subtract each of those from 180 to get the interior angles, and that's what we get here.

B, to justify why the quadrilateral is a trapezium, well, we can think about how a trapezium is a quadrilateral with one pair of parallel sides.

Now it looks like that, but how do we prove that the top side and the bottom side are parallel? Well, co-interior angles between parallel lines sum to 180 degrees, and we can see that 127 plus 53 makes 180, and so does 104 and 76, so the sides with lengths three and seven are parallel.

So why not stop here? Well, that shows that there is at least one pair of parallel sides, but we haven't proven that it's not parallelogram.

It doesn't look like a parallelogram, but let's just justify that it's not a parallelogram, and show that it doesn't have two pairs of parallel sides.

So 127 plus 104, that makes 231.

They do not add up 180, neither does 53 and 76.

So we can see that the size of lengths five and 4.

1, they are not parallel because those angles, those co-interior angles between them do not add up to 180 degrees.

Therefore, we can see it has exactly one pair of parallel sides and it's a trapezium.

Great work so far.

Now let's move on to the second learn cycle, which is finding and using the sum of exterior angles.

Let's start by taking another look at some of the shapes we've seen so far.

Laura looks at the triangle that she drew with the turtle, the equilateral triangle, and she notices that the three exterior angles I used sum to 360 degrees, because we have 120 degrees three times.

She then wonders, "Is this just for equilateral triangles or do the exterior angles of other triangles also sum to 360 degrees?" What do you think? Let's explore that together.

Andeep looks at a triangle that he drew with the turtle and he says, "The three exterior angles I used also sum to 360 degrees," because we have 143, 127, and 90, they sum to 360.

So that's two different triangles with the exterior angles summing to 360 degrees.

He says, "Is this just for triangles or do the exterior angles of other polygons sum to 360 degrees?" What do you think about that? Hmm, let's take a look at this some more.

Izzy looked at the quadrilateral that she drew, and she notices the four exterior angles that she used also sum to 360 degrees, 97, 49, 104, 110.

So we've got three different shapes here, two triangles and one quadrilateral, where the exterior angles keep summing to 360 degrees.

Why do you think that happens? Pause the video while you consider this, and press play when you're ready to continue.

Well, exterior angles in any polygon always sum to 360 degrees, no matter how many sides it has.

And there's a few different ways we can see why this is the case.

For example, when drawing a polygon with turtle graphics, the turtle completes one full turn before returning to its starting position, and we can act that out with an object of some sort.

If you take an object such as a pencil and put it in a starting position, and then move that object in straight lines to create some kind of shape, some kind of polygon, and then return it to its starting position, you should see that the object completes one full turn before returning to its starting position.

Or you can try this by standing up and going into some sort of big empty space.

Make a note of which way you're standing and which way you're pointing to begin with, and then walk around that space to trace out some sort of polygon and return to your starting position, and notice that you complete one full turn while you're walking around in that space.

Another way we can visualise this is to imagine zooming out from these shapes.

As we zoom out, we get further away and these shapes appear to be getting smaller to us.

So as we do that, each polygon would tend towards appearing to be a single point.

And imagine what the angles will do as we do that.

As we zoom out and these polygons appear smaller and smaller until they get to a point, we can see that the exterior angles complete one full turn around that point, and angles around a point sum to 360 degrees.

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

Exterior angles in a polygon sum to? Write down what you think should go in that blank.

Pause while you do it and press play when you're ready to continue.

Exterior angles in a polygon sum to 360 degrees.

In which diagram have the exterior angles definitely not been measured correctly? Pause the video while you work on this, and then press play when you're ready to go through the answer.

Okay, we can check this by adding up the exterior angles and seeing if they sum to 360 degrees.

In the case of B, no, they don't.

Those exterior angles do not sum to 360 degrees, therefore they cannot have been measured accurately because they should sum to 360 degrees.

For A and C, we don't know for certain that they definitely have been measured correctly.

They do sum to 360.

Well, it might be that more than one vertex was measured incorrectly and just happened to be that the errors cancel each other out, and you end up adding up to 360.

But we know for certain that B definitely was not measured correctly.

Now, this fact that exterior angles sum to 360 degrees can be used to prove that interior angles in a triangle sum to 180 degrees.

We've previously learned that angles in a triangle sum to 180 degrees, and you may have seen some different ways of proving it before.

But there are sometimes multiple ways you can prove something, which is quite delicious about mathematics.

So let's use this particular fact to try and prove now that angles in a triangles sum to 180 degrees.

If we have a triangle and we label the angles A, B, and C, let's now write an expression for the exterior angle at each vertex.

Well, it'll be 180 subtract whatever the interior angle is.

So 180 subtract A, 180 subtract B, 180 subtract C.

Now, we know that the exterior angles sum to 360 degrees, so let's write down an equation with that.

Now this equation shows the three angles summing to 360, and the brackets are being used so we can see clearly each of the three angles, but we don't really need the brackets in this equation here.

So we can rewrite it like this without the brackets, and then we can rearrange the left hand side, so that we have all of the constants together and all the algebraic terms together, and it'll look something a bit like this.

And then if we simplify the equation a little by adding those 180s together to get 540, it'll look something like this.

And then next we could add A, B and C to both sides of this equation, so we get something that looks like this, and then subtract 360 from both sides of the equation, and we get A plus B plus C equals 180.

And that shows that the interior angles A, B, and C add to 180 no matter what those interior angles are.

Okay, let's use this fact now to find a missing angle.

I'm gonna do a question on the screen and then I'm gonna give you a very similar one to try yourselves.

So find the value of X.

We have a quadrilateral.

We've been given two exterior angles and an interior angle, and we need to find the value of X, which is at one of the exterior angles.

Now, there are multiple ways to do this, but one way could be to find the exterior angle at this vertex here.

That'll be by doing 180 subtract 85 to get 95, so that exterior angle must be 95 degrees.

And we know that those four exterior angles should sum to 360, so if we add up the ones that we know so far, we get 334 degrees, and then subtract that from 360 and we get X as 26.

Now, an alternative way to have done that could have been to find the interior angles at each vertex, and then when you've got the interior angle at the vertex with X on it, you could subtract from 180 degrees to get X.

The thing to bear in mind with that is the interior angles sum to different amounts depended on the number of sides, so you just got that tricky thing to deal with.

Whereas the exterior angles of any polygon always sum to 360 degrees.

And here's one for you to have a go at.

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

Okay, let's go through this.

So we could get the exterior angles of 60 and 80.

Remember that exterior angles sum to 360, so work out the sum of the ones we know, subtract it from 360 to get 110 for the value of X.

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 of those diagrams. Pause the video while you have a go at this, and then press play when you're ready to go through some answers.

Okay, great job with that.

Let's now go through some answers.

So the value of A is 145, because those three should add up to 360.

For B, you may need to work out the exterior angles first of 140 and 41, but then you should get 113 for B.

And for C, you could work out the interior angles and remember that they sum to 540 degrees, and then get the exterior angle for C, or you can get the exterior angles and remember they sum to 360.

Either way, you should get 68 for C.

And for D, we could work out the exterior angle at the 30 degrees, which will be 150, and then make an equation with 150 + D + 2D + 3D is 360.

Solve it and we'll get the value of D as 35.

Fantastic work today.

Let's now summarise what we've learned in this lesson.

An exterior angle is an angle on the outside of a polygon, but it's not the entire outside.

It's between an extension of an edge and its adjacent edge.

An exterior angle and its adjacent interior angle sum to 180 degrees.

Exterior angles sum to 360 degrees in any polygon.

And the sum of exterior angles can be used to prove that angles in a triangle sum to 180 degrees.

We've seen that today.

And finally, the sum of exterior angles can be used to find missing angles in a polygon.

Great work with that.

Thank you very much.