# 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

## Introductory Videos

https://youtu.be/51K3uCzntWs?t=842 \\ https://youtu.be/j3QSD5eDpzU

## 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, is the following statement:

*If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.*

Though some resources state:

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

This postulate 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.

There are also other types of geometry that don’t follow this postulate, like hyperbolic geometry and spherical geometry, which say that there are more than 2 parallel lines, or there are no parallel lines, respectively.

## 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.