Questions:
Non-Euclidean Geometry on a Sphere
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 Euclidean 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?