Ollege Geometry neutral geometry and Euclidean geometry

Neutral, Euclidean, hyperbolic, and spherical geometries have many interesting characteristics. It is important to understand the differences and limitations inherent in each. In this task, you will explore an interesting, counterintuitive result that will illustrate the differences found in these geometric systems.

Task:

A. Discuss the differences between neutral geometry and Euclidean geometry.

B. Explain the importance of Euclidas parallel postulate in the development of hyperbolic and spherical geometry.

Note: Euclidas parallel postulate states the following: a?For every line l and for every external point P, there exists a unique line through P that is parallel to l.a?

C. The sum of the angles in a triangle varies according to the geometry in which the triangle lies.

1. Prove that the statement a?There exists a triangle with a sum of angles greater than 180 degreesa? is true in spherical geometry.

2. Prove that the statement a?The sum of the angles in any triangle is 180 degreesa? is true in Euclidean geometry.

Note: The attached a?Parallel Postulate to Triangles Diagrama? may prove useful in relating the parallel postulate to triangles.

3. Prove that the statement, a?Rectangles do not exist,a? is true in hyperbolic geometry.

Note: You might want to look up Clairautas Axiom and the Universal Hyperbolic Theorem.

D. If you choose to use outside sources, include all in-text citations and references in APA format.

Note: Please submit word-processing documents as *.pdf (Portable Document Format) files.