# Euclidean and Non-Euclidean geometry

Task:

A. Discuss differences between neutral geometry and Euclidean geometry.

B. Explain the importance of Euclid's parallel postulate and how this was important to the development of hyperbolic and spherical geometries.

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 by example 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.

Note: The attached "Parallel Postulate to Triangles Diagram" may prove useful in relating the parallel postulate to triangles. The only "given" in this diagram is triangle ABC (i.e., the creation of line NM is a required step in your proof).

3. Prove by contradiction (i.e., indirect proof) that the statement, "Rectangles do not exist," is true in hyperbolic geometry.

Note: You may use the hyperbolic triangle angle sum theorem as a given fact in this proof.

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#### Solution Preview

Please see attached for diagrams

A. Discuss differences between neutral geometry and Euclidean geometry.

As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is set aside. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry.

B. Explain the importance of Euclid's parallel postulate and how this was important to the development of hyperbolic and spherical geometries.

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."

The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, ...

#### Solution Summary

Discuss differences between neutral geometry and Euclidean geometry.

Euclidean three space

Suppose instead of working on the Euclidean plane we study geometry on a sphere in (Euclidean) three space. We interpret point to mean any point on the sphere and we interpret line to mean any great circle on the sphere (that is any circumference of the sphere).

a. Is Euclidâ??s parallel postulate true in this setting?

b. How would you define the angle which two great circles make at a point where they

intersect on the sphere?

c. How would you define a triangle on the sphere? Give some examples.

d. Is the sum of the angles in a triangle greater than, less than, or equal to 180 degrees in this setting?

e. Does the angle sum depend on the area of the triangle? How?

f. How does part d relate to showing that the parallel postulate implies the triangle postulate?