Lesson video

Hello, my name is Dr.

Rowlandson and I'll be helping you with your learning during today's lesson.

Let's get started.

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

This lesson is called checking and securing understanding on chains of reasoning with angle facts.

And by the end of today's lesson, we'll be able to reason mathematically using our knowledge of angle facts.

Here are some previous keywords that'll 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 before pressing play to continue.

This lesson is broken into three learning cycles, and we're going to start with justifying answers with a single angle fact each time.

Let's begin by reminding ourselves some of the angle facts we know, and let's start with this fact here.

Angles around a point are conjugate.

In other words, they sum to 360 degrees.

And we can see an example of that with this diagram here.

We have two angles that meet at a point and they sum to 360 degrees.

Let's now adapt this diagram and consider what other angle facts we remember.

For example with this diagram, when two or more angles meet at a single point on a line, they are supplementary.

In other words, they sum to 180 degrees.

So that means these two angles sum to 180 degrees.

Let's adapt this diagram a little bit more, like this.

When two lines intersect, the angles which are opposite each other on the vertex are vertically opposite.

So these two angles are both opposite on that same vertex, so they are vertically opposite and so are these angles as well.

And vertically opposite angles are equal to each other.

Let's now add another line segment to this diagram, like this.

Now, we have two line segments and another line segment, which is a transversal because it intersects the other two.

We have two intersections, each with lots of angles on.

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

For example, these two angles are both in the bottom right of the intersections.

They're both below their own line and to the right of the transversal.

Therefore, they're in matching positions, so they are corresponding angles.

So are these angles, these are corresponding, these angles are corresponding, and these angles are corresponding.

And corresponding angles are equal when the lines are parallel.

So in each of these cases, we've seen corresponding angles.

If the lines are parallel, those corresponding angles are equal.

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

For example, here we have a pair of alternate angles.

One angle is below its line and to the left of the transversal.

The other angle is above it's line and to the right of the transversal, so they're in opposite corners on different intersections on the same transversal.

Therefore, they are alternate angles.

These are also alternate, so are these, and so are these.

And alternate angles are equal when the lines are parallel.

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

For example, these two angles are co-interior with each other.

They are both on the right of the transversal and they are both in between the other two lines.

So they are co-interior.

These are co-interior as well.

They're both on the left of the transversal and in between the other two lines.

Co-interior angles are supplementary, or other words, they sum to 180 degrees when the lines are parallel.

So let's check what we've learned.

Here we have five diagrams labelled A to E and five angle facts.

You need to match each diagram with the angle relationship that it shows.

Pause the video while you do that and press play when you are ready to see some answers.

Let's take a look at some answers.

Diagram A is an example of alternate angles on parallel lines being equal.

Diagram E is an example of this fact, angles meeting at a point on a straight line sum to 180 degrees.

Diagram C is an example of a next fact.

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

Diagram B is an example of the next fact, that corresponding angles on parallel lines are equal.

And diagram D is an example of this fact, vertically opposite angles are equal.

Relationships between angles can be used to find missing angles and angle facts can be used to provide justifications for our answers.

For example, here we have a diagram with one angle given to us and another angle that we don't know and it's labelled X, and we need to find the value of X and justify our answer with reasoning.

Well, start by thinking about what is the angle relationship between these two angles we can see here.

They are corresponding with each other and what do we know about corresponding angles when they're on parallel lines? We know that the angles are equal, therefore, X is equal to 120 because corresponding angles on parallel lines are equal.

So our answer includes the value of X, 120, and our justification, which is the angle fact we've used.

Let's now change this question bit at a time and consider how our answer might vary each time.

For example, if the question looks like this, what angle relationship can we now see? Well, those angles are alternate and they're on parallel lines, which means X is 120 because alternate angles on parallel lines are equal.

If the question looks like this, well, these two angles are co-interior and between parallel lines, which means X is 60 because co-interior angles on parallel lines are supplementary.

In other words, they sum to 180 degrees and 180 subtract 120 is 60.

How about if we're given a different angle? How about if we're given that this angle is 60 degrees and we need to work out the value of X? Well, the relationship between these two angles we're working with is that they are vertically opposite.

They're on the same vertex and they are opposite each other.

Therefore, X is 60 because vertically opposite angles are equal.

And how about this time? What's the relationship between these two angles? Well, they meet at a single point and form a straight line, so they must sum to 180 degrees.

That means X is 120 because angles that meet at a point on a straight line are supplementary, and 180 subtract 120 is 60.

So let's check what we've learned.

Here you've got a diagram, find the value of X, justifying your answer with reasoning.

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

X is 135 because co-interior angles on parallel lines are supplementary or you can write this sum to 180 degrees and 180 subtract 45 is 135.

Here's another question.

You've got a diagram and the question says, why is X, 45? Pause the video while you choose from one of those options and press play when you're ready for an answer.

The answer is B, X is 45 because corresponding angles are equal on parallel lines.

So what if the diagram has more than one transversal? For diagrams with more than one transversal, you may need to perform additional calculations to find missing angles.

For example, here we have a diagram contains a pair of parallel lines and two transversals.

And we need to find a value of X, justifying our answer with reasoning.

So how might we do that? Well, we can almost see alternate angles.

In fact, if you merged a 60-degree angle and a 50-degree angle together, and ignore that line segmenting that goes through them, that angle would be alternate to the angle labelled X.

So we can use that fact to help us.

Alternate angles on parallel lines are equal.

Therefore, if we add 60 and 50 together, we get 110, which means X is equal to 110.

Here's another example.

This time we can nearly see co-interior angles.

If we merged the angle which is labelled 60 degrees with the angle labelled X degrees together, we would have an angle that is co-interior with this 70 degrees and we know that co-interior angles on parallel lines sum to 180 degrees.

So that means the angle that we can see on the right there is equal to 180, subtract 70, which is 110.

So if the entire of that angle is 110 and we know part of it is 60, we could do 110 subtract 60 to get 50 for our value of X.

So let's check what we've learned.

Find the value of X in this diagram, justifying your answer with reasoning.

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

X is equal to 47 and here is our justification.

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

This task contains one question and here it is, find the value of each unknown, justifying your answer with reasoning.

Pause the video while you do this and press play when you're ready to see what the answers are.

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

Let's start with these ones.

A is equal to 78 because corresponding angles on parallel lines are equal and that one corresponds with the 78 that's above it.

B is 73 because alternate angles on parallel lines are equal, and it's alternate with the 73 that is below it and slightly to the right.

C is 36 because angles that form a straight line sum to 180 and we can see but it forms a straight line with that 144-degree angle.

So 180 subtract 144 is 36.

D is equal to 82 because it is vertically opposite the other angle we can see which is 82, and vertically opposite angles are equal.

E is 68 because it's co-interior with that angle which is 112 degrees and co-interior angles on parallel lines sum to 180 degrees.

So 180 subtract 112 is 68.

And F, well, F plus 38 is 105 because the entire of that angle F and 38, that is alternate with the 105-degree angle.

So if we know F plus 38 is 105, F must be 105 subtract 38, which is 67.

And then what about these ones? G is equal to 99 because co-interior angles on parallel lines sum to 180.

And 180 subtract 81 is 99.

H is equal to 72 because alternate angles on parallel lines are equal.

That's what we can see there.

I is equal to 64 because alternate angles on parallel lines are equal.

Doesn't necessarily matter about that other line segment we're not using in that diagram.

We can see that the angle labelled I is alternate with the one labelled 64 degrees.

And then we can see J is 104 because corresponding angles on parallel lines are equal and it corresponds with the 104 in the top left corner.

And as for K, well, it's co-interior with both the 104 degrees and the 59 degrees, but it's only with the 59 degrees that it's co-interior between parallel lines.

So that means K plus 59 must be one 120, and that means K is 121 and our reason is because co-interior angles between parallel lines sum to 180 and when we subtract them, we get 121.

Great work so far.

Now, let's move on to the next part of today's lesson, which is justifying answers by combining angle facts.

Sometimes you may need to use multiple angle facts to find a missing angle.

One angle fact might not be enough.

Now, if you do find other angles before you find the one that you're asked for, you could write those down on the diagram or you could reference them using three-letter notation.

And if you don't have letters written on the diagram for you, you could always write them in yourself.

For example, let's find the value of X in this situation and justify our answer with reasoning.

Now, the angle which is labelled X and the angle which is marked as 120 degrees, there's no direct relationship between those.

They're not alternate with each other, they're not corresponding, they're not co-interior or vertically opposite or they don't meet at a point on a straight line, so we may need to work out another angle before we work out the value of X.

There are multiple different routes you can take to get this answer.

For example, one solution could be to do the following, you could work out angle DCB and say that's 120 degrees because it's corresponding with the other 120 degrees, and corresponding angles on parallel lines are equal.

Now we know that angle, we could work out the value of X because our new angle and X are together on a straight line and that means they sum to 180 degrees.

Therefore, X equals 60 because angles that form a straight line sum to 180 degrees and 180 subtract 120 is 60.

We could have done that in a different way.

For example, we could have worked out angle FCE.

That is 120 degrees because it's alternate to the 120 degrees given to us, and alternate angles on parallel lines are equal.

Now we know that angle, we could work out X in the same way.

X is 60 because angles that form a straight lines sum to 180 degrees.

And 180 subtract 120 is 60.

We could have done it a different way.

We could have worked out angle BCF because that one is co-interior with the angle given to us, which is 120.

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

So 180 subtract 120 is 60.

Now we know that angle is 60, we can work out the value of X because those two angles, the one labelled X and our new angle, they are vertically opposite each other and vertically opposite angles are equal.

So X equals 60.

There are other ways we can do this.

We could do it by extending one of the line segments, for example.

We could extend the line segment from A to F to create another point, G, and then we could work out angle GFC, which is 60 because it forms a straight line with the angle which is given to us.

And if we know that one is 60, we could work out the value of X because our new angle and X are corresponding, therefore, they're equal.

So what's nice about these problems is that there are lots of different routes you can take to get to your answer.

And there are more routes than the ones we've seen so far.

But what's really important is you explain step by step how you get from the information you are given to the information that you find out.

Here we have an example that involves a reflex angle, reflex angles on figures with parallel lines can also be found using different methods.

For example, if we want to find a value of X in this diagram and justify our answer with reasoning, we can do it in a few different ways.

We could extend some line segments which breaks our angle that we're trying to find into three parts and work out each part separately using corresponding, alternate angles, and all the angle facts that we know.

For example, this angle is 120 degrees because it's corresponding with the angle which is given to us, and corresponding angles on parallel lines are equal.

This angle is also 120 degrees because it's alternate to the angle given to us and alternate angles on parallel lines are equal.

This remaining angle is 60 degrees because angles forming a straight line sum to 180 degrees.

So we could do 180 subtract 120 to get 60.

And that could be subtracting either of those 120s from 180 and then we could add the three angles together to work out the total value of X, which is 300.

Another way we could have done that is by thinking about angles around a point.

We could start by working out the acute angle at that same point, which is 60 degrees because it's co-interior with the 120 given to us and co-interior angles on parallel lines sum to 180 degrees.

And then once we know that, we can consider the angles around a point sum to 360 degrees.

So X is equal to 300 because angles around a point sum to 360 degrees and 360 subtract 60 is 300.

So let's check what we've learned.

Justify why X is equal to 135 by first finding angle EFB.

There are other ways to do it, but try and create a justification that finds angle EFB first and then use that to justify why X is 135.

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

Well, here's our justification.

Angle EFB is 135 degrees because angles forming a straight line sum to 180 degrees and 180 subtract 45 is 135.

Then once we know that, we can say X is also 135 because corresponding angles are equal on parallel lines.

So why do we keep this question the same but change the route in which we answer it? Justify why X is 135 by finding angle CEF first, instead.

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

Well, here's an answer.

Angle CEF is 45 degrees because corresponding angles are equal on parallel lines.

And then we know that X equals 135 because angles formed in a straight line sum to 180 degrees and 180 subtract 45 is 135.

Let's adapt this question ever so slightly now by extending one of the line segments to create point G.

Now, justify why X is 135 by finding angle GED first.

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

Here's an answer.

Angle GED is 45 degrees because alternate angles are equal on parallel lines.

Therefore, X is 135 because angles formed in a straight line sum to 180 degrees.

And 180 subtract 45 is 135.

So we've answered the same question in multiple different ways where each time we've worked out one angle before finding the value of X and each time we've given a reason for our answers.

So over to you now for task B.

This task contains one question and here it is, find the value of each unknown, justifying each answer with reasoning.

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

Okay, let's now go through some answers.

In each of these questions, the answer could be justified in lots of different ways and it'll be impractical to go through every single way in this video.

So instead, let's take a look at one example of an answer for each question, but yours might differ to this one.

Here's an example answer for how to get A equals 117.

Pause the video and take a read of this answer, make sense of it, and then take a look at your own and see how clear yours is in comparison to this one.

And then press play for the next part.

Let's take a look at B.

B is equal to 292 and here's an example of a justification that gets to that.

C is equal to 46 and here's an example of how you can get that and justify it.

And D is equal to 221 and here's an example of an answer that justifies why.

E is equal to 56 and here's an example answer for that one.

And F is 99 and here is an example of a justification for this one.

And then we have I, H, and G, Well, G is 45, you could say because corresponding angles on parallel lines are equal.

H is 65 because corresponding angles on parallel lines are equal.

And I is 70 because vertically opposite angles are equal.

But there are other ways to get some of those as well, especially once you've worked out one, you can have more different ways to working out others.

You're doing great.

Now, let's move on to the third and final part of this lesson, which is investigating by drawing extra parallel lines.

Here we have Aisha.

Aisha starts with a parallelogram and cuts a shape out of it.

Her first cut makes an angle of 33 degrees with the bottom edge and when she cuts along it, her shape looks a bit like this.

Her second cut makes an angle of 64 degrees with the top edge and when she cuts along that, the shape looks a bit like this.

Aisha looks at this and says, "I think I know what this angle is without measuring it." How might she know that? How does Aisha know the value of X now in this situation, based on the information we can see here? Pause the video while you think about this and press play when you're ready to continue.

Well, complex geometry problems can be made simpler by ignoring parts of the diagram that are unlikely to contribute towards the solution.

Aisha says, "The parallel lines on the left and right are not useful here." So let's ignore them.

And it can sometimes be helpful to draw extra lines as well on a diagram.

Aisha says, "I'll break the X degrees angle into two parts by drawing a line through it which is parallel to the top and bottom lines which we know are parallel to each other." It looks a bit like this.

So now we can see there are two unknown angles which are not necessarily equal to each other and those two unknown angles, sum to get X, which is the angle we're trying to work out.

How can we get those two unknown angles? Aisha says, "The top part of this angle is alternate to the 64-degrees angle." And its parallel lines so we know that they are equal.

She says, "The bottom part is alternate to the 33-degree angle." And because they're parallel lines they're equal.

So now we know both parts of that angle which was labelled X.

Aisha says, "X is equal to 64 plus 33." Which is 97 degrees, so the angle is 97 degrees.

Let's check what we've learned.

Here we have a diagram with three parallel lines and lots of angles given to you.

One angle is labelled A degrees and what I want you to do is work out what is the value of A.

You got four options to choose from.

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

The answer is 50.

The angle which is labelled A degrees is alternate to angle labelled 50 degrees between parallel lines.

So what's the value of B? Pause the video while you choose and press play when you're ready for an answer.

The answer is A, 35.

And the same reason can be given.

The angle labelled B degrees is alternate to the angle labelled 35 degrees, and we have parallel lines so they're equal.

So with that in mind, what is the value of C in this situation? Pause the video while you work this out and press play when you're ready for an answer.

C is equal to 85.

It is the sum of the two angles you just worked out, but you could work it out by drawing an extra line going through the angle which is parallel to the other two parallel lines, and then use alternate angles to find a total value of C, which is 85.

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

This task contains one question and here it is, it's in three parts.

In part A, you have four diagrams and you need to find the value of the unknown in each diagram.

Then for part B, I want you to try and notice something.

Look at the value you got for A and the angles you were given, and the value you got for B and the angles you were given, and C, compared to the other angles you were given.

Explain what happens each time and also try and explain why it happens.

If you want to investigate this more and you have access to this slide deck, you can click on the link which takes to a GeoGebra file where you can adapt the diagram and move things around and see whether what you think is always true or just sometimes true.

And in part C, justify this more using algebra by writing an algebraic expression for the highlighted angle.

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

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

A is equal to 66.

B is equal to 111.

C is equal to 249.

And D is equal to 81.

In part B, you need to think about what you notice about the values of A, B, and C compared to the two angles you are given and explain why that happens.

Well, with A being and 66, that is the sum of 25 degrees and 41 degrees.

B is 111 and that is the sum of 80 and 31.

And C is 249, which is the sum of 100 and 149.

So it looks like in each situation, they are the sum of the other two angles which are given.

We could take a look at that on the GeoGebra file now and see if it continues to be the case when we adapt the diagram even more.

If you open up the GeoGebra file, you might see something that looks a bit like this.

And in this situation, we can see that the angle which is 31.

7 degrees and the angle which is 30.

9 degrees, sum to angle in between them, which is 62.

6 degrees.

What happens if we change those angles on the top and bottom but keep the angle in the middle the same? Well, we get something a bit like this, the top and bottom angle change, but no matter where they are, they still sum to 62.

6 degrees.

What happens if we change the angle in the middle? Well, all of them change now, but wherever it goes, we can see that the angle at the top and the angle at the bottom sum to the angle in the middle.

Even if we move the angle all the way over here and we now have lots of very big angles, well, 154.

2 plus 151.

3 is equal to 305.

5.

So for an answer, we could say each angle is a sum of the other two angles.

And the reason why is, if you draw an extra parallel line through the angle, each part of the angle will be alternate to one of the angles given.

And alternate angles in parallel lines are equal.

And then part C, you need to write down an algebraic expression for the highlighted angle.

Well, we're given, one angle is X degrees, the other angle is Y degrees.

We have the same situation, again, we can break it into two parts where each part is alternate to one of those two angles given and we get a total angle of X plus Y degrees.

Fantastic work today.

Now, let's summarise what we've learned.

Alternate, corresponding, and co-interior angles can be identified in diagrams containing pairs of parallel lines or two or more parallel lines.

Facts about angles and parallel lines can be used to find unknown angles, but also other angle facts can be helpful for finding missing angles on a diagram of parallel lines as well.

Even if the angle fact works in diagrams that don't have parallel lines, you can still use them in diagrams that do have parallel lines.

For example, angles around a point sum to 360 degrees.

You can use that in a diagram that has parallel lines or not.

Combinations of facts about angles can be used in succession to find multiple unknown angles.

And there may be more than one way to find a missing angle or justify your answer.

Well done today.

Have a great day.