How to measure the angle between two parallel lines?

Question

In some geometries, parallel lines "meet/touch/coincide" at infinity. This being the case, there must necessarily be an angle between them. I was wondering what the "value" of this angle would be. Is it always $\pi/2$? Is it $0$? Is it infinite? is it $2\pi$? Or is there some formula which makes the angle variable depending on the perpendicular distance between the lines?

I'm particularly interested in answers that approach the question from multiple different geometries, including geometries where parallel lines don't meet (in which case the question becomes, "what is the angle between two lines which don't meet?"). As mentioned, the concept of "angle" is meaningless in projective geometry. What does this question look like from the perspective of hyperbolic, euclidean, and elliptical geometries?

(It has been a while since I've done serious mathematics and my terminology might be off. I've put words which I'm not sure about in scare quotes. Feel free to edit.)

Answer 1

More accurately, two distinct lines in a projective plane are never parallel.

This being the case, there must necessarily be an angle between them.

Not necessarily.... why would there be? Angles do not play a role in projective geometry. As Wikipedia mentions:

It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant with respect to projective transformations”

Similarly there is no notion of distance. The thing that takes their place is called the cross-ratio.

Answer 2

We speak of the shortest geodesic lines in each geometry.There need be no angle between the parallels.

Euclidean

Parallels never meet, or always meet at a point at infinity.

Elliptic

3D models No parallels. Geodesics intersect at a point always. Parallels on a sphere are not geodesics, except the equator.

Hyperbolic Plane models

Poincare models half-plane and disk models. Parallels meet at a point at infinity, on the boundary axis or circle. There are non-intersecting hyper-parallels.

3D model Constant negative Gauss curvature surface. Two parallel sets with no self intersection in each set .Two geodesic parallel asymptotes through each saddle point from either of two sets.