what do terms like lines and angles mean after Euclid's parallel postulate is relaxed?

Can such non - Euclidean geometries actually be imagined visually or can they be worked on only analytically like 4 dimensional worlds? If they can be imagined, and we relax Euclid's parallel postulate, what exactly do the previous postulates and definitions even mean? As apparently by relaxing the parallel postulate, we achieve geometries where the sum of the interior angles of a triangle isn't 180 degrees anymore but then the triangle itself in such worlds is curved and the angles are not sharp. Now if the angles themselves are not sharp and resemble a curve how do we give a certain degree or radian measure to it? And lastly does a geometry exist where a 'straight line' in the Euclidean sense is non existent, if so what restraint does that system have that says that such a thing is impossible. I mean what mathematical equation or axiom of the system disallows straight lines or sharp angles? Thank you very much for your time!