Lesson video

Hello, my name is Dr.

Rowlandson, and I am happy to be helping you with your learning during today's lesson.

Let's get started.

Welcome to today's lesson from the unit of geometrical properties with polygons.

This lesson is called problem solving with polygons, and by the end of today's lesson, you'll be able to use your knowledge of polygons to solve some problems. Now, for some of the problems we're gonna see today, you'll see the polygon very clearly within the diagram, but for other problems, you won't see a polygon to begin with, but you may draw a polygon in order to find some missing angles.

Here are some previous keywords that are gonna be extremely helpful for us to solve these problems during today's lesson, so you may wanna pause the video while you remind yourselves the meanings of these keywords before pressing play to continue.

This lesson contains three learning cycles, with each learning cycle focused on a different style of problem.

In the first learning cycle, we're gonna solve problems with angles on parallel lines.

And in the second learning cycle, we're gonna solve missing angle problems in polygons.

And then finally, in the third learning cycle, we're gonna express relationships between pairs of angles or groups of angles using algebra.

But let's start off with solving problems with angles on parallel lines.

Here we have a figure.

The figure below is made from a pair of parallel lines connected via two zig zags.

Now, before we go any further, I should probably stress that zig zag is not a mathematical term.

It's just a description for how this figure looks.

It's got a bit of a zig zag shape to it.

So for this problem, each zig zag is one of the additional vertices that lie between the parallel line segments.

So we can see this one has two zig zags because it's got two vertices between the parallel line segment on the left and the parallel line segment on the right.

Our problem is to find a sum of the angles marked on the underside of this figure.

Now, this is quite a tricky problem because there is so little information given to us in this figure.

None of the angles are given to us anywhere on this figure, so we've got to think very carefully about how we can deduce some angles out of this.

Now, we will work through it together shortly, but before we do, pause the video and just have a think about this problem yourself.

Think about how you might get started with this, or if there are any strategies you can think of that might help us see some of the angles on this diagram, and then press play when you're ready to continue.

So I wonder what we thought we might do to get started.

Well, one thing we could do is draw some additional lines on this diagram.

It might help us see some angles.

One way we could do that could be to extend the two parallel line segments that are given to us.

Now we've done this, I wonder if we could find any angles on this diagram.

Can you spot anywhere where there's an angle that you now know the value of? Perhaps can you spot anywhere where there is an angle that is 180 degrees.

This angle here is 180 degrees.

Because it's half a turn, it is one angle on a straight line, so it's 180 degrees.

So just by extending those two line segments, we've been able to deduce one angle that's on this diagram.

So what other lines could we draw? Well, we could draw additional parallel line segments through the remaining vertices.

So we have now four lines that are parallel to each other.

With that in mind, I wonder if we can spot any more angles on there.

Right at the bottom of this diagram, we've got this angle here is 180 degrees as well.

So we've got two angles.

We now just have five bits of angles left to consider.

So if we don't know what these angles are in particular, I wonder if we can think about what some of these angles might add up to.

Can you think about any angle facts that could be used to connect the remaining angles? Now quite a lot of our angle facts tend to tell us cases where angles sum to 180 degrees.

Can you spot any situations in the diagram where angles might sum to 180 degrees? For example, co-interior angles sum to 180 degrees.

Can you spot any co-interior angles? We have these angles here that are co-interior.

So these two angles, which are co-interior between parallel lines, must sum to 180.

Also angles in a triangle sum to 180 degrees.

Can you spot anywhere where that fact might help us? Well, here we have a triangle, and we know that these three angles here in this triangle must sum to 180 degrees, and two of those angles are ones that are highlighted.

But the problem with this is that the angle in the bottom right hand vertex of that triangle is not one of the ones we wanna find, but that angle is equal to the remaining part of an angle that we have in that diagram because those two angles are corresponding on parallel lines, so they must be equal.

So I think we've now accounted for all the angles or parts of angles that we need to find a sum of, and it appears that either an angle is 180 degrees or it sums with some other angles to make 180 degrees.

So in that case, how many lots of 180 degrees are there? Well, let's count them.

Here we have one.

This angle is 180 degrees.

Here's another.

Now this one is probably the most subtle to see.

We know that angles in that triangle sum to 180, but if we just switch the angle in the bottom right hand vertex of that triangle with its correspondent one, which is equal, we can then see that those three angles sum to 180 degrees.

We have a third lot of 180 degrees here with this angle, and then a fourth lot of 180 degrees with these co-interior angles.

So that means we have four lots of 180 degrees, so the sum of the marked angles must be 720 degrees.

So despite the fact that this problem provided us with such a little amount of information, with no angles explicitly stated for us, we'd still be able to find a sum of the angles marked on the underside of the figure by drawing on some additional actual lines and thinking about relationships between angles, in particular finding angles that sum to 180 degrees.

And then with that information, we've been able to solve the problem.

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

True or false? To find the sum of the marked angles in this diagram, you need to find each angle and then add them together.

Is that true or is it false? And then choose one of the justifications below.

Pause the video while you think about it, and then press play when you're ready to continue.

The answer is false.

If you think back to when we solved this problem, we didn't actually work out the size of any of those four angles marked on that diagram.

We found that parts of some of those angles were 180 degrees, but we didn't actually calculate the size of the whole angle.

We did this by thinking about which parts are 180 degrees and which parts of angles sum to 180 degrees, but we didn't actually work out any of the actual angles themselves.

So our justification is by using angle facts and grouping angles, it may be possible to answer questions without knowing the size of every angle.

Okay, it's your turn now with a similar style of problem.

Here's task A.

It has one question broken into three parts.

You need to find the sum of the angles marked on the underside of each figure.

In part a, you have one zig zag between the pair of parallel line segments.

In part b, you have three zig zags.

Now, you might remember that in the example we did together there were two zigzags, so at this point it might be good to consider the relationship between those three questions when there's one zig zag, two zig zags, and three zig zags.

Because then in part c it says, "A figure contains a pair of parallel lines connected via 10 zig zags.

What is the sum of the angles on the underside of the figure?" And you could do that by drawing a figure with 10 zig zags or you can try and think about relationships and patterns with our previous answers.

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

Right, let's see how we got on with that.

The answer to question a is 540 degrees.

The answer to question b is 900 degrees.

So when there's one zig zag, they sum to 540, when there were two zig zags earlier, they sum to 720 degrees, and when there's three zig zags, they sum to 900 degrees.

Can we spot a pattern with those numbers? It seems that every time we add on an extra zig zag, we add on an extra 180 degrees to our answer.

So we could to answer part c by adding on more and more lots of 180 degrees until we've got what it would be for 10 zig zags, or an alternative approach we could take to this could be to find an nth term of this arithmetic sequence.

And if you did that, you'd get 180n plus 360, where n represents the number of zigzags.

So when there's one zig zag, we do 180 times 1 to get 180, and then add on 360 and we get 540.

For three zig zags, 180 times 3 is 540, then add on 360 and you get 900.

So whichever way we do it, for 10 zig zags we should get 2,160 degrees.

How do you find that, then? Well, let's now move on to the second learning cycle, which is solve missing angle problems in polygons.

Here we have a diagram containing two intersecting pentagons, and we need to find the value of x.

And this is a complex angle problem, so before we start this, let's consider some pieces of advice.

One is that big problems are usually just lots of small problems stuck together, so let's try and break this problem down into lots of smaller, easier problems without worrying too much on just getting straight to the final solution.

A second bit of advice is to always consider trying to find the easiest angles first, the most straightforward angles you can.

You may not need those angles later on, but at least finding some angles can get you started with the question, and they might be helpful towards the next steps.

And the third bit of advice is try and simplify the diagram if you can by ignoring some parts of it while you're not working on them.

If you've got some sort of way of rubbing out some of the lines and then you can put them back again later, maybe with the original copy still visible, or using tracing paper or something like that that allows you just to look at one tiny part of it at time, that can be really helpful for simplifying the problem.

So at this point, pause the video and think about how you might get started with this question before pressing play to work through it with me.

Okay, let's start by focusing on a single part of this diagram.

We've faded out one of those pentagons so we can just focus on the left hand pentagon.

Now we can see just that, can we think about how we might find some angles in that regular pentagon? The interior angles of that regular pentagon could be found.

One method could be by using the sum of interior angles in a pentagon.

Interior angles in a pentagon sum to 540 degrees, so we could get each interior angle by dividing 540 by 5 to get 108.

An alternative method could be to use the fact that exterior angles sum to 360 degrees, so we could use 360 divided by 5 because it's a regular pentagon to get 72, and then use that with the fact that adjacent angles on a straight line sum to 180 degrees to get the interior angle of 108.

However you do it, we've got some angles in this diagram now.

All the angles in that regular pentagon are 108 degrees.

Then we can search for some angles which are equal to some other angles.

So we have the other regular pentagon, and all these interior angles will also be the same.

All of these are 108 degrees.

And then the more angles that we find, the easier it is to find the angle we want.

We're getting closer and closer each time to finding the value of x.

So what could we do next? Well, let's just focus on the overlapping shape in the middle of this diagram, this part.

What shape is that, and how could we use that to find the value of x? Well, one method could be to think about how angles in a quadrilateral sum to 360 degrees.

So if we do 360 subtract the two lots of 108, we get 144 degrees.

And if we think really carefully about how the edges of that pentagon on the left and right were parallel to each other, that would mean that these two edges are parallel to each other as well, and so are these.

That means this shape must be a parallelogram, and opposite angles in a parallelogram are equal.

So we have 144 degrees left between two angles, the top angle and the bottom angle, so let's divide it by 2 and we get x equals 72.

Another way we could have done that could have been to think about how co-interior angles in parallel lines sum to 180 degrees.

So we have 108.

If we subtract that from 180, we get 72.

So x is 72 that way.

Okay, let's check what we've learned then with that.

The angle marked a is angle DEF.

Can you please find the value of a? Pause the video while you have a go at this, and press play when you're ready for an answer.

The value of a is 135.

Here's another question.

We have the same octagon, but we are now focusing on just the bottom right hand corner of it.

We have a triangle within that octagon.

The angle b marked is angle EFD.

Can you please find the value of b in that triangle? Pause the video while you have a go at this, and then press play when you're ready for an answer.

Well, it's an isosceles triangle with one angle being 135 degrees.

The other two angles are equal to each other, so they must both be 22.

5 degrees.

The value of b is 22.

5.

So then let's look at this part of the diagram.

Here we have a line segment going through the octagon which is parallel to one of its edges.

You've got the 135 as the interior angle at DEF.

Can you please find the value of c, which is at the angle EFC? Pause the video while you try this, and press play when you're ready for an answer.

Well, these two angles are co-interior of each other between parallel lines, so they must sum to 180 degrees.

That means the value of c is 45.

And now let's look at this part of the diagram.

We've got the same line segment as the last question, which is parallel to the edge of the octagon, and we have that other line segment we had a couple of questions ago with that isosceles triangle, and you need to find out the angle between those two line segments, the angle DEC.

And to help you with that, I've displayed the angles that you've previously found on this diagram.

Pause the video while you use this information to find the value of d, and then press play when you're ready for an answer.

Well, if angle EFC is 45 degrees, we can see that angle DEC is part of that angle, and the other part of it is angle EFD.

That means the angle that we want plus the 22.

5 degrees from EFD must make 45 degrees for EFC.

So to find the value of d, we'll use a subtraction and we'll get 22.

5.

Okay, it is time for you to apply this yourself now with task B.

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

The diagram shows a regular hexagon intersecting a regular octagon.

You need to find a value of y.

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

And here is question two.

The diagram shows a regular pentagon and a regular octagon, which share two common vertices.

You need to find the value of x.

This is quite a tricky problem, but you've solved some parts of this problem through our checks for understanding earlier, so see if you can remember what we did earlier or take a similar approach to get yourself started with this question.

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

Right, let's go through some answers.

Question one.

We need to find the value of y.

Each interior angle in a regular octagon is 135 degrees, so we could start there.

Each interior angle in a regular hexagon is 120 degrees, so that could be our next step.

And then if we look at the shape that is made by the overlap, that's a pentagon, and the interior angles are pentagon sum to 540 degrees.

We know three of those angles, and the remaining two angles are both equal to y, so they're both equal to each other.

So with that information, we can make an equation which is the sum of the three angles we know plus the 2y equals 540.

We can simplify it and solve it to get y equals 75.

And for question two, the value of x is 13.

5.

Now, there are loads of different ways you can start this question and lots of different routes you can take to get to the value of x, so if you've got 13.

5, you can be pretty confident that you did something right along the way.

So big well done with that.

If you would like to explore some different ways of doing this, on the slide deck on this slide there's a link to a GeoGebra file that takes you through a few different solutions if you want to explore it.

Well done so far.

Let's now move on to the final learning cycle of this lesson, which is expressing relationships between angles using algebra.

So we've seen a few situations where angles are unknown, and we're unable to work out the actual size of the angles, but that's okay because when angles are unknown, algebraic statements can be used to express connections between angles.

Let's take a look at some of those together now.

For example, in this diagram, we don't know the size of either of these angles, but what we could do is express y in terms of x.

Can we think what that would be? Well, first, you are gonna wanna think about what angle fact connects those two angles, and then one approach you could take from here could be to imagine you knew the value of x and think about what would you do then to get the value of y, and then turn that into an algebraic statement using x and y.

Perhaps pause the video while you think about this, and press play when you're ready to work through it together.

Well, the angle fact that connects those two angles is that adjacent angles on a straight line sum to 180 degrees, so if you knew the value of x, you would subtract it from 180 degrees to get the value of y.

So that means we could write that y is equal to 180 subtract x.

Here's another one, very similar, but has an extra detail added in to make it a little bit more complex.

How could we express y in terms of x this time? Pause the video while you think about what you might do differently in this one, then press play when you're ready to work through it together.

Once again, we're gonna use the fact that adjacent angles on a straight line sum to 180 degrees.

So to get the value of y, we would subtract the value of x and the 30 degrees, so we could write adjacent angles in a straight line sum to 180 degrees, that's our justification, and y equals 180 subtract 30 subtract x, which could be simplified to get y equals 150 subtract x.

That means if I then told you the value of x, you'd just subtract it from 150 to get the value of y.

How about now? What would you do differently this time? Pause the video while you think about this one, and then press play when you're ready to continue.

Well, once again, we're going think about how adjacent angles on a straight line sum to 180 degrees, but this time we have two xs and the y are the adjacent angles on that straight line.

So because we have two lots of x, y is equal to 180 subtract 2x.

Here's a different diagram.

This time we have an isosceles triangle with x and y labelled.

Can you express y in terms of x this time? Think about what angle fact you might use here.

And then again, if you knew the value of x, what calculation would you do to get the value of y? And then consider how you might generalise that in terms of algebra.

Pause the video while you think about this, and press play when you're ready to work through it.

Well, some facts that might help us are that base angles in an isosceles triangle are equal, which means the angle in the bottom left vertex of this triangle is equal to the angle in the bottom right vertex.

They are both equal to x.

And also that angles in a triangle sum to 180 degrees.

So if we knew the value of x, we would double it 'cause there's two of them, and then subtract it from 180 to get the value of y.

Therefore, we could write y equals 180 subtract 2x.

And how about this diagram? It's very similar to the last one, but with a little extra detail included.

How could you express y in terms of x this time? Pause the video while you think about it, and press play when you're ready to work through it together.

Well, let's imagine we knew what the value of x was.

What would we do to get to the value of y? Well first, that angle marked x is part of the right angle that we can see there, and the angle that makes up the other part is at one of the vertices of the isosceles triangle, so we could, if we knew the value x, work out the angle at the bottom right hand vertex of the isosceles triangle.

So that means the base angle of the triangle, or one of the base angles, is 90 subtract whatever x is.

So we could write that as an expression 90 subtract x degrees for that angle.

And that would mean that the angle in the bottom left corner of the isosceles triangle would also be 90 subtract whatever x is degrees because base angles in an isosceles triangle are equal.

And then if we knew those two base angles, we would add them together and subtract it from 180 to get the value of y.

So let's do that with the algebra, but let's do it in small steps.

Let's start by writing an expression that is for the sum of those two base angles of the isosceles triangle.

Two lots of 90 subtract x is identical to 180 subtract 2x, and then we can consider that angles in a triangle sum to 180 degrees.

So to get the value of y, we would do y equals 180 subtract the expression that we've just written for those two angles in that triangle.

So y equals 180 subtract, and then in brackets, 180 subtract 2x.

And then when we simplify this expression, we get y equals 2x.

Okay, so before you go ahead and put this into practise in the next task, let's check how well we've understood it.

Here we have a diagram with x and y marked on it.

Which algebraic statement correctly expresses y in terms of x? You've got three options to choose from.

Consider maybe what justification you might use alongside that option, but pause, have a go, and press play when you're ready for an answer.

Our answer is c, y is equal to 360 subtract x, and our justification would be that angles around the point sum to 360 degrees.

There are only two angles around that point, so if you knew x, you would subtract it from 360 to get the value of y.

Here's another one.

Which algebraic statement correctly expresses y in terms of x? Once again, you've got three options to choose from.

Pause the video, make a choice, and press play when you're ready for an answer.

This answer is c, y equals 270 subtract x.

You might be thinking, "Where does that 270 come from?" Well, that's the simplified answer.

Initially, y would be equal to 360, which is the angles that are on a point, subtract the 90 we can see in the diagram and then subtract the x, but when we do 360 subtract 90 we get 270, so it simplifies to this.

And here we have a pair of parallel lines with x and y in it.

Which algebraic statement correctly expresses y in terms of x this time? Pause the video while you make a choice, maybe think about what justification you could use as well, and then press play when you're ready for an answer.

This time our answer is b, y equals 180 subtract x.

The reason for that is those two angles are co-interior in parallel lines, and co-interior angles in parallel lines sum to 180 degrees, so if you knew the value of x, you'd subtract it from 180 to get the value of y.

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

This task contains one question broken into six parts, and here it is.

In each question, you need to express y in terms of x.

In other words, write y equals and then some sort of expression with x in it.

Once you've done that, if you wanna push yourself a bit further, you could also go back and write your justifications in for each one.

Which angle fact did you use to connect your x and y together? Pause the video while you have a go at this, and press play when you're ready for some answers.

Okay, let's see how we got on with that.

In part a, we have adjacent angles on a straight line sum to 180 degrees.

That means y is equal to 140 subtract x.

In part b, we have angles in a triangle sum to 180 degrees, so we have y equals 90 subtract x.

In part c, we have angles in a quadrilateral sum to 360 degrees, so we simplify that to get y equals 200 subtract x.

In part d, we have that exterior angles in a polygon sum to 360 degrees, so we have y equals 210 subtract x.

And then in part e, we've got two angle facts will help us here.

One is that corresponding angles in parallel lines are equal, and that would tell us that the angle on the bottom left of that triangle is equal to y.

And then also we have that angles on a triangle sum to 180 degrees, so we get y equals 90 subtract x.

And then part f, once again, a couple of angle facts can help us.

The first would be that adjacent angles on a straight line sum to 180 degrees.

So where you can see x is the exterior angle of the triangle at that point, the interior one would be 180 subtract x.

Another factor can help us could be that the base angles in a isosceles triangle are equal, so where you have y in the bottom left vertex, you've also got y in the top right vertex, and also all three angles of that triangle will sum to 180 degrees.

So when you put it all together and simplify it, you'll get y equals x over two, or y equals a half x.

Well done with that.

You've applied loads of different angle facts during today's lesson and flexed lots of great problem solving strategies to solve some quite tricky problems during this lesson.

Let's summarise now what we've learned during this process.

Problems can be solved using facts about angles on parallel lines.

Problems can be solved using interior and exterior angles of polygons, and angle problems can be solved and the solutions justified by using these angle facts.

It is possible to write algebraic statements about connected angles, and that is particularly helpful when you don't know the actual size of an angle, but what you do know is the connections between them.

So you can turn that into some sort of algebraic statement that can help you later on or just simply express the connection between those angles in a really formal mathematical way.

And then finally, your knowledge of algebraic manipulation can be helpful for simplifying these algebraic expressions too.

Well done today.

Thank you very much.