# Geometry

**Geometry** is the field of mathematics dealing with figures in a given space. It is one of the two oldest branches of mathematics, along with arithmetic (which eventually branched into number theory and algebra). The geometry usually studied is

## Contents

## Euclidean Geometry

*Main article: Euclidean geometry*

The most common type of geometry used in pre-collegiate mathematics competitions is Euclidean geometry. This type of geometry was first formally outlined by the Greek mathematician Euclid in his book *The Elements*.

### Parallel Postulate

*Main article: Parallel Postulate*

The fifth postulate stated in the book,

*“Through any line and a point not on the line, there is exactly one line passing through that point parallel to the line”*

was the subject of a controversy for many centuries, with many attempted proofs. It is much less simple than the other postulates, and more wordy. This postulate is the basis of Euclidean geometry.

## Non-Euclidean Geometry

Non-Euclidean geometry are geometries in which the fifth postulate is altered. Types of non-Euclidean geometry include:

## Student Guides to Geometry

## Main Concepts

- The notion of dimensions is fundamental to geometry. N-space is a term related to this concept.
- A point is a geometric structure with no area, length, width, or dimension. Its only property is space. It is said to be zero-dimensional.
- A line is generally taken to mean a straight line, which is the locus of points on the Cartesian plane satisfying a linear function. It has length and position, but no other properties. It is one-dimensional. A line segment means a finite segment of a line, while a ray is a line infinitely extending in only one direction.
- A plane is a line, but in a Cartesian space. It as length, width, and position. It is two-dimensional. The point/line/plane sequence can be extended to spaces and higher dimensions.
- An angle is a structure formed by the intersection two rays at their endpoints. It is measure in either degrees or radians, though the less-common metric unit gradian is also used.