
 NonEuclid: Postulates and Proofs 
  Most pairs of points (A and B) in Spherical Geometry, lie on one and only one great circle; however if A and B happen to be antipodal (on opposite ends of any single axis), then there are an infinite number of different great circles that pass through them. 
  In Euclidean Geometry, for any triangle ABC, there exists a unique parallel to BC that passes through point A. Additionally, it is a theorem in Euclidean Geometry that when two parallel lines are cut by a transversal, then the opposite interior angles are congruent; therefore, ∠NAB ≅ ∠ABC and ∠MAC ≅ ∠ACB. 
  In Hyperbolic Geometry, however, there are an infinite number of lines that are parallel to BC and pass through point A, yet there does not exist any line such that both: ∠NAB ≅ ∠ABC and ∠MAC ≅ ∠ACB. 
 www.cs.unm.edu /~joel/NonEuclid/proof.html (1392 words) 
