# Euclidean GeometryEuclid’s Axioms

Before we can write any proofs, we need some common terminology that will make it easier to talk about geometric objects. These are not particularly exciting, but you should already know most of them:

A **point***size* or *shape* themselves. They are labelled using capital letters.

In Mathigon, large, solid dots indicate interactive points you can move around, while smaller, outlined dots indicate fixed points which you can’t move.

A **line***width*.

Lines are labeled using lower-case letters like *a* or *b*. We can also refer to them using two points that lie on the line, for example

A **line segment**

A **ray***line* and a *line segment*: it only extends to infinity on one side. You can think of it like *sunrays*: they start at a point (the sun) and then keep going forever.

When labelling rays, the arrow shows the direction where it extends to infinity, for example *does* matter.

A **circle****radius**

## Congruence

The two shapes on the right basically look identical. They have the same size and shape, and we could turn and slide one of them to exactly match up with the other. In geometry, we say that the two shapes are **congruent**

The symbol for congruence is

Here are a few different geometric objects – connect all pairs that are congruent to each other. Remember that *more than two* shapes might be congruent, and some shapes might not be congruent to *any* others:

Two line segments are congruent if they

Note the that *“congruent”* does not mean *“equal”*. For example, congruent lines and angles don’t have to point in the same direction. Still, *congruence* has many of the same properties of *equality*:

- Congruence is
**symmetric**: ifX ≅ Y then alsoY ≅ X . - Congruence is
**reflexive**: any shape is congruent to itself. For example,A ≅ A . - Congruence is
**transitive**: ifX ≅ Y andY ≅ Z then alsoX ≅ Z .

## Parallel and Perpendicular

Two straight lines that never intersect are called **parallel**

A good example of parallel lines in real life are *railroad tracks*. But note that more than two lines can be parallel to each other!

In diagrams, we denote parallel lines by adding one or more small arrows. In this example, ** a ∥b∥c** and

**. The**d ∥ e

*“is parallel to”*.

The opposite of *parallel* is two lines meeting at a 90° angle (right angle). These lines are called **perpendicular**

In this example, we would write *a* *b*. The *“is perpendicular to”*.

## Euclid’s Axioms

Greek mathematicians realised that to write formal proofs, you need some sort of *starting point*: simple, intuitive statements, that everyone agrees are true. These are called **axioms***postulates*).

A key part of mathematics is combining different axioms to prove more complex results, using the rules of logic.

The Greek mathematician *father of geometry*, published the five axioms of geometry:

**First Axiom**

You can join any two points using exactly one straight line segment.

**Second Axiom**

You can extend any line segment to an infinitely long line.

**Third Axiom**

Given a point *P* and a distance *r*, you can draw a circle with centre *P* and radius *r*.

**Fourth Axiom**

Any two right angles are congruent.

**Fifth Axiom**

Given a line *L* and a point *P* not on *L*, there is exactly one line through *P* that is *L*.

Each of these axioms looks pretty obvious and self-evident, but together they form the foundation of geometry, and can be used to deduce almost everything else. According to none less than *“it’s the glory of geometry that from so few principles it can accomplish so much”*.

Euclid published the five axioms in a book *“Elements”*. It is the first example in history of a systematic approach to mathematics, and was used as mathematics textbook for thousands of years.

One of the people who studied Euclid’s work was the American President

This is just one example where Euclid’s ideas in mathematics have inspired completely different subjects.