*World of Mathematics*. ©2005-2006 Thomson Gale, a part of the Thomson Corporation. All rights reserved.

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## Euclidean Geometries

Euclidean **geometry**, sometimes called parabolic geometry, is a geometry that follows a set of **propositions** that are based on Euclid's five postulates (see **Euclid's axioms**), as defined in his book *The Elements*. More specifically, Euclidean geometry is different from other types of geometry in that the fifth **postulate**, sometimes called the **parallel postulate**, holds to be true. Non-Euclidean geometry replaces this fifth postulate with one of two alternative postulates and leads to **hyperbolic geometry** or elliptic geometry. There are two types of Euclidean geometry: **plane geometry**, which is two-dimensional Euclidean geometry, and **solid geometry**, which is three-dimensional Euclidean geometry.

Euclid's five postulates can be stated as follows:

1. It is possible to draw a straight line segment joining any two points.

2. It is possible to indefinitely extend any straight line segment continuously in a straight line.

3. Given any straight line segment, it is possible to draw a...

This section contains 666 words(approx. 3 pages at 300 words per page) |