Euclidean and hyperbolic geometry have a common set of axioms:
- Side-angle-side congruency for rectangles (SAS)
- The Existence Postulate
- The Incidence Postulate
- The Ruler Postulate
- The Plane separation postulate
- The Protractor Postulate
It is known that from these axioms you cannot prove the Euclidean parallel postulate, and hence we also have interesting geometry in hyperbolic geometry.
Given these axioms, one can show that the existence of at least one rectangle is equvalent to the Euclidean parallel postulate. If parallel lines were equidistant in hyperbolic geometry it would imply the existence of a rectangle, a contradiction.
You might want to look up the Saccheri Quadrilateral and the Lambert Quadrilateral. If I remember correctly, these quadrilaterals were studied to try and prove the Euclidean parallel postulate, without success of course.