Lesson video

Hello there.

You made a great choice with today's lesson, it's gonna be a good one.

My name is Dr.

Rowlandson, and I'm gonna be supporting you through it.

Let's get started.

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

This lesson is called checking and securing understanding of interior angles.

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

Here are some previous keywords that will be useful during today's lesson, so you may want 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 off with finding interior angles in polygons.

Interior angles in a triangle sum to 180 degrees.

In other words, these three angles we can see here, they sum to 180 degrees.

This fact can be proven by drawing a line that intersects one of the vertices, and is parallel to the opposite side.

Let's do that together now.

We can look at angle DAC and angle ACB and see that they are alternate between parallel lines, which means they are equal.

Those two angles have been highlighted by putting an extra arc on each of them.

Also, angles OAB is equal to angle ABC, as they are also alternate between parallel lines and they're highlighted with the two extra arcs in them.

Now let's look at those three angles which are bunched together around vertex A.

They rest on a straight line.

Therefore, those three angles, OAB, BAC, and DAC sum to 180 degrees because they form a straight line.

Therefore, angles ABC, BAC, and ACB sum to 180 degrees.

That symbol you just saw there, that is a shorthand symbol for the word therefore.

If you see that, that's what it means.

Let's now apply this to an example, and you'll try one yourself just like it.

Here we have a triangle, we need to find the value of X.

Well, we know that those three angles should sum to 180 degrees, so let's use that fact to make an equation.

Then let's simplify this equation, and then let's solve it to get X equals 52.

So that missing angle is 52 degrees.

Here's one for you to try.

Pause the video while you work out the value of Y, and then press play when you're ready for an answer.

The answer is 49, and here is your working to get to it.

Now, we've answered both of these using an equation, but we could also do this by using a single calculation.

For the example on the left, we could do 180, which is sum of the interior angles, subtract the two angles that we have summed together.

That means we do 180 subtract the sum of 93 and 35, and that would give 52 again.

So could you reattempt the question you've just done, but rather than write down the answer, write down a single calculation that gets you that answer? Pause video while you do that, and press play when you're ready to see what it is.

Okay, we could do 180 subtract the sum of 43 and 88, and that would give you 49 as well.

Let's adapt these questions now.

Let's make this question a little bit more complex, and find the values of X, Y, and Z.

Well, the value of X could be worked out in the same way.

Do 180 subtract the sum of the other two angles in that triangle, and that would give 52 for the value of X.

How do we then use that to find the value of Y and Z? Well, angle X and angle Y form a straight line.

Therefore they sum to 180 degrees, which means Y is equal to 180, subtract 52, which is 128.

Now we know that we have multiple ways of getting angle Z.

Probably the most straightforward way is to recognise that angle X and angle Z are vertically opposite each other, therefore they are equal to each other.

That means angle Z is 52 as well.

Could you do the same with this question? Could you please work out the values of X, Y, and Z? Pause while you do it, and press play when you're ready for an answer.

Here are your answers.

Y is 49, that's what you worked out earlier.

X is equal to Y, so they are both 49.

And Z is equal to 180, subtract 49, which is 131.

Isosceles triangles contain a pair of equal interior angles.

And that can be helpful for us to know, because this fact can be used to find missing angles in isosceles triangles.

Let's take a look at some examples.

Let's find the value of X in this isosceles triangle.

Let's start by considering which angles are equal to other angles.

The angle at the top, which is labelled 71, is between a side which is equal to one of the other sides and the side which is not equal to the other sides.

And the angle which is at the bottom left of that triangle is the same.

It is between one of the equal sides and the third remaining side.

Those two angles are both equal to each other.

Whereas the angle labelled X, that's between both of the equal sides, that means that's different to the other two.

So if we want to work out the value of X, we can make an equation.

We can do X plus two lots of that 71 degrees and that would equal 180, the sum of the interior angles.

We could simplify that equation, and then solve it to get X equals 38.

How about if we change this ever so slightly? Here we have the same triangle, but I've changed which information is given to us.

This time, rather than being given one of the equal angles, we're given the other angle, the 38 degrees.

And what we need to do is find the value of Y, which is one of the two equal angles.

Consider how that might change the equation we solve.

This time, we would do two lots of Y plus the 38 equals 180, because both of those angles, the one on the top and the one on the bottom left, are equal to Y.

Two Y's plus 38.

We could then simplify the equation by subtracting 38 from both sides, and solve it by dividing both sides by two to get Y equals 71.

We knew it was 71 because we could see it's the same as the other triangle, and I told you it's the same as the other triangle, but because we had different information, we solved it in a slightly different way.

X and Y can also each be found by using a single calculation as well, and we can do that by considering what calculations we did step by step when solving the equation.

For example, on the left, X could be calculated by doing 180 subtract two lots of 71.

If you look at the equation we solved, we did two times 71 to begin with, and then we subtracted it from 180.

So that's the same as the calculation we can see there.

On the right, if we look at what we did in the equation, we first subtracted 38 from 180 and then we divided by two, we could do that as one calculation by doing 180, subtract 38 in brackets, and then divide it by two to get 71.

Let's check what we've learned.

Here we have three triangles and three equations.

Match each equation with its appropriate triangle.

Pause the video while you do that, and press play when you're ready for answers.

Let's go over answers.

Equation A matches up with triangle F.

Equation B matches up with triangle D.

And equation C matches up with triangle E.

So let's think about calculations now.

Match each calculation with the unknown that it finds.

So write X, Y, or Z next to each of A, B, and C.

Pause the video while do that, and press play when you're ready for answers.

Let's take a look.

A would work out Y, 180 divided by three.

Because it's an equilateral triangle, all those angles are equal.

For B, that would work out the value of Z.

It is an isosceles triangle, and Z is the odd angle out.

So you'd be subtracting two lots of 61.

Whereas C would work out the value of X.

It's an isosceles triangle again, but X is one of the two equal angles.

So you'd be subtracting 61 and then dividing it by two to work out the value of X.

So we know that angles in a triangle sum to 180 degrees, but what about polygons that have more sides? Polygons with more than three sides can be split into triangles by choosing a vertex, and drawing a line segment to each other vertex.

For example, here we have a quadrilateral, a pentagon, and a hexagon.

Let's do just that.

With the quadrilateral, I could split it into two triangles like this.

A pentagon, I could take the top left vertex and split the shape into three triangles, like so.

And with a hexagon, I could choose, for example, the vertex on the right and draw a line segment to each other vertex to split it into four triangles like we can see here.

When polygons are split up this way, the number of triangles is always two less than the number of sides of its original polygon.

Let's take a look at that.

The quadrilateral has four sides, but we split it into two triangles.

The pentagon has five sides, and we split it into three triangles.

And the hexagon has six sides, and we split it into four triangles.

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

The interior angles of a set of triangles combine to make the interior angles of its original polygon.

For example, with this four-sided quadrilateral, which we split into two triangles, if we look at the interior angles of those two triangles, we can see they combine to get the interior angles of the quadrilateral.

And the same as well with the pentagon, and the hexagon as well.

This means that the sum of the interior angles for each set of triangles is equal to the sum of the interior angles of its original polygon.

The sum of the interior angles for two triangles is equal to the sum of the interior angles for the quadrilateral, for example.

That would be 360 degrees, two lots of 180.

With the pentagon, we have three triangles.

The sum of those interior angles will be three lots of 180, which is 540, which means the sum of the interior angles of the pentagon is also 540.

The hexagon was split into four triangles.

Four lots of 180 is 720, which means the interior angles of the hexagon sum to 720 degrees.

Therefore, the sum of interior angles for any polygon can be found by subtracting two from the number of sides, and multiplying it by 180 degrees.

We subtract two from the number of sides because that tells us how many triangles we make.

Four sides, we split it into two triangles.

Five sides, we split into three triangles.

And we multiply by 180 because each triangle contributes 180 degrees towards the sum of the interior angles.

For example, with the four-sided, the quadrilateral, we do four subtract two times 180.

For the pentagon, we do five subtract two times 180.

And for the hexagon, we do six subtract two times 180.

These calculations can be generalised into a formula.

Whenever you find yourself doing the same calculations again and again, but just changing one thing depending on the question, it's usually a good indication that you can create some kind of formula out of it.

The thing we are changing each time is the number that we are subtracting two from, and that number is the number of sides.

So if we call that letter X, then we have an equation, which is the sum of the interior angles is equal to 180 multiplied by N subtract two, where N is the number of sides.

Let's do an example, and you can try one yourself.

What is the sum of the interior angles for a heptagon? A heptagon has seven sides, and we can work it out by splitting it into triangles.

We can take one of the vertices and draw a line segment to each other vertex, and we can see we have five triangles.

Five lots of 180 is 900, so it's 900 degrees.

Or we could use a formula, we could do seven subtract two, seven, the number of sides, subtract two, multiplied by 180, and we'd still get 900 degrees.

Here's one for you to try.

What is the sum of the interior angles for an octagon? Pause the video while you do it, and press play when you're ready for an answer.

Well, you could split it into triangles.

You would have six triangles, and you would end up with 1,080 degrees.

Or you could use the formula, eight subtract two, multiply by 180, you're still doing six multiply by 180 and you still get 1,080 degrees.

Here's another example.

Find the size of each interior angle of a regular heptagon.

Well, to begin with, we can find the sum of the interior angles.

And you can do that any way you like, here is using the formula.

We get 900 degrees, and then because all those angles are equal, we could divide that 900 degrees by seven to get 128.

6 degrees.

The are more decimals, but that has been rounded to one decimal place.

So could you please find the sum of each interior angle in a octagon, which we can see here is regular.

Pause the video while you do that and press play when you're ready for answers.

Okay, you could do that by first finding the sum of the interior angles and then dividing it by eight to get 135 degrees.

And here we have a polygon that is not regular, we need to find the value of X.

Well, we can find the sum of the interior angles.

We can see that there are seven angles, therefore some of the interior angles would be 900 degrees.

We could then find the sum of the angles that we know, we could add them together and get 800 degrees.

That means X plus the 800 from the other angles equals 900.

So the value of X would be 900 subtract 800, which is 100.

That missing angle is 100 degrees.

Here's one for you to try.

Find the value of Y, pause the video while you do it, and press play when you're ready for an answer.

The answer is 112, and there's your working.

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

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

Pause the video while you do this, and press play when you're ready for question two.

And here is question two, pause the video while you do this and then press play when you're ready to go through some answers.

Here are the answers to question one.

Pause while check this with your own, and press play when you're ready for more answers.

And here are the answers to question two.

Pause while you check, and press play to continue with the lesson.

Well done so far, now let's move on to the second part of this lesson, which is using interior angles in polygons to find out other information.

Interior angles can be used to determine properties of a polygon.

For example, a polygon has interior angles which sum to 3,240 degrees.

How many sides does this polygon have? Well, we can think about this formula here.

That 180 multiplied by N subtract two is equal to the sum of the interior angles, which in this case is 3,240.

And N is the number of sides.

What we need to do now is solve this equation.

We could divide both sides by 180.

We could then add two to both sides, and we get N equals 20.

Let's think about the right hand side of each of these equations and what they tell us.

The 3,240 is the sum of the interior angles.

What about the 18? The 18 is the number of triangles we'd have if we split the shape into triangles from one vertex.

And the 20 is the number of sides, which we know is always two more than the number of triangles.

Let's look at another example.

A regular polygon has interior angles of size 162 degrees.

How many sides does the polygon have? Well, once again, we can make an equation out of this situation, an equation a bit like this.

At the top of that fraction, we have 180 multiply by N minus two.

That works out the sum of the interior angles.

And then to get each angle, we then divide it by how many angles there are, or how many sides there are.

That is N, so that's why we're dividing it by N.

And that would tell us the size of the interior angle, which we know is 162 degrees.

So let's solve the equation.

We can multiply both sides by N, and then we can expand the brackets, add 360 to both sides of the equation.

Subtract 162n from both sides of the equation, and then divide both sides of the equation by 18 to get N equals 20.

Let's check what we've learned.

A polygon has interior angles which sum to 3,960 degrees.

How many sides does the polygon have? Pause while you choose, and press play when you're ready for an answer.

C, 24, and there's your working to get to it.

Here we have Sofia.

Sofia is working out how many sides a regular polygon has, based on one of its interior angles.

Her working so far can be seen below, in this white box we can see here.

What is the size of each interior angle of the polygon? Pause the video while you do this, and press play when you're ready for an answer.

It has 156 degree angles.

So in that case, how many sides does this polygon have? Pause the video while you do this, and press play When you're ready for an answer.

N equals 15, so it has 15 sides.

Finally, use the information you've got here to find the sum of its interior angles.

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

The answer is 2,340.

Let's look at some other things we could work out using interior angles.

Firstly, here we have a diagram, which looks like it's part of a polygon.

Is it possible to draw a regular polygon with interior angles of 160 degrees? Let's take a look.

If it is possible, then how many sides would it have? Well, we could work that out by using the same steps we did earlier.

We could create an equation like this, where we've got these sum of interior angles, 180 multiply by N minus two, divided by the number of angles, N, which gives you the interior angle of 160.

And what we wanna do is work out, what would the value of N be if this interior angle was at 160 and it was regular? Well, we would solve it by multiplying both sides by N.

And then we would expand the brackets, add 360 to both sides, subtract 160 N from both sides, and divide both sides by 20 to get N equals 18.

The fact we've got a whole number there means that it can be regular.

It can be a regular 18 sided polygon, so long as all the sides are the same length and all the angles are 160 degrees.

How about if it's 161 degrees? Let's think how that would change things.

If it was possible, how many sides would it have? Well, we could do the same again, but this time, put the equation equal to 161.

We could solve it by multiplying both sides by N, expanding the brackets, adding 360 to both sides, and subtracting 161 N to get the final part, which is 19 n equals 360.

So if we divide by 19, we get n equals 18.

947 and more decimals.

Now, a shape cannot have a decimal number of sides.

So that means, no, it cannot be regular.

Let's compare the solutions to each of these questions side by side.

Look at the parts that are the same, and look at the parts of the solution that are different.

Can you identify the similarities and differences between these two sets of working? Pause the video while you think about this, and press play when you're ready to continue.

So which parts are the same? Well, we always start in the same way, on the left hand side of this equation, that is 180 multiplied by N minus two over N.

We always then still have 180 multiplied by N minus two on the next pipeline of the equation.

The next line of the equation, each time has 180 N subtract 360, and the next line in the equation always 180 N equals something plus 360.

And finally, we always have something equals 360 on this line of the equation as well.

Those parts are the same each time.

So what's different? Well, the original number that we equate the left hand side of this equation to differs each time, and that number is the interior angle that is given to us.

And then we can see how that difference tracks all the way through our working, until we get to a point where we have either 20 N equals 360, or 19 N equals 360.

Laura says in both cases, the solution ends with a similar equation, so many lots of N equals 360.

She says the coefficient of N in each equation is equal to whatever 180 subtract the interior angle is.

Laura says this could be a more efficient place to start with similar problems, rather than start from the very beginning, or you could start from the very beginning if you want to.

Here we have Lucas, Lucas is trying to make a ring by alternating squares with equilateral triangles.

He says, I wonder if the shapes will join up perfectly to make a ring or if there'll be a gap at the end? How could Lucas find this out before he draws any more shapes? Well, Lucas uses reasoning about angles to determine whether continuing the pattern would result in forming a ring.

He says, if the pattern does connect perfectly at the end, then there will be a regular polygon inside the ring.

Therefore, if I can work out the angle at each vertex, then I can determine whether a regular polygon is possible with that angle.

So let's do that.

He says, this vertex shares the same point as two vertices of squares and a vertex of an equilateral triangle.

So angles in a square are 90 degrees.

Angles in an equilateral triangle are 60 degrees.

And we know that angles around a point sum to 360 degrees.

So to work out the angle inside this possibly regular polygon, inside the ring, we could do 360 subtract the sum of the angles that we know, and that gives 120 degrees.

And now we have a problem a bit like the ones we've been solving so far.

Lucas says, I can now determine if it's possible for a regular polygon to have an interior angle of 120 degrees.

He can do it like this.

Start off with his equation, where he puts equal to 120 degrees.

Solve the equation by multiplying both sides by N, expand the brackets, adding 360 to both sides.

And then we could subtract 120 N from both sides, or we could factorise that and think about it as 180 subtract 120 lots of N equals 360, if that's where he starts.

We get 60 N equals 360, divide by 60 and we get N equals six.

It's a whole number of sides.

That means, yes, it does form a ring, it'll have a regular hexagon inside it.

It'll look like this.

Let's check what we've learned.

In which diagram, or diagrams, show a vertex that could be part of a regular polygon? Pause the video while you choose from A, B, and C, and then press play when you're ready for an answer.

The answers are A and B, C does not.

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

This task contains eight questions, and here are questions one to six.

Pause the video while you do these, and press play when you're ready for the rest of the questions.

Here are questions seven and eight.

Pause the video while you do these, and press play when you're ready for answers.

Here are the answers to questions one, two, and three.

Pause while you check, and press play to continue.

And here are the answers to questions four and five.

Pause while you check, and press play to continue.

Here is an explanation for question six.

Pause while you read this, and press play to continue.

And finally, the answer to question eight.

Pause while you check this, and press play for the next part of the lesson.

Fantastic work so far.

Now, let's move on to the third and final part of this lesson, which is investigating angles on a circular geoboard.

A geoboard is a mathematical tool to explore aspects of geometry.

It is often a board containing pegs laid out in a pattern.

Patterns can vary between boards, but it could look something a bit like any of these here.

And polygons can be made on a geoboard by wrapping elastic bands around the pegs, like we can see with these examples here.

Jacob is using a circular geoboard that contains 15 points equally spaced out around a circle, and a point in the centre.

He makes a triangle.

He says, I wonder what size the interior angles are in my triangle.

How might Jacob work out the size of each interior angle? Let's see what Jacob does.

He says, the points around a circle are equally spaced, so the angles at the centre of each sector are equal.

That means we can do 360 degrees, angles around a point, divided by 15, and that would give 24 degrees.

He then says, the angle I formed at the centre of the circle is equal to four lots of 24 degrees.

That means that angle is 96 degrees.

He then says, the two sides of this triangle are radii of the circle, which means they are of equal length.

That means that this must be an isosceles triangle, and the two remaining angles are equal.

So using that fact and the fact that interior angles and triangles sum to 180 degrees, we could do 180 subtract 96, and then divide our answer by two to get 42 degrees for each of those angles.

So now he knows all of his angles.

Let's check what we've learned.

Here we have a diagram that contains 15 points equally spaced around a circle, and a point in a centre.

We've got a triangle formed inside it.

What is the value of A? Pause the video while you work this out, and press play when you're ready for an answer.

A is 120, and there's our working.

What are the values of B and C then? Pause the video while you work this out, and press play when you're ready for an answer.

B and C are both 30.

Here we have Laura, Laura makes a different triangle on the geoboard.

She says, I haven't used the point in the centre, so I wonder how I could work out the size of the interior angles of my triangle this time.

How might Laura work out the size of each interior angle? What could she do? Pause the video while you think about this, and press play when you're ready to continue.

Laura draws a radius to each vertex of her triangle, like so.

This makes three isosceles triangles.

I could work out each of the three angles at the centre in the same way Jacob did, by using fifteenths of 360 degrees.

360 divided by 15 is 24 degrees.

We can multiply it by four for this angle here, which is four sectors.

This angle is five sectors, so we can multiply it by five.

And the other angle is six sectors, so we can multiply it by six.

We now have the three angles at the centre of each of these three isosceles triangles.

I can now find the remaining two angles in each isosceles triangle by subtracting each angle at the centre from 180 degrees, and then divide them by two, like we do when we're finding missing angles in an isosceles triangle.

So for this triangle, we do that and we get 42 degrees for each one.

For this triangle, we do that and we get 30 degrees for each one.

And for the remaining triangle, we do that and we get 18 degrees for each one.

She says, I could then add together the pair of angles at each vertex of my triangle.

So 30 plus 18 is 48 degrees.

18 plus 42 gives 60 degrees.

And 42 plus 30 gives 72 degrees.

So let's check what we've learned.

What are the values of A, B, and C? Pause the video while you work this out, and press play when you're ready for an answer.

Here are our answers.

What is the value of X? Pause while you do it, and press play when you're ready for an answer.

X is 36, and here's how you get the answer.

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

This task contains one question, and here it is.

Pause the video while you do this, and press play when you're ready for answers.

Here are the answers to this task.

Pause while you check these against your own, and then press play when you're ready for today's summary.

Wonderful work today, now let's summarise what we've learned during this lesson.

Interior angles in a triangle sum to 180 degrees.

That means any polygon can be split into triangles to determine the sum of its interior angles.

The sum of interior angles can also be found by substituting the number of sides for N into the expression, 180 multiplied by N minus two.

The sum of the interior angles is the same, regardless of whether or not a polygon is regular.

And a missing interior angle in any polygon can be found if enough information is known about the rest of the angles.

And finally, the number of sides of a regular polygon can be determined by the size of its interior angles.

Well done today, have a great day.