neutral geometry
Imagine doing geometry while deliberately refusing to take sides on the one question that splits Euclidean from non-Euclidean worlds: how many parallels pass through a point. Neutral geometry is exactly this disciplined restraint. You keep all of Hilbert's axioms — incidence, order, congruence, continuity — but you say nothing at all about parallels, neither asserting the parallel postulate nor denying it. Whatever you can prove in this state of suspended judgement is true in every geometry, flat or curved alike.
Quite a lot survives. The triangle congruence criteria (SAS, ASA, SSS, AAS) are all neutral theorems. The isosceles-triangle theorem, the exterior-angle theorem (the exterior angle of a triangle is greater than either remote interior angle), the existence of perpendiculars and of at least one parallel through an outside point, the triangle inequality — none of these needs the parallel postulate. A famous deeper result, the Saccheri-Legendre theorem, is neutral too: the angle sum of any triangle is at most 180 degrees, never more. What you cannot prove neutrally is the punchline that the sum equals exactly 180 — that single statement turns out to be equivalent to the parallel postulate, so it sits on the boundary, decidable only once you take a side.
Neutral geometry is the proving ground for understanding what the parallel postulate actually adds. It shows precisely which classical theorems are 'free' (true regardless) and which are 'paid for' by the fifth postulate. It also clarifies the equivalences: assuming any one of 'there exists a rectangle', 'some triangle has angle sum 180', 'parallel lines stay equidistant', or Playfair's axiom, in the presence of the neutral axioms, gives you all the others and full Euclidean geometry. Strip them all away and you are left with the common trunk from which both Euclidean and hyperbolic geometry branch.
In neutral geometry you can prove the exterior angle of a triangle exceeds each remote interior angle, and that every triangle's angles sum to at most 180 degrees — both with no mention of parallels. But you cannot prove the sum is exactly 180; the moment you can, you have secretly assumed the parallel postulate.
The common trunk of all geometries — true before you choose how many parallels there are.
'Neutral' and 'absolute' geometry mean the same thing; the modern term is 'neutral' because the geometry takes no side. Note that elliptic geometry is usually excluded, since it also alters the order axioms, not just the parallel one.