Lesson video

Hello and thank you for choosing this lesson.

My name is Dr.

Rowlandson and I'm excited to be helping you with your learning today.

Let's get started.

Welcome to today's lesson from the unit of angles.

This lesson is called, Checking and Securing, Understanding of Exterior Angles.

And by the end of today's lesson, we'll be able to find the exterior angle of any regular polygon or deduced exterior angle of an irregular polygon.

Here are some previous keywords that will be useful during today's lesson, so feel free to pause the video if you want to remind yourself what any of these words mean and then press play when you're ready to continue.

This lesson contains three learning cycles and we're going to start with finding and using 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.

If you'd like to have a go at this activity for yourself before getting started, feel free to click on the link at the bottom of this slide and take you to a copy of it.

Let's see what it does.

We have a button here, so it says forward.

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

Like this.

We have a turn left and a turn right button.

They rotate the turtle and the angle slider determines the angle of turn.

For example, turn left 45 degrees, turn right 45 degrees.

Laura wants to programme the turtle to draw an equilateral triangle with sides of length four units.

If you'd like to try this out for yourself before watching Laura do it, feel free to click on the link and have a go at creating this triangle.

If not, feel free to keep watching.

Let's see what Laura does.

She says, "Each interior angle in an equilateral triangle is 60 degrees." So she commands the turtle to do the following things.

First, "I'll instruct the turtle to move forward four units." So she does, and it does this.

She then says, "If I don't turn the turtle, then its next move will continue in a straight line." I want the turtle to turn so that there is a 60 degree angle inside of the triangle." So can you think what angle she'll want the turtle to turn left in this situation? Hmm.

She says, "I want the turtle to turn 120 degrees," and so she does, and the turtle is now pointing in the right direction.

The angle that the turtle turns at this point is the exterior angle of that vertex.

And the interior angle and exterior angle at a vertex form a straight line.

Therefore, they are supplementary.

They sum to 180 degrees.

60 plus 120 is 180 degrees.

Laura repeats the same two instructions two more times to complete her triangle and return the turtle to its starting position.

So moves forward four, turn left 120 degrees, move forward four, and then to get it pointing in the same way again, turn left 120 degrees.

Laura says, "The turtle completed one full turn as it drew the triangle.

The angles that it turned must sum to 360 degrees," which you can see here, they do.

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

For example, this is an interior angle.

However, an exterior angle is an angle on the outside of a polygon between an extension of an edge and its adjacent edge.

So if we extend that bottom edge like so, we have an exterior angle on the outside.

An exterior angle at a vertex could go in one of two directions depending on which edge is extended.

So at this vertex we extended the bottom edge, but I could have extended the other edge that meets that vertex.

Instead, an exterior angle would be here.

Exterior angles in any polygon always sum to 360 degrees.

We can see examples here where on the left, we have three angles, which are all 120 degrees.

They're all exterior angles, and yes, they sum to 360 degrees.

On the right, we have a quadrilateral with four exterior angles and they also sum to 360 degrees.

This can be seen when drawing a polygon with turtle graphics, as a turtle always completes one full turn before returning to its starting position.

Like so.

And then if we do it with the other turtle, it does one full turn before it gets back to its starting position.

This can also be seen when zooming out of diagrams as each polygon tends towards being a single point.

Let's do that now.

Let's imagine zooming out from these two polygons.

They would appear to get smaller as we do it because they're further away, and we'd see that the angles sum to 360 degrees, because they rotate around a single point.

And angles around a point sum to 360 degrees.

Interior angles of a polygon sum to whatever 180(n-2) is where n is the number of sides.

This fact can also be used to prove that exterior angles sum to 360 degrees.

For example, this diagram shows part of a polygon with n sides, which means we don't know how many sides it has, but we can see that one angle is unknown.

It's called X1 degrees.

Another angle is also unknown.

It's X2 degrees and would have other angles labelled in a similar way.

Let's prove that exterior angles sum to 360 degrees using this.

We could start by extending these edges to draw exterior angles and think what the expression would be for each.

Remember the interior angle and exterior angle sum to 180 degrees.

So the exterior angle here would be 180 - X1.

And here it would be 180 - X2.

Both times they're in degrees.

That means the sum of exterior angles would be 180 - X1 + 180 - X2, and then same with X3 and four and so on for however many angles there are until we get to the point where we have 180 subtract Xn, where n is the number of angles.

We don't need the brackets here, so we could write it like this.

And then we can see we've got so many lots of 180.

We would have n many lots of 180.

That's how many angles there are.

So we could simplify that to get 180 lots of n.

And then minus one can be taken out as a factor because we are subtracting X1, subtracting X2, subtracting X3 and so on.

So if we take minus one out as a factor, we'd have 180n subtract and now the sum of all the other angles, X1, X2, X3, and so on.

This is the sum of the interior angles 'cause X1 is an interior angle, so is X2 and so on.

Now the sum of exterior angles can be written as 180(n-2).

So I could rewrite that part in brackets with X1, X2 and so on as 180(n-2).

That means we now have this.

We could expand the brackets and then we could simplify and doing so would leave us just with 360.

Let's check what we've learned.

In which diagrams is an exterior angle marked? Pause video while you choose and press play when you're ready for an answer.

The answer is C and D.

In both cases, one of the edges has been extended and the angle we are looking at is on the outside between the extension and it's adjacent edge.

Exterior angles in any polygon sum to blank.

What goes in the blank? Pause video while I write it down and press when you're ready for an answer.

The answer is 360.

Exterior angles in any polygon sum to 360 degrees.

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

What goes in the blank? Pause video while I write it down and press play when you're ready for an answer.

The answer is 180.

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

Let's work through an example of finding a missed angle together and then you've got a similar example to try yourself.

Find the value of X.

Well, let's take a look.

We could start by finding this exterior angle here by doing 180 - 70 to get 110, 110 degrees.

Now we know that all four of those exterior angles will sum to 360 degrees.

So we could write this equation.

We could then simplify this equation by adding together the angles we know to get this, X + 316 = 360, and we could solve this equation by subtracting 316 from 360 to get X = 44.

In other words, we first worked out the missing exterior angle that we didn't know to begin with, and then we added up the exterior angles we knew and subtract it from 360.

Here's one for you to try.

Find the value of x.

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

Well, your two exterior angles are 65 degrees and 87 degrees.

So that means your value of X can be worked out from this equation, which is 98.

Exterior angles in any polygon always some to 360 degrees.

The exterior angles in a regular polygon can be calculated therefore by dividing 360 degrees by the number of vertices because all the exterior angles will be the same size.

For example, let's find the size of each exterior angle for a regular hexagon.

It has six sides.

Well, we know the sum of exterior angles is 360 degrees, so we know that the number of exterior angles is six.

That means we'll do 360 degrees divided by six to get 60 degrees for each exterior angle.

The number of sides of a regular polygon can be calculated when the size of the exterior angles are known.

For example, a regular polygon has an exterior angle of each size 72 degrees.

How many sides does it have? Well, let's think about what we know.

We know that the sum of exterior angles is 360 degrees.

We know that each exterior angle is 72 degrees.

So what we wanna think about here is how many of those 72 degrees go into the 360 degrees? So we'll do 360 divided by 72 to get 5, which means this shape has five sides.

It's a Pentagon.

Exterior angles can be used to determine whether a polygon could be regular.

For example, a polygon has interior angles of size 160 degrees.

Can it be regular? Well, we could do this by first working out the exterior angle, doing 180 degrees, substract 160 degrees to get 20 degrees.

And then we can think to ourself, if it was regular, how many sides would it have? And we'd do that by doing what we've just done.

We'd do 360 divided by 20 and that would give 18, which means we'd have 18 sides.

A polygon can't have 18 sides.

So, so long as all the angles are the same, then yes, it could be regular.

How about this then? If the polygon has interior angles of 161 degrees, can it still be regular? Let's do the same thing again.

Let's find the exterior angle by subtracting from 180.

That'll give 19 degrees.

And then let's say if it was regular, how many sides would it have? We get that by doing 360 divided by 19.

That gives 18.

947 and some more decimals.

Now a polygon cannot have a decimal number of sides, therefore it is impossible for this polygon to be regular if one of its interior angles is 161 degrees.

So no, it cannot be regular.

Let's check what we've learned.

Can a polygon with an exterior angle of 30 degrees be regular, and if so, how many sides would it have? Pause video while you work this out and press play when you're ready for an answer.

The answer is yes, it can be regular.

It can be a regular 12 sided polygon, a dodecagon.

How about if we change the angle now to 35 degrees? Can a polygon with an exterior angle of 35 degrees be regular? And if so, how many sides would it have? Pause the video while you do it and press play when you're ready for an answer.

The answer is no, it cannot be regular, because when you divide 360 by 35, you get a non integer answer.

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

This task contains six questions.

And here is question one.

You've got a series of statements with some blanks.

You need to fill in each blank with either the word always, sometimes, or never.

Pause video while you do it and press play when you're ready for more questions.

And here are some more questions.

You've got questions two to six.

Pause video while you work through these and press play when you're ready for answers, Here is your answer to question one.

Pause video while you check this and press play when you're ready for more answers.

And here are your answers to questions two to six.

Pause video while you check this and press play when you are ready for the next part of the lesson.

Well done so far.

Now let's move on to the next part of the lesson, which is solving angle problems with regular polygons.

Here we have a figure that is made from a regular hexagon and a regular decagon.

We need to find the size of the shaded angle.

Let's think about how we might do this.

Aisha says, "Sometimes it can be helpful to draw extra lines or extend lines." Let's do that with a common etch.

It'll look a bit like this.

She says this has split the angle into two parts.

Each part is an exterior angle to one of the two shapes.

It's worth knowing those two parts aren't equal to each other, but they are equal to the exterior angles of those polygons.

So let's work it out.

The exterior angle of a regular polygon is 60 degrees and the exterior angle of a regular decagon is 36 degrees.

So the total angle, which is marked there, would be the sum of these, which is 96 degrees.

Let's check what we've learned.

Here we have a figure that is made from a regular octagon and a regular decagon.

Find the size of the shaded angle.

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

The answer is 81 degrees.

The exterior angle of the regular octagon is 45 degrees, the exterior angle of the regular decagon is 36 degrees, and the sum of those makes 81 degrees.

Here we have the figure that we had earlier, a regular hexagon and a regular decagon, but this time there is a line segment that goes from a vertex of one shape to a vertex of the other shape, and there's an angle marked, which we want to find the size of.

Let's think how we could do that.

Aisha says, "Since the hexagon and decagon share a common side, all the sides of both shapes must be the same length.

So the triangle is isosceles." Let's mark it on like this.

Aisha says, "I've worked out one of the angles in this triangle already, so I now need to work out the other two angles of the isosceles triangle, which are equal." Let's do that.

We could do 180 substract 96, and then divide that by two to get 42 degrees for each angle in isosceles, which means our shaded angle is also 42 degrees.

Let's check what we've learned.

The figure is made from a regular octagon and a regular decagon.

Find the size of the shaded angle and one angle has been given to you.

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

Answer is 49.

5 degrees, and those are working.

Here we have a figure that is made from a regular hexagon and a regular decagon, but this time one shape is inside the other.

We have an angle shaded.

Let's find the size of that angle.

Aisha says, "I could extend the common edge and consider how the exterior angles could help in this situation." We know that the exterior angle of a hexagon is 60 degrees and we can work it out if we don't know that.

The exterior angle of a regular decagon is 36 degrees.

So the shaded angle is a difference between these two angles.

If we do 60 subtract 36, we get 24 degrees.

Let's check what we've learned with that.

Find the size of the shaded angle.

Pause video while you do this and press play When you're ready for an answer, The answer is nine degrees.

We can do the exterior angle of the octagon, subtract the exterior angle of the decagon, which are both regular.

Here we have a figure made from a regular hexagon and a regular decagon again, but this time they are overlapping each other.

We need to find the size of this shaded angle here.

Now we can see that they are overlapping each other, but they do share one common vertex.

So let's think about how we could do it.

Aisha says, "I could extend the common edge and consider how exterior angles could help." Well, we know that angle there is 36 degrees.

It's the exterior angle of a regular decagon.

The other angle is co-interior with it, and it's between parallel lines.

We can see that from the arrows indicated on the shapes.

So we know that co-interior angles between power lines sum to 180 degrees.

That means we can do 180 subtract 36 to get 144 degrees.

Let's check what we've learned with that.

Here's a figure made from a regular octagon and regular decagon.

They are overlapping.

They share a common vertex.

Can you find the size of the shaded angle? Pause while you do it and press play when you're ready for an answer.

The answer is 135 degrees.

And there's you're working.

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

This task contains one question, and here it is.

Each figure is made from a regular octagon and a regular hexagon, and what you need to do is find the size of each marked angle.

Pause the video while you do this and press play when you are ready for some answers.

Okay, let's go through some answers.

Part A, the answer is 105 degrees.

We get it by finding the sum of the exterior angle of the octagon and the hexagon, which are both regular.

In part B, the line segments have the same orientation as those in part A, so the angle will be the same.

It is 105 degrees.

In part C, we could get an answer of 15 degrees by subtracting the interior angle of the hexagon from the interior angle of the octagon.

In part D, we can get an answer of 75 degrees.

We can do that by finding the exterior angle of the regular octagon, the exterior angle of the regular hexagon, and noting that angles in a triangle sum to 180 degrees.

So subtract them from 180 and you get 75.

In part E, hmm, this answer is 37.

5.

We can get it by working out the 105 degrees from parts A or parts B, noticing that they share a common edge, these two shapes.

So that means the sides are the same length, which means the triangle is isosceles.

Therefore, if you do 180, subtract the 105, which you know, and divide by two, you get each angle in isosceles, which is 37.

5.

And then with F, the answer is 165 degrees.

You can do it by finding the exterior angle of the regular hexagon, which is 60 degrees.

You can find the 105, which is the outside part, which we worked out in parts A and B and adding them together to get 165 degrees.

Wonderful work so far.

Now onto a third and final part of the lesson, which is investigating problems with turtle graphics.

Here we have Sophia.

Sophia is using GeoGebra to programme a turtle to draw shapes.

This one's a bit different to the one we saw earlier.

We can see there's a slider for distance and a slider for left turn only, but not a slide for right turn.

And there's also a slider for number of repetitions.

So the way this one works is that the turtle will always move forward and then turn left and it'll repeat those instructions the number of times that you ask it to repeat.

If you want to have a go at this yourself before watching Sophia use it, feel free to click on the link at the bottom of this slide.

Sophia says, "If I set the turtle to move forward and turn the same amount each time, then all the sides and angles will be the same." So it should draw a regular polygon.

Let's see what happens.

Sophia sets a turtle to move forward three and turn 50 degrees to the left, and she sets that to happen seven times.

It looks a bit like this.

Hmm, that doesn't quite look like a regular polygon.

Sophia says, "I've drawn seven sides so far.

The turtle is nearly back to the start, but there is a small gap." So it's not a regular polygon, it's not even a polygon.

Sophia sets a turtle to move forward three and then turn 50 degrees left, and she sets it this time to eight times.

Let's see what happens.

It looks like this.

Hmm, still not quite right.

Sophia says, "The eighth side overlaps the first one.

Why can I not get it to draw a regular polygon with a 50 degree turn?" Well, the sum of exterior angles of a polygon is equivalent to one full turn, 360 degrees.

For regular polygons, 360 is a multiple of the exterior angle then.

For example, we know that one full turn is 360 degrees, if the size of each exterior angle is 50 degrees.

Well, if we divide 360 by 50, it should theoretically work out the number of sides, but we get 7.

2, which is a decimal, and a polygon cannot have a decimal number of sides.

So it can't have 7.

2 sides.

So a regular polygon cannot have an exterior angle of 50 degrees.

It can only be a regular polygon if 360 is a multiple of the exterior angle, which in this case, it's not.

So if we can't draw a regular polygon with a left turn of 50 degrees, what could we do? Sophia considers using a different number of repetitions.

She says, "I wonder what would happen if I increase the number of repetitions." So it's more of an eight.

"Would the turtle ever return back to its starting position?" Let's take a look at this.

The turtle doesn't return to its starting position after one full turn because 360 degrees is not a multiple of 50 degrees.

She says, "Two full turns would be 720 degrees.

That's two lots of 360.

Could it return to its starting position by doing two full turns? Ah, that's not a multiple of 50 degrees either." So no.

"I wonder how many full turns it would take to find a multiple of 50 degrees." Well, three full turns would be 1080 degrees.

That's not a multiple of 50.

Four full turns would be 1,440, which is not a multiple of 50, which is why we get a decimal there.

Five full turns is 1,800, which is a multiple of 50, which is why when we divide 1,800 by 50, we get a whole number, 36.

Sophia says, "1,800 is a multiple of 50, so the turtle will return to its starting position after five full turns, and it will draw 36 line segments in the process." So Sophia sets a turtle to move forward three and turn 50 degrees left 36 times.

And this is how it looks.

It does five full turns.

So it turns 1,800 degrees altogether and it draws 36 line segments.

Sophia says, "I've spotted something.

1,800 is the lowest common multiple of 50 and 360." Lowest common multiple is sometimes abbreviated to LCM.

She says, "So I could use the LCM, the lowest common multiple, to solve similar problems more efficiently." Sophia says, "The five full turns can be found by doing 1,800 divided by 360, which gives five.

And the 36 repetitions can be found by doing 1,800 divided by 50, which gives 36." So let's check what we've learned.

Sophia sets the turtle to move forward three units and then turn left 80 degrees.

She wants to know how many repetitions to use so that the turtle returns to its starting position.

So let's break this problem down into smaller problems. What is the lowest common multiple of 360 and 80.

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

The answer is 720.

So if the turtle's gonna turn 720 degrees, how many repetitions should Sophia use? Pause the video while you work this out and press play when you're ready for an answer.

Sophia should use nine repetitions, which we can get by dividing the 720, which the turtle is about to turn in total by 80, which is what it turns each time to get nine.

How many full turns would the turtle make? Pause video while you work this out and press play when you're ready for an answer.

A turtle would make two full turns.

Each full turn is 360 degrees and 360 goes into 720 twice.

It would look a bit like this.

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

This task contains one question, and here it is.

Sophia is using GeoGebra to programme a turtle to draw shapes.

And for each angle, theta, work out the number repetitions, n, that Sophia would need to use so that the turtle returns to its starting position and also work out the number of full turns that the turtle will complete.

In other words, in part A, Sophia sets the turtle to move forward three and then turn left 75 degrees, and she wants to keep doing that until it gets back to its start position.

How many times would she need to repeat that process each time? And that's the value of n.

And do the same with the angle of 32 degrees and then 170 degrees.

If you would like to, you may wish to draw the shapes yourself on GeoGebra or explore your own shapes using a similar process using the link below.

Pause video while you work through this and press play when you are ready for answers.

Okay, let's take a look at some answers.

Part A, if the angle was 75 degrees, she would need to repeat it 24 times.

So that's de value of n.

That we'll complete five full turns and the image would look a bit like this.

For part B, if the angle was 32 degrees, she would need to repeat the instructions 45 times and that would complete four full turns and the image would look something a bit like this.

And finally, part C, if the angle was 170 degrees, she would need to use 36 repetitions and that would complete 17 full turns and the image would look something a bit like this.

Fantastic work today.

Now let's summarise what we've learned.

An interior angle and exterior angle pair at the same vertex are supplementary.

In other words, they add to 180 degrees.

The sum of exterior angles for any polygon is 360 degrees.

The sum of exterior angles of a polygon can be demonstrated using angles around a point like we saw earlier, but the sum of exterior angles of a polygon can be proven using interior angles on a straight line.

Thank you very much.

Have a wonderful day.