# Euclidean GeometryGeometric Definitions

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. 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”*.