Lesson video

Hello there, and welcome to today's lesson.

My name is Dr.

Rowlandson and I'll be guiding 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 angle notation and drawing on measuring angles.

And by the end of today's lesson, we'll be able to draw and measure any angle and be confident with various forms of angle notation.

Here are two previous keywords that will be useful again during today's lesson.

So you may want to pause the video if you want to remind selves what these words mean and then press play when you're ready to continue.

This lesson contains three learning cycles, and we're going to start off with measuring angles.

An angle describes how much something has turned.

So in these three images here we can see things turning.

In the image on the left we have a door which turns around its hinge.

In the image in the centre we have a wheel which turns around the centre of the wheel.

And in the image on the right-hand side we have a person who is turning around on the spot there.

Angles are measured at a point, at a vertex or a centre of rotation.

So with these two images here, we can see with the wheel on the left, that wheel is turning around its centre, which is the centre of rotation.

With the image on the right we have two line segments that meet at a point called a vertex, and the angle is measured around that vertex.

The size of an angle describes the amount of turn between two lines or line segments.

In other words, if I was to turn that wheel, how much would it need to turn so that the line segment which is highlighted white, which kind of goes upwards from the centre rotation, how much would it need to turn so that it is in the same direction as the other line segment which is highlighted? With the image on the right you can see there's one line segment above the vertex and one line segment below and to the right of the vertex.

How much would I need to turn one of those line segments so that it is pointing in the same direction as the other line segment? Angles can be measured in degrees, and in this case they are both 112.

5 degrees.

But there are other units of measuring angles which you may learn about if you continue studying mathematics in the future.

One way you can measure an angle is by using a protractor.

A protractor is an instrument for measuring angles and it looks something a bit like the image on the left.

However, protractors can have different features, so they may look different to each other.

To measure an angle with a protractor, we do the following.

First, place the centre spot of the protractor on the vertex and then turn the protractor so that one of its zeroes is over the line or one of the lines that forms the angle.

In this case we can see that the line segment that is above the vertex, that one is covering one of the lines that is in line with the zero on the protractor.

Check which scale has the zero on that line? We can see in this case it is the zero on the outside scale.

And then look for where the other line intersects the scale and read its value.

So we're using the outside scale 'cause that's where zero is, and we're measuring clockwise in this case and we can see that the angle is 112.

5 degrees.

You can see the line is between 112 degrees, 113 degrees, and when using this protractor, we do need to estimate a little bit, but it's 112.

5 degrees.

But if we measured it differently using some different, more accurate instrument, we might see it's not 0.

5, it might be 0.

4 or 0.

3.

But with this protractor, the best we can do is say it's 112.

5.

It's between 112 and 113.

Now, angles can be measured either clockwise or anticlockwise.

It doesn't matter which way you measure your angle, it all depends on how you position your protractor while you're measuring it.

For example, these two images both show the same angle.

With the image on the left, the protractor is positioned so that the line segment which is above the vertex is in line with the zero on the outside of the scale.

Whereas on the right-hand image we can see that the line segment which is below and to the right of the vertex, that one is in line with a zero on the inside scale of the protractor.

Now, that affects which way we will take our readings when we measure the angle.

With the image on the left we can see that the outside scale goes clockwise.

So we're measuring that angle clockwise.

With the image on the right we can see that the inside scale goes anticlockwise, so that's why we are measuring that one in the anticlockwise direction.

But in each case we get the same reading, 112.

5 degrees.

Now, there are a few little things that might cause some problems when you're trying to measure an angle.

For example, if the line segments do not reach the scale of your protractor, it can make it difficult to make an accurate reading because you can't see exactly where that line segment intersects the scale.

But what you could do in this case is you may need to extend the line segments before placing your protractor over the angle.

So, for example, we could take our protractor away.

Use a ruler or a straight edge to accurately extend each of those line segments so they're a little bit longer, and then place your protractor back over it again.

The angle hasn't got bigger.

The line segments have, but the amount of turn from one line segment to the other has remained the same.

So the angle's the same size and you can measure it now using your protractor.

Andeep says, "If you can't extend both lines, for example, if one of them is near the edge of the page, you could just extend the one from which you are taking the reading from." So let's check what we've learnt.

In which diagram or diagrams has the protractor been positioned correctly to measure the angle marked? Pause the video while you make your choices and then press Play when you're ready for an answer.

The answer is B.

This is the only diagram where a spot in the middle of the protractor is placed over the vertex.

So here's another one.

In which diagram or diagrams has the protractor been positioned correctly to measure the marked angle? Pause the video while you make your choices and press Play when you're ready for an answer.

The answers are A and C.

In both of these cases, the protractor is lined up so its spot is on the vertex, the line segment or one of the line segments is in line with a zero on a protractor, but also, the over line segment is somewhere along its scale on the protractor, whereas in B that's not the case.

Here we have a diagram that shows part of a protractor measuring an angle.

What is the size of this angle? Is it A, 58 degrees, B, 62 degrees, C, 122 degrees, or is it D, 138 degrees? Pause the video while you choose and press Play when you are ready for an answer.

The answer is A, 58 degrees, and the reason why is because the zero is on the inside scale and that scale is going anticlockwise, and the line is between the 50 and the 60.

And if you read it off, you get 58 degrees.

Protractors that look like the ones we've used so far only measure up to 180 degrees.

But what do you do when an angle is greater than 180 degrees? Well, an angle that is greater than 180 degrees is called a reflex angle, and we can see one in the diagram on the right.

Izzy, Jun and Laura each measure their reflex angle using a different method.

Let's see what they do.

Izzy says, "I extended the line or one of the lines." In this case we can see the line segment which was below and to the right of the vertex, that has been extended to the left of it and we can see the extension in grey.

Izzy says, "A half turn is 180 degrees.

So I measured the rest of the angle and added it to 180 degrees." So the part which is below the protractor, that's a straight line, it's a half turn, which is 180 degrees.

And then the part which is between that and the other line segment, that is 66 degrees.

So if we add them together, we get the total angle of 246 degrees.

Jun's method is a bit different.

Jun says, "A full turn is 360 degrees.

So I measured the inside angle and subtracted it from 360." We can see that the inside angle is 114 degrees.

So if we subtract that from 360 degrees, we get the same answer of 246 degrees.

Laura said, "I measured it using my circle protractor, which measures all the way up to 360 degrees," because you can get protractors that do measure a full circle like the one we see here.

Now, sometimes it is possible to measure many different angles at a vertex.

Here we can see a vertex which has many line segments coming off it.

It's got line segment DH, going to the left, DE, going upwards, DF, going upwards and to the right, and DG, going to the right and down a little bit.

So there are lots of different angles that can be measured around this vertex.

Here we have Sam and Lucas.

Sam says, "Lucas, can you please measure the size of the angle at D?" (Dr.

Rowlandson humming) Can you see what problem Lucas might have with Sam's request here? Lucas says, "Which one? There are many angles around that point D." So how could Sam better describe which angle Lucas should measure? Pause the video while you think about this and press Play when your ready to continue, What do you think? Let's take a look together.

There are different ways to specify particular angles by using angle notation.

For example, if representing angles with an angle marker, you can represent an angle with a lowercase letter such as, a.

In this case we can see that this angle which is highlighted is, a degrees.

But it is also common convention to label angles using Greek lowercase letters such as theta or alpha.

Another way of specifying angles does not require angle markers, and that is to use three letter notation.

An angle can be defined by three points, described by using capital letters, that highlight the two line segments that bound the angle.

And it looks a bit like this.

We have a symbol that represents it's an angle we're talking about.

The symbol looks a bit like an angle, and then we have three letters, and these three letters are a bit like a route or pathway.

We go from G to D to F.

Let's consider what line segments we use when we do that.

When we go from G to D, we have this line segment which is highlighted here from G to D.

And then when we go from D to F, we go down this line segment, which is highlighted now from D to F.

So the angle we're talking about is the angle bound by those two line segments which we could label as alpha, but it's the same angle as angle GDF.

So, Sam says, "Lucas, can you please measure the size of angle EDG?" And Lucas says, "Sure, it is," and now he goes and measures it.

By using the letters EDG, he's highlighted which line segments he's using, E to D and D to G.

He places a protractor over it and measures it as 112 degrees.

Now, this angle could've also been described as angle GDE as well.

So it can sometimes be helpful to specify whether you're referring to the reflex angle or the acute/obtuse angle whenever possible.

Usually when people are referring to the reflex angle, they specify reflex angle, and if not they just tend to say, the angle.

So let's check what we've learnt.

Which notation describes the highlighted angle here? You've got four to choose from.

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

The answers are A and C.

And here's another question.

Describe the highlighted angle using the three letter notation.

Pause video while you write it down and press Play when you're ready for an answer.

Here are your answers.

You can either say it's angle EMG or you can say it's angle GME.

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

This task contains two questions and here is question one.

Each diagram shows a polygon inscribed inside a circle.

And what you need to do is measure the size of each of the marked angles.

Pause video while you do that and press Play when you're ready for question two.

And here is question two.

You have a diagram containing many line segments, and what you need to do is measure each of the angles described in parts A to G.

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

Okay, let's see how we got on then.

Here are the answers to question one.

Check these angles against the ones that you measured yourself.

Now, when measuring with a protractor, it can sometimes be a bit tricky to be completely accurate, so if you are one degree out, I wouldn't worry about it too much.

Okay, and here are the answers to question two.

Pause the video while you check these against your own and press Play when you're ready to continue.

Okay, well done with measuring angles.

Now let's move on to drawing angles.

Here we have Aisha.

Aisha wants to use her protractor to draw an angle of 75 degrees.

What steps could she take to do this? Pause the video while you think about what Aisha might do to draw an angle of 75 degrees, and press Play when you're ready to watch what she does.

Let's see what Aisha does.

She says, "I'll start by drawing a line segment using my ruler." So she gets her ruler and she draws a line segment and it doesn't really matter how long that line segment is.

She then says, "I'll place my protractor at one edge of my line segment, measures 75 degrees, and mark a point on my page.

So she does that.

She marks a point where 75 degrees is on her protractor.

She then moves away her protractor and says, "I'll then use my ruler to draw a line through the point I marked from the vertex," which she does like this.

"Finally, I'll mark and label my angle," which is done like this.

Here we have Jacob.

Jacob wants to use his protractor to draw an angle of 285 degrees.

So that's a reflex angle.

What steps could Jacob take to do this? Pause the video while you think about this.

Think about why this is trickier than Aisha's one and then consider what he could do, and press Play when you're ready to see what Jacob does.

Okay, let's see what Jacob does.

He says, "A full turn is 360 degrees, and 360 subtract 285, is 75.

So, I could draw a 75 degree angle and then mark the reflex angle at that point." So, he does the same thing as what Aisha did earlier to measure a 75 degree angle, but when he moves away his protractor, he labels the 285 degrees instead.

So let's check what we've learnt.

Sophia wants to draw an angle of 52 degrees and has drawn her first line segment.

In which diagram or diagrams is her protractor in a correct position to do that? Pause the video while you choose and then press Play when you're ready to see your answers.

The answer is A and C.

In both of these cases, the spot in the middle of a protractor is at the edge of the line segment.

Alex has drawn an angle of 230 degrees.

What is the value of a in this case? Pause the video while you write down an answer and press Play when you're ready to see what the answer is.

The answer is 130.

That angle is 130 degrees.

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

This task contains one question and here it is.

You need to draw angles with the sizes shown in each part of the question.

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

Okay, let's see how we got on here.

On the screen we can see an example of each angle, but an angle can be in any orientation and the line segments can be of any length.

So, yours might look like the ones on a screen, but they might look differently.

The best thing I can suggest you do is either measure your angle yourself again and double check it's exactly right or as close to being exactly right as you possibly can with a protractor, or, if you have someone nearby, ask them to measure your angles instead.

Pause the video while you do that and press Play when you're ready for the third and final part of today's lesson.

Okay, you're doing great so far.

You can measure angles.

You can draw angles.

So now let's apply what we've learnt about angles to a game that I love playing which is snooker and pool.

Here we have a diagram of a snooker table or a pool table, and we have a cue ball which is the white circle somewhere around the middle of the table.

The diagram shows the path of the cue ball as it bounces off the side of the snooker table or the pool table.

The angle between the path of the cue ball and the side of the table in this case is 40 degrees.

Now, usually the cue ball will bounce off the side of the table with the same angle.

So we can see here we have the dashed line which shows where the cue ball will go after it's bounced off the side of the table, and the angle between that dashed line segment and the side of the table is the same angle, 40 degrees.

So the main thing to bear in mind for this part of the lesson is that those two angles are usually the same as each other.

One angle is between the path of the cue ball bouncing onto the table and the side of the table, and the other angle is between the path of the cue ball bouncing off the table and the side of the table.

Now, anyone who plays snooker or pool a lot may know that these angles are not always the same.

Sometimes the angle that the cue ball bounces off the side of the table can be affected by the way that the ball is spinning.

However, that can make the mathematics behind these problems quite complicated, so for the problems in this lesson, let's assume that the path of the ball is not affected by the spin.

So these two angles are always the same in all the problems in this lesson.

Now, here we have Alex who is playing pool and there are three pool balls on the table.

We have the cue ball at the top.

We have Alex's ball that he's aiming for, the target, which is at the bottom there.

And in between those two balls we have another ball, his opponent's ball, which is an obstruction.

He wants to hit his ball with a cue ball, but there is another ball obstructing his path.

So how is he going to do it? How is he going to hit his ball with the cue ball? Well, he plans to hit his ball by bouncing the cue ball off the side of the table.

Now, there are different ways Alex could do this, but the shot that Alex is about to take is shown on the diagram here.

Will the cue ball hit Alex's ball after one bounce? What do you think? Pause the video while you consider this and maybe consider how we might investigate this from a mathematical point of view based on what we've learned about angles in today's lesson, and then press Play when you're ready to continue.

Okay, let's take a look at it together.

Using this diagram we have here, what we could do is measure the angle between the path that the cue ball that's about to take and the side of the table.

If we did that using a protractor, for example, with this image here, we would get that the angle is 75 degrees.

We could then measure the same angle from the side of the table at the same point, but going in the other direction.

And draw a line segment going from the vertex between the path of the cue ball and the side of the table, through that point of the protractor to see what path the cue ball is likely to take after it bounces off the side of the table.

And if we do that, we can see that as the cue ball approaches Alex's ball, it will miss it entirely.

So that means the cue ball will not hit Alex's target ball after one bounce.

So let's check what we've learnt.

Here we have another situation, a pool table with just a cue ball on it.

Alex bounces the cue ball off the side of the pool table.

Assuming the path of the ball is not affected by spin, what is the value of x? Pause video while you write it down and press Play when you're ready for an answer.

The answer is 55 degrees.

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

This task contains one question and here it is.

We have a pool table with a cue ball.

We have Izzy's ball which is pink and solid, which is on the right-hand side of this pool table.

And we have an obstruction, her opponent's ball, which is purple with a white stripe through it, and that is in between the cube ball and Izzy's ball.

So, Izzy is playing pool.

She wants to hit her ball with the cue ball, but the direct path is obstructed by another ball.

So she plans to bounce the cue ball off the side of the table once and then hit her ball.

She considers four possible shots which are all labelled A to D.

So, what you need to do, using a copy of this diagram, a protractor, a ruler and a pencil preferably as well, work out which of these shots will work.

In which cases will the cue ball hit Izzy's target ball? And throughout this problem, assume that the path of the cue ball is not affected by the spin.

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

Okay, let's take a look at this together and let's start with shot A.

If you measure the angle between the path of the cube ball and the side of the table for shot A, it would be 50 degrees.

So if you measure the same angle coming off the table, you would see that the cue ball would hit Izzy's target ball.

So, so long as she hits that ball with enough power and doesn't cause the ball to spin in any kinda way, it would hit Izzy's target ball.

Now, shot B looks a bit tricky because the cue ball will just about miss that obstruction, but will it hit her target ball? Well, the angle between that path and the side of the table is 85 degrees.

So if you measure the same angle coming off the side of the table, the path would look like this and the cue ball would definitely miss her target ball in that situation.

So no, not B.

So what about shot C? Well, the angle between the path as it comes onto the table is 60 degrees.

So if we measure the same angle as it comes off the side of the table, it would go over here in this direction.

So it would be quite far away from the target ball.

Now, you may want to consider how Izzy could improve that shot? She could aim a bit further to the right, but the problem there is the cue ball might end up going into the pocket.

So that's a tricky shot.

So then what about shot D? The angle between the pathway and the side table is 80 degrees.

So measure the same angle coming off the table, it would be 80 degrees as well, and it looks like it would hit Izzy's target ball, and it would just about not hit the obstruction as well.

So that one would work but that looks like a very tricky shot.

So here are your answers.

A would work and so would D.

Fantastic work today.

Now let's summarise what we've learnt.

We've learned about what angles are.

An angle is a measure of turn and is often displayed as the rotation between two line segments.

And we've also learned about using a protractor.

A protractor is used to measure the size of angles.

It can also be used to draw angles.

And it measures in a unit called degrees, but there are other units available for angles as well.

And finally, we've learned about notation with angles.

And angles can be referenced using three letter notation, for example, using the angle BCA.

Great job today.

Thank you very much.