Terms and Notation – G1

My name is, Jason wrote,

and my affiliation is the University of Oxford,

and this is going to be a course on an introduction to geometry.

Let's start with our first mini lecture,

which is on the basic terms and notation in geometry.

So the starting point

is we're going to

just take

some dots

in the plane so the flat piece of paper is called a plane,

and these dots are called points.

We label these points, usually by capital letters like this ABC.

We can then connect these points by straight lines,

and we label those lines a B to indicate that this connects the points A and B.

Now we can relate

pairs of lines.

So, for example, if I take two horizontal lines like this,

then these are pointing in the same direction,

and so are called parallel lines.

If instead

I take a vertical line like this,

then you see it's pointing and kind of the opposite way to my horizontal lines.

These are then called perpendicular lines,

and perpendicular lines

will always meet

at 90 degrees. So if I look at the angle here, then it will be 90 degrees

and we denote that by a little kind of box like that,

which shows that it's a 90 degree angle.

So more generally, if I have two

lines like this,

which both meet a horizontal line,

they will be parallel if the angle that they

make with the horizontal line is the same.

And if we have two lines like this in the plane,

then they will be perpendicular lines if they meet at a right angle.

So that again that angle there is 90 degrees, which we call a right angle.

Now we're not just interested in lines were interested in more

other shapes.

For example,

if I take

collection of points in the plane like this

and label them A, B, C, D and E,

then I can connect them up

by straight lines in such a way

that my pen

never leaves the paper

and I get back to the point where I started.

The shape that I've created out of the lines

is called a polygon,

and the points that were used to make up the

polygon are called the Vertex is of the polygon,

and these lines

that were used to make up the polygon

are called its

edges

So,

in fact, the polygon that I've drawn here has five sides,

and a five sided polygon

is called

a Pentagon,

which is something we're going to see

in more detail later. In this course,

a particularly interesting class of polygons are the three sided ones,

which we know are called triangles.

Let's suppose that we label the Vertex is ABC.

Then we right Triangle ABC

for this triangle.

And then we know that its edges

or sides

are labelled a B

for the line connecting A and B

B C for the line connecting B and C

and see a

for the line connecting C and A.

But we're also interested in the angles inside

the triangle, for example, this angle here, which I'm going to label X

this angle here, which I'm going to label y

and this angle, which I will call, said.

So how do we denote

these angles?

Well, it's a slightly strange notation.

I'll explain it to you.

So if I want to say what is the angle X

it? We say it's the angle

c A B.

So what we do is we put

the Vertex where we're measuring the angle in the middle

as the middle letter,

so C A B means that we're looking at the angle at a

all. We could equally have written B A. C.

So just to check that we know what's going on,

the angle, which I've called why

is the angle

ABC

and said,

is the angle B C a.

Excellent.

Now, of course,

there are many different types of triangles and

polygons in general that I could draw,

but some are more interesting than others.

For example,

if I have a polygon

with all sides

the same length

and angles the same,

then this is what is called a regular polygon.

So, for example,

I could draw

four

straight lines like this,

all meeting at 90 degrees so they are all perpendicular to each other.

Then, of course,

if all of these sides of the same length this would be a square,

and we will see later in this course that

we're very interested in the regular five sided sake.

That's the regular Pentagon.

Now, if I take a square, then I see that it has lots of nice properties,

in particular has lots of what's called symmetry.

So, for example, I can draw a line down the middle of the square

like this. This dotted line.

And I see that if I reflect

in this line,

then the square

doesn't change.

You see,

I could also do the same thing

with a horizontal line through the middle of the square like that.

Can I do more?

Well, yes.

You see, I could also have picked

a diagonal line

like that

if I reflect in that diagonal again. The square doesn't state that doesn't change

and similarly like this.

So we have lots of reflection symmetries.

But we can do something even more If we label

The Vertex is of the square A, B, C and D,

then we can rotate

to square around.

We can send a to B B to C

C. Two d

and D to a.

This is a rotation symmetry

of the square.

Now, just to show that this is a special feature,

let's suppose that we take a rectangle instead.

So we take a long rectangle like this.

Well, then

we can definitely reflect this

in the horizontal

and we can reflect it in the vertical like that.

But we can't rotate it,

you see, because if I do that rotation

that I said before

then what will I end up with? We'll end up with a tall and

tangle like this,

which doesn't

have

the same symmetries.

And so,

just as a question for you,

let's try to ask,

What

are the symmetries

of a regular

Pentagon?

So in this mini lecture,

we have seen the basics of geometry in terms of its terms and notation.

Next time we'll see ruler and compass constructions.