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 advanced angle facts.

And by the end of today's lesson, we'll be able to recognise equal corresponding angles, equal alternate angles and supplementary co-interior angles.

Here are some previous keywords that will be useful during today's lesson, so it may be helpful to pause the video if you want to remind yourself what any of these words mean before pressing play to continue.

This lesson is broken into three learning cycles, and we're gonna start with identifying relationships between angles.

Let's begin by reminding ourselves what a transversal is.

A transversal is any line that intersects two or more lines at distinct points, distinct meaning different.

And here we have three examples of figures that all contain a transversal.

Can you spot which line, or line segment, is the transversal in each case? Well, let's take a look.

In example one, this is a transversal.

It intersects the other two lines.

In example two, this is a transversal.

It intersects but doesn't necessarily cross each of those two lines, but it still intersects them.

And in example three, this is a transversal.

It intersects four other line segments.

In some cases it goes through the line segment.

In other cases, it just touches the side of the line segment.

But either case is fine, it's a transversal.

A pair of angles are corresponding if they are in the matching corner of the respective intersection on a transversal.

So, for example, here we have a pair of lines and a transversal that intersects both of those lines, and the angles which are highlighted are corresponding.

Jun describes this as saying, "Both angles are in the bottom right corner of their intersection." Sofia describes this by saying, "They are both on the right of the transversal and below its other line." In other words, they are in the matching positions on each of those intersections.

Let's look at some other examples of corresponding angles.

So let's check what we've learned.

Which diagram, or diagrams, show a pair of corresponding angles highlighted? Pause the video while you make your choice, and press play when you're ready for an answer.

The answer is b.

Here we have another diagram.

On the bottom intersection, there is one angle that is highlighted and labelled e.

On the top intersection there are four angles labelled a to d, and what I would like to know is which angle is corresponding to angle e? Pause the video while you write it down, and press play when you're ready for an answer.

The answer is, a.

Let's look at alternate angles now.

A pair of angles are alternate if they are in the opposite corners of the respective intersection on a transversal.

For example, we have a pair of lines and a transversal, and two angles highlighted here, which are alternate angles.

One angle, the one at the top, is below its line, and on the right of the transversal, below and on the right, whereas the other angle is the opposite of those words, it's above its line and on the left of the transversal.

Those are alternate angles.

Jun describes these angles as being alternate, because, "One angle is in the bottom right of its intersection, while the other angle is in the top left of its intersection." Whereas, Sofia says, "They are both between the pair of lines but on different sides of the transversal." Let's take a look at some other examples of alternate angles.

Now this one's a bit trickier to see, because we have a line segment that goes through one of the angles, but if you ignore that line segment that is in grey and just look at the angle between the transversal and its own line, we can see that, that is alternate to the one at the bottom.

So let's check what we've learned.

Which diagram, or diagrams, show a pair of alternate angles highlighted? Pause the video while you choose, and press play when you're ready for an answer.

The answer is, A and B.

Here's another question.

You have angle e highlighted, which out of a, b, c, and d is alternate to angle e? Pause the video while you choose, and press play when you're ready for an answer.

The answer is, c.

A pair of angles are co-interior if they are on the same side of the transversal and in between the other two lines.

For example, we can see a pair of co-interior angles here.

Jon describes this as, "Both angles are on the right of the transversal and inside the other two lines." And Sofia says, "The word, interior, usually refers to something being inside." So in this case, with co-interior angles, they are together on the same side of the transversal, and they are inside the other two lines.

So let's take a look at some more examples of this.

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

Which diagram, or diagrams, show a pair of co-interior angles highlighted? Pause the video while you choose, and press play when you're ready for answers.

The answer is C.

Here we have angle e highlighted, and four other angles labelled a to d, which angle is co-interior with e? Pause while you choose, and press play when you're ready for an answer.

The answer is d.

Let's now talk about how notation can help us when talking about angles.

Three letter notation can be used to specify pairs of angles.

For example, angle EGB and angle DBC are corresponding.

They're both in matching positions on their intersections.

Angles EGB and ABG are alternates.

One is on the right of the transversal, whereas the other is on the left of the transversal, and one is above its line while the other is below its line.

Angle EGB and angle GBD are co-interior.

They are both on the same side of the transversal on the right, and they are in between the other two lines, so they're co-interior.

Sofia says, "All these statements would still be true if there was an extra line segment from B to E." Even though that line segment from B to E goes through the angle we've marked, it doesn't change the fact that the angle GBD is co-interior with angle EGB, or any of the other facts as well.

Because when we talk about angle GBD, we talk about all of that highlighted angle, ignoring that line segment from B to E.

So let's check what we've learned.

Which angle is corresponding to angle DBA? Write down your answer using three letter notation, Pause while you do that, and press play when you're ready for an answer.

The answer is angle EDC, or you can write that as angle CDE.

Which angle is co-interior to angle DBA? Pause while you write it down, and press play when you're ready for an answer.

The answer is angle CDB, or you can write that as angle BDC.

Which angle is alternate to angle DBA? Pause while you write it down, and then press play when you're ready for an answer.

The answer is angle FDB, or you could put angle BDF.

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

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

For each diagram, you need to state whether the marked angles are alternate, corresponding, or co-interior.

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

And here is question two.

You have five questions, a, b, c, d, and e.

Each question contains a diagram with one angle highlighted, and on each diagram you need to mark three angles.

One of those angles needs to be alternate to the highlighted angle.

So if it's possible to do so, mark the angle which is alternate to the highlighted angle and label it a.

One of the other angles needs to be corresponding to the highlighted angle.

So if it's possible to do so, find the angle which is corresponding to the highlighted one, and mark it and label it b.

And the third angle is meant to be co-interior to the highlighted one.

So if it's possible to do so, find the angle which is co-interior to the highlighted one, mark it and label it c.

In some cases it may not be possible to find all three of those angles in relation to the highlighted one.

So if there's one which is impossible to do, just leave that as it is.

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

And here are questions three and four.

In question three, you have a diagram with lots of letters, and you need to use three letter notation to write down every pair of co-interior angles that can be found on that diagram.

You then need to use three letter notation to write down every pair of corresponding angles, and then every pair of alternate angles.

And then for question four, you need to write down a sentence or two that describes in words, how you can tell if a pair of angles are co-interior.

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

Here are your answers to question one.

Pause and check these against your own, and then press play when you're ready for more answers.

Here are the answers to question two.

Pause while you check, and then press play to continue.

Then question three.

For part A, you need to write down every pair of co-interior angles.

Well, here they are.

You may have written the order of your letters differently.

For example, that first angle, angle HBD, you may have written angle DBH, and that's fine.

So long as that middle letter is the same, and the other two letters are just switched, that's okay.

And the same with the other ones as well.

In part B, you need to write down all the pairs of corresponding angles.

Here they are.

And in part C, you need to write down every pair of alternate angles, and here they are.

In question four, you need to describe in words how you can tell if a pair of angles are co-interior.

Here's an example of an answer.

They are in-between the pair of lines, which may be parallel, and they need to be on the same side of the transversal.

Great work so far.

Now let's move on to the next part of the lesson, which is all about looking at what happens to these angle relationships when the lines are parallel.

Here we have a pair of parallel lines and a transversal.

When a transversal intersects parallel lines, corresponding angles are equal.

So if we look at these two highlighted angles here, we don't know what they are, which is why we've labelled them with algebra, but what we do know, is that they are corresponding, and the lines that they're on are parallel to each other; they're going in the same direction.

That means that those two angles are equal to each other, which is why we've labelled them with the same letter x.

Also, when a transversal intersects parallel lines, alternate angles are equal too.

These two angles highlighted are both x, because they are alternate and on a pair of parallel lines.

And when a transversal intersects a pair of parallel lines, co-interior angles are supplementary.

In other words, they sum to 180 degrees.

So in this diagram here, we don't know what the value of x is, so we don't know the size of that angle highlighted at the top vertex.

But what we do know, is the angle which is highlighted below it in the other intersection, is co-interior to it, which means it's equal to 180 subtract whatever x is in degrees.

That means that each angle on this diagram is either one value or another.

It's either x, or whatever the value of x is, or the angle is 180 degrees subtract x.

Here we have Jacob.

Jacob is identifying which angles equal each other in the figure we can see on the screen.

He shows the angles which are equal by using the same number of arcs.

Jacob says, "These angles are all equal to each other, and there are different ways I can justify this." For example, here we have a pair of corresponding angles, so we know that they're equal to each other.

Here we have a pair of corresponding angles, so we know they're equal to each other, but how do we know that each pair of corresponding angles is equal to the other pair or we need to connect them in some way? We could do it by saying that these two angles are vertically opposite, so they're equal to each other.

Now all these angles are connected, we can see that they're all equal to each other.

We could have also used the vertically opposite angles on the other vertex, or a different fact.

We could say that these are alternate angles.

Now all four angles are connected to each other in one way or another.

Or we can highlight these pairs of alternate and corresponding angles.

Via one route or another, we can justify why one angle is equal to another angle, which is highlighted.

And here's another way.

Jacob says, "The other angles are all equal to each other too." That's why he's used two arcs, and he could justify this in the same way.

So now we have all angles labelled either with one arc or two arcs.

That means that each of the angles take either one value or another.

Jacob says, "Pairs of angles which are different to each other are supplementary." In other words, they sum to 180 degrees.

And there are different ways to justify this too.

For example, you could show these two angles here, which are co-interior between parallel lines, so therefore they are supplementary.

Or you can say that these two are co-interior, or you can talk about angles that are adjacent on a straight line, how they sum to 180 degrees.

And you could do that with other pairs of angles as well.

These relationships between angles on parallel lines can be used to find missing angles when the other angle is known.

For example, here we have a pair of parallel lines.

You got one angle labelled x and another angle labelled 75 degrees, and we need to find the value of x, justifying our answer with reasoning.

Now, finding the value of x is probably the easier part of this question, because with a diagram like this, x is either going to be 75, or 180 subtract 75.

And we can see that it's an acute angle, and the one above it is an obtuse angle, so you could probably guess that it's 75.

But what's trickier here is providing a justification for why our answer is true.

Why is x 75 degrees? We can't just rely on guesswork alone.

We need to think about what fact justifies what our answer is.

X is 75, because corresponding angles on parallel lines are equal and these two angles, which are highlighted, are corresponding, and they're on parallel lines, therefore they are equal, which means x must be 75, so the angle is 75 degrees.

Now, Izzy says, "The reasoning we use will differ depending on which angle is known." For example, here we have a pair of corresponding angles, but if we change the question slightly, so it looks like this, we now have a pair of alternate angles.

So x is still 75, but this is because alternate angles on parallel lines are equal.

And this time, we're not given the 75 degrees, we're given a angle which is 105 degrees, but we know it's co-interior with the other angle, which is labelled x, that between parallel lines.

So our justification here is that x is 75, because co-interior angles on parallel lines are supplementary, and 180 subtract 105 is 75.

So let's check what we've learned.

Find the value of x; justify your answer with reasoning.

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

The answer is x = 60, because co-interior angles in parallel lines are supplementary, and 180 subtract 120 is 60.

So, what happens if this one is labelled x instead? Find the value of x now, and justify your answer with reasoning.

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

The answer is 120, because corresponding angles on parallel lines are equal.

How about if this angle is labelled x degrees instead? Find the value of x, and justify your answer with reasoning.

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

The answer is 120, because alternate angles on parallel lines are equal.

Angle relationships can also be used to determine whether a pair of lines are parallel.

Here we have a pair of line segments that look parallel, but we don't necessarily know for sure.

We have Andeep, Laura, and Sam, who are going to help us out.

Andeep says, "The lines look parallel." Laura says, "They might look parallel, but their directions might be ever so slightly different." And Sam says, "We could check using a transversal." So, we have a transversal now drawn through that pair of lines.

Let's think how we could check using that transversal.

We could measure each of the angles made between the transversal and its other line segments, and then look at how big those angles are in relation to each other.

Andeep says, "The lines are parallel, because corresponding angles are equal." We can see that these two highlighted ones are 105 degrees.

Laura says, "The lines are parallel, because the alternate angles are equal to each other." Here the highlighted ones are both 75 degrees.

And Sam says, "The lines are parallel, because the co-interior angles sum to 180 degrees.

And here we can see the highlighted ones are 75 and 105, which sum to 180.

So, those angle facts about alternate angles being equal, corresponding angles being equal, and co-interior angles being supplementary, are only true when the lines are parallel.

So, if those facts are true, then we know the lines are parallel.

Here's another example: Determine whether the diagram contains a pair of parallel lines.

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

Well, these lines are not parallel, and Andeep, Laura, and Sam are gonna tell us why.

Andeep says, "These lines are not parallel, because the corresponding angles are not equal." And we can see that with the highlighted angles here.

Laura says, "The lines are not parallel, because the alternate angles are not equal," and we can see that with these highlighted angles.

And Sam says, "These lines are not parallel, because the co-interior angles do not sum to 180 degrees." 58 plus 121 is not 180.

So, those lines are not parallel.

Now, not all angle facts can determine whether lines are parallel.

For example, we have a diagram here with two angles which are adjacent to each other on a straight line, 75 degrees and 105 degrees.

Andeep says, "Angles on a straight line sum to 180 degrees, and these angles do that." But Laura says, "That only tells us that the line on the right is straight, not that it's parallel to the line on the left." Sam says, "We need to know at least one angle on the other vertex to determine whether the lines are parallel." So let's check what we've learned.

Here we have a diagram that has not been drawn accurately.

You've got three statements, A, B, and C, and you need to decide which statement is true.

Pause while you choose, and press play when you're ready for an answer.

The answer is B.

The figure does not contain a pair of parallel lines.

One justification you could give, could be that those two angles are co-interior, but they do not sum to 180 degrees, so they're not parallel.

Here we have another diagram that has not been drawn accurately.

The same statements again, which one is true? Pause while you choose, and press play when you're ready for an answer.

The answer is A.

The figure does contain a pair of parallel lines.

Those two angles are alternate, and we can see that they are equal, so the lines must be parallel.

Here we have three diagrams that have not been drawn accurately.

For which diagram, or diagrams, is it not possible to determine whether the lines are parallel? Pause while you choose, and press play when you're ready for an answer.

The answer is A and B.

In those diagrams, you cannot determine whether the line segment at the bottom is parallel to the line segment at the top, because you don't know an angle on each vertex.

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

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

You need to find the value of each unknown, justifying your answer with reasoning.

In other words, don't just write down the number for what the angle is.

Also write down the angle fact you used to justify the answer.

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

Here is question two.

None of these diagrams have been drawn accurately.

So you can't use a protractor or rely on sight to help you decide whether or not the lines are parallel.

What you need to do is use the numbers given to you to decide which diagrams contain a pair of parallel lines, decide which diagrams do not contain a pair of parallel lines, and decide which diagrams it is not possible to determine whether or not the lines are parallel.

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

Okay, let's go through some answers.

Question one, the value of a is 77, because corresponding angles on parallel lines are equal.

B is 77, because alternate angles on parallel lines are equal.

And C is 103, because co-interior angles on parallel lines are supplementary, and 180 subtract 77 is 103.

In part two, which diagrams contain a pair of parallel lines? Well, that would be B, G, H, and J.

Which diagrams do not contain a pair of parallel lines? Well, we know that for sure with diagrams, A, D, F, and I.

And then, in which diagrams is it not possible to determine whether the lines are parallel? That would be with C and E.

Those lines might be parallel, or it might not be.

We don't know with the information given.

You're doing great.

Now let's move on to the third and final part of this lesson, which is investigating angles with paper folding.

Here we have Lucas.

Lucas takes a rectangular piece of paper.

He folds it in half and then opens it up again, like we can see on the screen here.

So Lucas now has the piece of paper he started with, but there is a fold going vertically through the centre of it.

Lucas says, "I can see three vertical lines, and these are all parallel with each other." So, he then takes the piece of paper, and he folds it at a different angle and opens it up again.

He now has the piece of paper he started with, but with two folds visible on it, one going vertically and one going diagonally.

He says, "I can now see lots of angles.

I wonder how many angles I would need to measure before I could work out the rest?" I wonder why he doesn't need to measure them all.

How could he possibly know all the angles by just measuring a few of them? Pause the video while you think about this, and press play when you're ready to continue.

Well, let's see what he thinks.

Lucas considers which angles he already knows.

He says, "I know that each corner of the piece of paper is a right angle, because it's a rectangle." So he knows some angles without measuring any.

He then says, "These are also right angles, because the fold in the middle is parallel to the sides of the rectangle." He says, "I could justify this further with the fact that co-interior angles between parallel lines sum to 180 degrees, and in each case, 180 - 90 = 90." Lucas then considers the relationship between the remaining angles, and he labels them with letters a to h.

He says, "I wonder which angles I could work out if I knew the value of a." Pause the video and think about this.

If you knew the value of a, would you be able to work out the value of b, c, d, and so on? Could you work them all out, or would you need to know more? Pause while you think about it, and press play when you're ready to continue.

Let's see what Lucas thinks.

He says, "A and b are supplementary, in other words, they sum to 180 degrees, because the angles form a straight line." So if he knows a, he can work out b.

He says, "A and c are also supplementary, and so are a and g.

And this is because each pair are co-interior between parallel lines." So he knows a, he can work out b, c, and g.

He says, "A and d are equal, because they are corresponding angles in parallel lines." He says, "A and f are also equal, and so are a and h, and this is because each pair are alternate angles in parallel lines." And then he says, "There are lots of ways he can now work out e.

It's vertically opposite to c, it's corresponding with b, and alternate to g.

So it's equal to all three of those angles," we just said.

And he also says, "There are other ways I could work out e by using supplementary angles." But that means we now have all the angles highlighted.

"So," he says, "I only need to measure one of the acute or obtuse angles, and then I could work out the rest." So let's check what we've learned.

A rectangular piece of paper is folded as shown below with those six steps, in the same way Lucas did.

And a larger version of the end product can be seen on the right with the angles labelled a to h.

The angle labelled a is equal to 140 degrees.

Which other angles are also 140 degrees? Pause the video while you write down the letters, and press play when you're ready to see what the answers are.

D, f, and h, are all 140 degrees as well, and can be justified in lots of different ways.

So, in that case, what are the sizes of angles b, c, e, and g? Pause the video while you work it out, and press play when you're ready for an answer.

40 degrees.

We can get it by doing 180 subtract 140.

So, it's over to you now for Task C.

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

A rectangular piece of paper is folded as shown below, and is a larger version of the end product, and it can be seen with one angle given.

And what you need to do is find the size of the remaining angles on the paper.

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

And here is question two.

And this time, it's gonna be you who does the paper folding.

Take a rectangular piece of paper, make a fold that is parallel to one of its sides, and then unfold it again.

And then make another fold that is not parallel to any of its sides, and then unfold it again.

Label any angles which are right angles to begin with.

And then label one of the acute or obtuse angles x degrees.

And then finally, write an expression for each of the remainding angles in terms of x.

Pause the video while you do this, and press play when you're ready to look at some answers.

Okay, here are your answers to question one.

Pause while you check these against your own, and then press play when you're ready to look at question two.

And here is question two, with an example of what the paper might look like at the end.

However, your folds might be in different positions, or one of them might be at a different angle.

Fantastic work today.

Let's summarise what we've learned during this lesson.

A pair of angles are corresponding if they are in the matching corner of the respective intersection on a transversal.

They are equal if the lines are parallel.

If the lines are not parallel, they won't be equal.

A pair of angles are alternate if they are in the opposite corners of the respected intersection on a transversal, and they are equal if the lines are parallel as well.

A pair of angles are co-interior if they are on the same side of the transversal and in between the other two lines.

And they are supplementary, or in other words, they sum to 180 degrees if the lines are parallel.

Great work, thank you very much, and have a nice day.