$5.00 Neutral and Euclidean Geometry
A. Discuss the differences between neutral geometry and Euclidean geometry.
B. Explain the importance of Euclid’s parallel postulate in the development of hyperbolic and spherical geometry.
Note: Euclid’s parallel postulate states the following: “For every line l and for every external point P, there exists a unique line through P that is parallel to l.”
C. The sum of the angles in a triangle varies according to the geometry in which the triangle lies.
1. Prove that the statement “There exists a triangle with a sum of angles greater than 180 degrees” is true in spherical geometry.
2. Prove that the statement “The sum of the angles in any triangle is 180 degrees” is true in Euclidean geometry.
3. Prove that the statement, “Rectangles do not exist,” is true in hyperbolic geometry.
Note: You might want to look up Clairaut’s Axiom and the Universal Hyperbolic Theorem.
D. If you choose to use outside sources, include all in-text citations and references in APA format.