Lesson Video: Lines and Angles Mathematics

Video Transcript

Lines and Angles

In this video, we’re going to learn how to identify and name points, lines, line segments, rays, and angles. And importantly, we’re going to learn how to read and write the mathematical notation that we can use to describe these. Right, where should we start?

Let’s begin at this position here. It’s a very definite position, isn’t it? We could’ve started in the top corner here or even this position down here, but we didn’t. We chose the exact spot here. We call an exact position like this a point. And because there are lots of possible points out there, to show which one we’re talking about, we can label it with a letter. So a point is an exact position marked with a dot. And if we want to refer to it or talk about it in maths, we just say the letter. This is point 𝐴.

Now, to understand some of the other words we’re learning about in this video, we’re going to need to draw some more points. Let’s put one here, we’ll call this 𝑋, and another one here, we’ll call this one 𝑌. Now, we could take our ruler out and connect these two points together with what looks like a line. Well, in everyday life, we call it a line. We’d say I’ve just drawn a straight line with my ruler. But in maths, when we’re talking about lines and angles, a line is a very definite thing. And it might surprise you to realize that this isn’t one, because although we’ve connected up our two points 𝑋 and 𝑌, we could carry on with our ruler past 𝑌, all the way to the end of the blackboard, and just keep going on and on and on. And who says that our line has to stop at 𝑋 as well? We could carry on with our ruler in the opposite direction, off the blackboard, onto the wall, on and on.

A line continues on and on in both directions. And to show this, we can draw arrowheads at either end. Think of a number line. Numbers carry on going on and on, and we draw arrowheads on either end, don’t we, to show that they carry on. We’ll come back to the idea of a line in a moment. But for now, if we know that a line continues endlessly on and on in both directions, what do we have here? It’s not a line, is it? It has two endpoints 𝑋 and 𝑌. Well, hopefully you now understand this is only part of a much longer line. It’s what we call a line segment.

A line segment is part of a line with two endpoints. You can see these two points we’ve labeled 𝑋 and 𝑌. And if we wanted to write down or refer to a line segment using maths notation, we’d need to write the two letters for the endpoints. So that’s 𝑋 and 𝑌. And then above them, we just draw a mini version of a line segment, like this. This idea of drawing a little mini version is going to crop up again and again in this video.

Now, as we’ve just said, a line segment has two endpoints. But is it possible to draw part of a line that only has one endpoint? Might not sound possible, but you know it is. Let’s draw two more points again to help us. Let’s have this one here. We’ll label this 𝐶. And we’ll put one here. Just to show that we don’t always have to have the next letter in the alphabet, let’s label this one 𝐹.

Now, let’s imagine that we start at point 𝐶 and we connect it up with our ruler to point 𝐹. But we want to show that it’s going to continue going, past 𝐹 all the way, perhaps off the blackboard, onto the wall, on and on and on. What we’ve drawn is endless, but only in one direction. It started at point 𝐶. It might sound confusing, but point 𝐶 is actually the endpoint of what we’ve drawn. We started there, but hopefully you can see why we called it an endpoint. Doesn’t carry on beyond 𝐶, does it? And what we’ve drawn passed through point 𝐹 and continued in this one direction. What we’ve got here is what we call a ray.

To help you remember, think of the Sun’s rays for a moment. They start off from a very definite position. We could call this the endpoint of the Sun. But then the light from the Sun travels on and on. So a ray then is part of a line with one endpoint, which is actually the starting point, that continues in one direction. And the way that we’d write this if we wanted to refer to it using mathematical notation would be to write the two letters of the two points. The first one is our endpoint, and the second one is the point that our ray goes through. So we’ll write 𝐶𝐹. And then above them, this time, we draw a mini version of a ray. And we draw it pointing to the right to show that the ray starts at 𝐶, travels through 𝐹, and goes on and on and on. And we’d read this as ray 𝐶𝐹.

Hopefully, we’ve talked enough about lines now to know that everything we’ve drawn so far isn’t a line. I think it’s time to draw a line; don’t you? We’ll start with two points. Let’s call them 𝐸 and 𝐷. And we’re going to connect them up but also show that our line continues in both directions, not just in one direction. In other words, 𝐸 and 𝐷 aren’t endpoints at all. They’re just points that our line travels through along the way.

So in maths, a line is a straight path that continues in both directions and doesn’t end. Notice how we said “in maths” then. Thankfully, we don’t use this definition in everyday life, do we? Imagine your teacher asking you to line up on the playground and then you realizing that you have to continue in both directions on and on and on into the distance.

Now, I wonder whether you’ve worked out how to write the notation for a line. Our line passes through points 𝐸 and 𝐷. So we’re going to write those two points to begin with: 𝐸, 𝐷. And then above them, we’re going to draw a mini version of a line with arrowheads at both end. And when we see this symbol, we’d say line 𝐸𝐷.

We’re gonna come back to these definitions in a second. But for now, let’s try answering some questions to see if you can remember them.

In the given figure, which of the following would represent the straight line passing through 𝐵 and 𝐶?

The given figure that our question mentions is this diagram here. We can see that it’s made up of four points labeled 𝐴, 𝐵, 𝐶, and 𝐷. And these are connected together by two paths. We’ve got a line segment that connects 𝐴 and 𝐵, and we’ve got another path that seems to start at 𝐵, pass through 𝐶 and also 𝐷.

Our question asks us how we would represent the straight line that passes through points 𝐵 and 𝐶. But before we look at our possible answers, let’s think about what this question means because at the moment we can’t see a straight line that passes through points 𝐵 and 𝐶. In maths, a line is something very definite. It’s a straight path that continues in both directions and doesn’t end. If we look at our diagram, we can see that we do have an endpoint.

At the moment, the path that goes through points 𝐵 and 𝐶 starts at point 𝐵. It does pass through 𝐶. And we can see by the arrowhead that it does continue on and on, but only in one direction. If we’re looking for the straight line that passes through 𝐵 and 𝐶, we’re looking for something that continues in both directions.

Now, it’s still there, even though it’s not drawn on the diagram. Let’s use our pink pen to show how this line is going to continue. It’s going to carry on past 𝐵 on and on. That’s better. Now, we can imagine what a straight line that passes through both 𝐵 and 𝐶 would look like. But how would we represent this?

Underneath the diagram, we can see five possible answers. We can work out which one’s correct because we know that a straight line continues in both directions. Two of our possible answers have a short line above them with an arrowhead at one end. These are mini versions of rays, not straight lines. We know this because a ray starts from an endpoint and then just continues in one direction. That’s why there’s one arrowhead. The last answer doesn’t have any arrowheads. This represents a line segment, in other words, part of a line between two endpoints. The correct answer of course is this one here.

The symbol at the top is a mini version of a line. Of course, it’s continuing in both directions. That’s why there are two arrowheads. And the letters 𝐵 and 𝐶 tell us two points that our straight line passes through. This is the correct symbol that represents a straight line that passes through points 𝐵 and 𝐶. And we’d read it as line 𝐵𝐶.

Think about rays. Name this ray using symbols. What is the endpoint of the ray?

This question’s all about rays, but what are they? A ray is a type of path that starts at a point, which we call the endpoint, and then continues on and on in one direction. If we look at our first picture, we can see this, can’t we? Point 𝐵 is the endpoint. This is where our ray begins. It then passes through point 𝐴. And this arrowhead on the end shows us that it continues on and on.

In the first part of this question then, we need to name this ray using symbols. Imagine we had a page with lots of rays on it and we wanted to show that we were talking about this particular one. How could we refer to it using symbols? Well, firstly, we’d write the letter of the endpoint, which in this case is 𝐵, then the point that the ray passes through, which in this case is 𝐴. And then above these two points, we’d need to draw a mini version of a ray with an arrowhead pointing in the direction that the ray continues. And if we saw this written, we’d read it as ray 𝐵𝐴.

In the second part of our question, we’re given another ray. And our question asks us, what is the endpoint of the ray? We can see that it begins at point 𝐶 and continues in one direction through point 𝐷. Do you think point 𝐷 might be the endpoint? Well, it’s not at the end, is it? Point 𝐶 is at the end. Although it’s where our ray begins, we still call it the endpoint. The endpoint of this ray is point 𝐶.

We’ve used what we know about rays to help answer these questions. The correct way to label the first ray using symbols is as 𝐵𝐴 with a mini version of a ray on the top pointing in the direction that it continues. And then the endpoint of our second ray is point 𝐶.

Let’s come back to our definitions because, if you remember, this video is not just about lines. It’s about lines and angles. Now, from what we know of angles, they’re a way of measuring turn. And you might think that we draw them by drawing two lines. But you know we don’t. Don’t forget lines continue in both directions and don’t end. An angle isn’t formed from two lines. It’s actually formed from two rays. And there’s something interesting about them. Here’s one ray. We’ll label the endpoint with a letter 𝑀. Perhaps we’ll show it’s a ray as well by drawing another point that our ray passes through. And we’ll label that 𝑁.

Now, for us to show an angle, we need to draw another ray. But this one is going to have exactly the same endpoint. In other words, it’s going to begin at point 𝑀. It’s just going to continue in a different direction. Let’s draw one of the points that this ray continues through. And we’ll label it point 𝑃. But to show that we’ve got an angle here, we sometimes draw a curve. This shows the amount of turn we’re talking about.

An angle then is formed by two rays that share the same endpoint. We call this the vertex of the angle. So how would we refer to this angle using notation? Now, there’s a few ways to do this. Firstly, we could just concentrate on the vertex. We could just draw the symbol for an angle, which is a mini version of an angle, and then write the letter that corresponds to the vertex, which in this case is 𝑀, angle 𝑀. The other two ways of describing our angle depend on how we look at it. We always put the vertex in the middle. So we could write angle 𝑁𝑀𝑃 or angle 𝑃𝑀𝑁. And if we’re being clever, we could even use our knowledge of rays. We could describe it as the angle formed by ray 𝑀𝑁 and ray 𝑀𝑃.

Let’s try answering a question about angles now to put into practice what we’ve just learned.

Name the angle formed by ray 𝐹𝐶 and ray 𝐹𝐵.

In the picture, we can see several rays. We know that a ray starts from an endpoint and then continues in one direction. For example, the ray 𝐹𝐴 starts at this point here, continues through point 𝐴, and this arrowhead on the end shows that we’re talking about a ray because it continues on and on in a single direction.

Now, our question talks about two other rays that we can see in the diagram: ray 𝐹𝐶 and ray 𝐹𝐵. How do we know that these symbols represent rays? Well, if we think about the first one for a moment, we can see two letters. The first letter is our endpoint. The second letter is a point that our ray travels through. And importantly, on top of them, we can see an arrow pointing in one direction.

Let’s try and find this ray on the diagram. It begins at point 𝐹. It travels through point 𝐶 and continues out the other side, on and on and on. So where’s our second ray, ray 𝐹𝐵? Well, once again, we’re starting from the same point, point 𝐹. This time though, we’re going to travel through point 𝐵 on and on beyond point 𝐵. So our question mentions two rays, but did you notice they share an endpoint? And that’s point 𝐹 here. And we know that where two rays meet at a single endpoint, they make an angle.

Now, our question asks us to name this angle. We could look at the vertex and write the symbol for angle and then just the letter 𝐹. And we could say that we’re talking about angle 𝐹. But there are lots of angles in this diagram, and they’ve all got 𝐹 as their vertex. We’re going to have to list all three points, and we’re going to write the vertex in the middle. So if we want to name this angle, we could start by writing the angle sign and then write point 𝐵, then 𝐹, that’s the vertex of our angle, and finally 𝐶. We can represent the angle formed by rays 𝐹𝐶 and 𝐹𝐵 by drawing the angle symbol and then writing the letters 𝐵𝐹𝐶.

What have we learned in this video? We’ve learned how to identify and name points, lines, rays, line segments, and angles and how to represent these using symbols.