### 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.