Lesson video

Hello.

My name is Dr.

Rollinson and I'm excited to 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 missing angles.

And by the end of today's lesson, we'll be able to solve problems that require a combination of angle facts in order to identify the values of missing angles.

But this lesson will have a particular focus on providing explanations of reasoning and logic used.

Yes, we're gonna be finding missing angles, but we're also gonna be backing up our answers by providing justifications that show why our conclusions must be true.

Here are some previous keywords that you may be familiar with and these words will be very useful during today's lesson.

So you may wanna pause the video at this point while you just check you remember all these words mean before press and play to continue.

This lesson contains two learn cycles.

In the first learn cycle, we're gonna be focusing particularly on what things we can write when we are justifying chains of reasoning in angle problems. And then in the second learn cycle, we're gonna be applying that to find missing angles with justifications.

But let's start with justifying chains of reasonings with angles.

Missing angle problems on a complex diagram can be difficult because they can contain many unknown angles.

It's not just the angle that you're asked to find that might be unknown.

There might be lots of unknown angles on that diagram.

And you need to find those first before you can find the angle that solves the problem.

So here are some pieces of advice that may help you solve missing angle problems during today's lesson.

The first is to think of how a complex angle problem can be broken down into small steps.

Usually what looks like a very, very big problem is just often a combination of lots of small problems joined together.

So, by breaking a big problem down into lots of small steps and just focusing on one step at a time, it can make the whole problem seem a bit less daunting.

And then people often ask, where do you start with a big problem? Well, it can be helpful to start by just finding the most straightforward angles first.

Because the more angles you find, the easier it is to find new angles.

Each time you find a new angle, it can help you find other missing angles on the diagram.

We can also simplify a problem in our minds by ignoring parts of the diagram when we're not dealing with those particular parts and just focus on the parts of the diagram that we are dealing with.

Or we can draw additional lines in a diagram to help us see things that we couldn't initially see and that might help us find new angles as well.

But you may also be required to justify your reasoning for missing angles that you find, which is usually done by stating the angle facts that you used.

As a rule of thumb, if you ever use or find a angle that is not provided in the problem, we've gotta explain where that number came from.

And that's usually by stating an angle fact and providing some sort of calculation alongside it.

So these angle facts that'll be particularly helpful throughout today's lesson.

Let's see which ones we remember.

We've got some angle facts on the screen here.

We have blanks at the end of them.

Pause the video while you complete these angle facts and then press play when you're ready to look at the answers.

Okay, here they are.

How many of these did you remember? If there were any that you didn't remember or you got wrong, that's okay.

Just make a note of it now so you can keep referring back to it during today's lesson.

Let's apply all this now while we go through a complex angle problem together.

So here we have a circle and points A to L are equally spaced around the circle.

Point O is at the centre of the circle.

You need to find the size of every angle marked in the diagram, justifying your reasoning at every step.

So here are some bits of advice to keep in mind while we work through this together.

The first is a complex angle problem can be broken down into small steps.

We'll go just solve one little bit of this at a time.

And secondly, let's try and focus on just getting one angle first and then we might be able to use that to find other angles.

So let's start by simplifying this diagram and we can do that by temporarily ignoring parts of it that we're not working with.

So we've now just got this triangle on these three points in a circle.

With that in mind, which angles can be found in this diagram? Pause the video while you think about this and particularly think about what angle facts you might use to find those missing angles.

And then press play when you're ready to continue.

Well, we've got a triangle and we've got three angles in this triangle here.

So we might wanna start by finding the values of these three angles.

Now, we're not just gonna find the missing angles.

Each time we use some sort of fact that helps us, we'll go write that fact down 'cause that fact will be a justification and it'll be part of our solution.

So firstly, points A, E, I are equally spaced around the circle.

So the sides of the triangle are equal length and therefore triangle AEI is equilateral.

Angles in a triangle sum to 180 degrees and angles in an equilateral triangle are equal.

So, if we divide 180 by 3, that will give us 60.

So each of these angles must be 60 degrees.

So now, we found one angle on this diagram, angle IAE.

We can consider all the information that we've previously ignored and we can also draw some additional lines in this diagram if we want to as well.

For example, if we wanna get the angle at the centre IOE, how am I drawing an additional line to help us with that? If we draw an additional line segment from O to A, well, we then have three angles around a point and we know that angles around a point sum to 360 degrees.

Now, points A, E, and I are equally spaced around a circle.

So that means the three angles in the centre are equal.

And if I wanna work out each individual angle, I can do 360 degrees divided by 3.

That gives me 120.

That means this angle at the centre here, angle IOE, is 120 degrees.

Now, I know that that's 120 degrees, I can use it to find the other two little angles I've marked here.

So what can be helpful is to look out for lines that are equal length 'cause they can help us find angles which are equal size.

OE and OI are both radii of the circle, so they must be equal length.

Therefore triangle OEI is isosceles.

So, what do we know about an isosceles triangle? Well, we know angles in a triangle sum to 180 degrees and we know that base angles in an isosceles triangle are equal.

So let's do 180.

Subtract 120 to get 60 and let's divide that 60 by 2 to get 30.

That means each of those two small angles are both equal to 30 degrees.

We can now use these angles to find other angles.

So we previously worked out that angle EIA was 60 degrees.

That's the bottom left hand vertex of the equilateral triangle.

And we've just worked out that angle EIO is 30 degrees.

That's the bottom left vertex of the isosceles triangle.

So, if we find the difference between those two angles by doing 60 subtract 30, we get 30.

And that is the size of angle OIA.

And we can do the same on the other side, on the vertex of the bottom right of the triangles.

We know where AEI is 60 degrees and angle OEI is 30 degrees, if we find the difference between them, we get 30 degrees, which is the size of angle AEO.

If we know that the angle at the top of the isosceles triangle is 120 degrees and we know that angles around a point add up to 360 degrees, we can do 360 subtract 120 to get 240.

So that means angle EOI must be 240 degrees.

All of the marked angles in this complex diagram have now been found.

And while we were finding them, we were writing down the angle facts that justify why those angles must be what we said they were.

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

Look at the diagram and decide which statement can help you find the size of angle KOI based on the information given.

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

For this particular step of the problem, statement A can be most helpful.

Adjacent angles on a straight line sum to 180 degrees.

So with that in mind, what is the size of angle KOI? Pause the video while you work it out and press play when you're ready for an answer.

To get the size of angle KOI, we go do 180 subtract 120.

The 180 came from the fact we mentioned in the previous slide.

It means KOI is equal to 60 degrees.

So now, we know that angle KOI is 60 degrees.

Which statements can help you find the size of angle OIK and angle IKO based on this information? You've got three options to choose from.

Pause the video while you make your choices and press play when you're ready for an answer.

Okay, there are two angle facts here that will help us get the values of those angles that we want.

One of them is about line segments, KO and IO, our radii.

And that tells us that the triangle KOI is at least isosceles 'cause we know those two sides of triangle are equal.

And the other angle fact we'll need to use is the angles in a triangle sum to 180 degrees.

So now we've stated those facts.

What is the size of OIK and the angle IKO? Pause the video while you work these out and press play when you're ready for answers.

So we can get these missing angles by doing 180 which comes from our previous angle fact.

Subtracting 60 to get 120 and then we can divide that 120 by 2 because we've justified why those two angles are equal.

That means angle OIK is equal to angle IKO and they are equal to 60 degrees.

Also means that the triangle KOI is equilateral.

So let's now work through another complex angle problem.

And this angle problem involves parallel lines.

Here's information that is given to us.

AC and HE are parallel.

Triangle BFD is equilateral.

And line segments AB and BG are equal to each other.

We need to find the size of as many angles as possible on this diagram and justify your reasoning for each step.

That means every time we use or find a number that is not given to us, that is not 100, we'll gonna explain where that number comes from.

Let's remind ourselves of some pieces of advice that can help us along the way.

One is about complex angle problems can be broken down to small steps.

We're not gonna find all the angles in one go.

We're gonna find one little bit at a time.

Also we've gotta find one angle to begin with and then we can use that to find other angles.

And we'll find the most straightforward angles to begin with.

And another thing is to ignore parts of diagram that you're not working with.

So let's start by ignoring some of those lines on the left of the diagram.

Here, we now have a much simpler problem and we can find some missing angles.

Perhaps pause the video at this point and think about what missing angles you could find on this much simpler problem, and also what angle facts you would need to write down to justify why those angles are what you say they are.

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

Well, there are multiple different ways you could start this, but here's one way.

We could find this missing angle here.

We could state that co-interior angles on parallel lines sum to 180 degrees and then do 180 subtract the 100 to get 80, which means angle FBC is 80 degrees.

And then we can also bear in mind that alternate angles are equal.

Can we see some alternate angles in this diagram? Angle ABF must be equal to angle EFB because they're alternate and therefore they're both 100 degrees.

And we can see that angle BFH is equal to angle FBC because they're alternate so they're both 80 degrees.

We could have got these angles in a different way by doing adjacent angles in a straight line sum to 180 degrees.

And that's what's great about angle problems. There's lots of different routes we can take.

So now, we've worked out some angles.

We can then start to attend to other parts of diagram that we previously ignored.

So we've put those other lines back in.

So, what other angles can we work out now? Well, we've got these three angles in that triangle in the middle of the parallel lines.

And the question told us that triangle BFD is equilateral.

So if BFD is equilateral and angles in a triangle sum to 180 degrees, we can do 180 divided by 3 to get 60.

So that means all three of those angles are 60 degrees.

And then we can use those angles to find other missing angles.

There are a few cases in this diagram where we have two or three angles that are adjacent on a straight line.

So let's state the fact that adjacent angles on a straight line sum to 180 degrees and then let's use that in a few different places.

We can get this angle that's just been highlighted there by doing 180.

Subtract the sum of 80 and 60 to get 40.

So angle ABD is 40 degrees.

We can get this angle that's just been highlighted by doing 180 subtract the sum of 160.

So angle DFG is 20 degrees.

So we've got even more angles now and there are lots of different ways we can find the few remaining angles that we have.

One way could be to think about how alternate angles on parallel lines are equal.

Can you see which angle is alternate to the one that's just been highlighted? Angle FGD is equal to angle ABG because they're alternate and that means they're both 40 degrees.

And then that angle in between the parallel lines G, D, F.

How would you get that missing angle? We could do adjacent angles on a straight line sum to 180 degrees and then do 180 subtract 60 to get 120.

Or we could have done angles in a triangle, triangle DFG sum to 180 and then get it that way.

Either way angle GDF is 120 degrees.

And then what other parts of the diagram did we ignore? There's a line segment from A to G.

Let's find these missing angles.

How could we get these? Well, the question told us that line segment AB is equal to BG.

So that means the triangle ABG must be isosceles because it has two equal lengths in it.

And we know that angles in a triangle sum to 180 and the base angles in an isosceles are equal.

So we can do 180, subtract the 40 that's in that triangle, and then divide it by 2 to get 70 degrees at each one of these.

And then finally this last angle here, AGH, is alternate to the 70 degrees at GAB.

So that means those angles are equal and it must be 70 degrees.

All of the angles in this complex diagram have now been found and we have provided justifications for our reasoning at every stage of working as well by stating our angle facts.

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

Here, we have a diagram contain a pair of parallel lines.

Can you decide which statement will help you find the size of angle FCD based on the information given.

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

Well, statement B can help us.

Co-interior angles on parallel lines sum to 180 degrees.

So which statement can help you find the size of angle FCB based on this information? Pause the video while you make a choice and press play when you're ready for an answer.

Statement A can help us with this one.

Alternate angles on parallel lines are equal to each other.

And which statements can help you find the size of angle CFB based on the information given to you now? Pause the video while you make choices and press play when you're ready to continue.

Statements A and B can help us with this.

Angles in a triangle sum to 180 degrees and the base angles in an isosceles triangle are equal.

Which statement can help you find the size of angle GFB based on this information? Pause the video while you make a choice and press play when you're ready for an answer.

Statement A can help us here.

We have three angles that are adjacent on a straight line and adjacent angles on a straight line sum to 180 degrees.

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

This task contains two questions and here's question one.

You can see we've got a diagram with a pair of parallel lines.

There are a couple of line segments are equal to each other and you're given one angle of 105 degrees.

Alongside that, there is some working that has found lots of missing angles, but this working is not complete because the justifications aren't there.

You need to complete the working by writing a justification for each angle found.

That will involve writing a sentence to describe an angle fact and also a calculation that might back it up as well.

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

And here is question two.

Each diagram is a circle and contains 12 points that is equally spaced around it and also a point at the centre.

You need to find the value of each unknown and justify your reasoning at each step.

And any angles that you find in one diagram, they may help you with angles in the next diagram because you might notice that these diagrams are quite similar to each other and you can get from one diagram to the other by either rubbing out some lines or drawing some extra lines.

So pause the video while you work on this and press play when you're ready to go through some answers.

Okay, let's go through some answers.

Question one.

For each angle, I'm gonna provide a justification.

But if you've given a different justification, don't panic about it straightaway.

'Cause remember you can work out angles using different chains of reasoning.

So you may just want to show it to someone else if you can to see if they agree with you.

Angle BFG is 75 degrees because co-interior angles sum to 180 degrees and we have that calculation.

Angle FBC is 75 degrees because adjacent angles on a straight line sum to 180 degrees and we have that calculation.

Or we can say alternate angles on parallel lines are equal and therefore those two angles are equal.

Angle BCF is 75 degrees because base angles in an isosceles are equal.

Therefore angle BCF is equal to angle FBC.

And angle EFC is 75 degrees because alternate angles on parallel lines are equal, therefore those two angles are equal.

And angle CFB is 30 degrees because angles in a triangle sum to 180 degrees and we have that calculation or adjacent angles on a straight line sum to 180 degrees and we have this calculation.

And then in question two, each diagram is an adaptation of its previous one.

So we go work out one angle and we can use that to help us work out other angles.

Angle a.

The value of a is 60 and our justification is the outer point are equally spaced around a circle.

So the angles at the centre are equal and angles around a point sum to 360 degrees.

Once we've done that, angles b and c.

Well, the values of b and c are 60.

Our now justification is that radii are the same length in every direction on a circle.

So that means the triangle is either isosceles or equilateral.

Either way, those base angles are gonna be equal.

And angles in triangles sum to 180 degrees and we have our calculations to show how we got 60.

And then for d, it's 120 and our justification is adjacent angles on a straight line sum to 180 degrees.

Angle f and e are both equal.

They are both 30.

And radii are the same length in every direction on a circle.

So that means the triangle once again is isosceles and those base angles are equal.

And our angles are triangle sum to 180 degrees and we have our calculations to show how we got the 30.

And then finally, g is 90 because angles in a triangle sum to 180 degrees and we have that calculation to show how we got 90.

Or we can say that g is equal to the sum of c and e, and that is 60 plus 30 is 90.

(exhales heavily) That was some tough work, but we're now ready for learn cycle two, which is gonna focus on finding missing angles with justifications.

Here, we have a diagram that contains a square inside a regular pentagon.

And we need to find the size of each marked angle and justify each stage of working.

Before we start this, let's bear in mind some pieces of advice.

One is that complex angle problems can be broken down into small steps.

Another is we're gonna start by finding the most straightforward angles first.

And then any angles we find might help us find other angles.

Perhaps pause the video before we start this and consider how you might get into this question.

Which angles do you think you would find first? And then what might you do next? And also, what justifications would you provide at each step? Pause the video while you have a go at this and press play when you're ready to work through it together.

Well, we could start this question by just looking at the square inside.

The angles in a square are probably the most straightforward ones to find because they're all 90 degrees.

And then once we've got those angles, what might be the next most straightforward ones to find? We've got interior angles and a pentagon.

We could get to those by looking at the exterior angles.

We've not been asked to find the exterior angles, but they might help us along the way because exterior angles sum to 360 degrees no matter what the polygon is.

So if we do 360 divided by 5, we get 72.

So each exterior angle is 72 degrees and then the interior angle to that is adjacent on a straight line, so they sum to 180 degrees.

So if we do 180 subtract 72, we get 108, which means each interior angle in a pentagon, in this regular pentagon, is 108 degrees.

So I can put those there as well.

Now, before we crack on this even more, we could have done that in a different way.

We could have done that just by focusing on the interior angles of a pentagon.

We could use this calculation to find the sum of interior angles in a pentagon, which is 540 degrees, and then divide that by 5 to get each interior angle of 108 degrees.

Whichever way you get to it, we've got our same result there and we've got our justifications along the way.

Then what can we do next after that? Well, we could find these two missing angles here.

These are part of an isosceles triangle and angles in a triangle sum to 180 degrees and the base angles isosceles triangles are equal.

So that means we can do 180 subtract 108 to get 72, and divide that by 2 to get 36 degrees.

And then angle ABI is equal to angle BIA, and they are 36 degrees.

And then we can get these angles here.

Each of these angles are adjacent to two other angles on a straight line.

And we know that angles on a straight line, which are adjacent, sum to 180 degrees.

So we can do 180 subtract the sum of 90 and 36 to get 54.

That means angle HIF is equal to angle DBC, which is 54 degrees.

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

So the diagram shows a regular pentagon and a regular octagon.

Which calculation would find the value of x? Pause the video while you make a choice and press play when you're ready for an answer.

The answer is C.

360 divided by 8, and that would give us 45.

But what would our justification be for why we did 360 divided by 8? Pause the video while you choose between statements A, B, and C and press play when you're ready to continue.

The justification is B, exterior angles in a polygon sum to 360 degrees.

That angle x is the exterior angle on the octagon.

This diagram shows a regular pentagon inside a square.

Which calculation and justification finds the value of x? Pause the video while you make a choice and then press play when you're ready for an answer.

The answer is B.

We could do 360 divided by 5 because x is the exterior angle on the pentagon.

And exterior angles sum to 360 degrees.

Here, we have another diagram.

It shows a regular pentagon and a regular hexagon.

Which calculation and justification finds the value of x? Pause the video and choose between A, B, and C and press play when you're ready for an answer.

It would be A.

We could do 540 divided by 5 because interior angles in a pentagon sum to 540 degrees.

Which statement can help you find the value of y? Pause the video.

Make a choice between A, B, and C.

And press play when you're ready to continue.

The answer is B.

co-interior angles on parallel lines sum to 180 degrees.

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

This task contains two questions and you're gonna need to apply everything you've learned in this lesson to solve these problems. In question one, the diagram contains a regular pentagon inside a square.

You need to find the value of each unknown and justify each stage of working.

Now, you can find these in whatever order you want to, but you may want to use the order of the letters as a guide for which ones might be easy to find first and which ones might follow from other ones.

Okay, so 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.

This diagram contains a regular pentagon and a regular octagon.

You need to find the size of each of the angles marked in the diagram and justify each stage of working.

Now, once again, it's up to you if you wanna find the angles in whatever order you want to.

That's absolutely fine.

But you're not sure where to start or what might be a useful order to go through, then you might find a list of angles on the right hand side a useful order to work through.

Pause the video while you work through this and press play when you're ready to go through some answers.

Right.

Let's see how we got on with that.

Once again, I'm gonna provide some justifications to each answer as we go.

You might have something different depending on what order you answered the questions in or what route you took.

If you're a bit unsure, don't worry about it.

Just check with someone else if you can.

So the value of a is 72 because exterior angles sum to 360 degrees and 360 divided by 5 is 72.

B must be 108 because adjacent angles in a straight line sum to 180 degrees and 180 subtract 72 is 108.

C must also be 108 because interior angles in a regular pentagon are equal.

D must be 90 because all angles in a square are right angles.

And e must be 18 because angles in a triangle sum to 180 degrees and 180 subtract the sum of 19 and 72 is 18.

And finally, f must be 54 because adjacent angles on a straight line sum to 180 and 180 subtract the sum of 18 and 108 is 54.

And in question two, once again, your justifications might be different to mine depending on what order you found the missing angles in.

That's absolutely fine.

Let's take a look at one solution.

So angle MFG is equal to 45 degrees because exterior angles sum to 360 and 360 divided by 8 is 45.

Angle EFL is equal to 72 degrees because exterior angles sum to 360 and this time we're doing 360 divided by 5 'cause it's a pentagon to get 72.

And then angle LFM is 63 because adjacent angles on a straight line sum to 180 degrees and 180 subtract the sum of 45 and 72 is 63.

And then angles FGH, KLF, IKL are all equal and they're all equal to 108 degrees because interior angles in a pentagon sum to 540 degrees and 540 divided by 5 is 108.

And then angles DEF and FMI are equal.

They are equal to 135 degrees because interior angles in an octagon sum to 1,080 degrees and 1,080 divided by 8 is 135.

And angle MIK is equal to 126 degrees because interior angles in a pentagon sum to 540 degrees and 540 subtract the sum of those angles is 126.

And angle JIH is equal to angle MIK, which are both 126 degrees because they are vertically opposite, and that means they're equal.

Great work during today's lesson.

That was some pretty tough going.

Let's summarise what we've learned during the process.

The lesson has been about solving complex angle problems while providing justifications for each stage of working.

And we can do that by applying all the angle facts that we've learned up to this point.

For example, missing angles can be found by using facts about parallel lines such as alternate angles and co-interior angles and so on.

We can also use facts about angles in a triangle to help us find missing angles too, or that facts about interior angles and exterior angles in polygons, particularly when they're regular.

The fact that an exterior angle and an interior angle in a polygon when they are adjacent on a vertex sum to 180 degrees, that can help us along the way too.

We should also bear in mind some of the advice we've had along this lesson too.

For example, how a problem may be simplified by ignoring parts of diagram that you're not working with while you work on a different parts.

And then we could also draw additional lines in our diagram to help us see things that we couldn't initially see in the original problem.

Whatever way you do it, make sure you provide justifications each stage of your working.

Great job today.

Thank you.