Is Euclid's 5th Postulate Crucial for Defining Euclidean Geometry?

[radou]

Is Euclid's 5th postulate the basic thing which, if valid or not, makes a geometry Euclidean or non-Euclidean?

 

[HallsofIvy]

Fundamentally, yes. There is also the axiom (I confess I don't rember the number- it might be #2!) that asserts that there exist exactly one line between any two points. "Hyperbolic geometry", in particular, the geometry on the surface of a sphere, does not satisfy that but generally speaking, the distinction between Euclidean geometry and "elliptic geometry"- normally thought of as "non-Euclidean" geometry is the requirement that, through a given point, there exist exactly one line parallel to a given point (known as "Playfair's axiom). While elliptic geometry allows that there exist more than one axiom, hyperbolic geometry requires exist exactly one such line.