According to this new geometry (has more that a century), by a point can exist more that one line, so according to parallel definition this more that 1 line would be parallels to the first line but not parallels between theirs (they have 1 commun point).
In same form would occur the same in the infinites points that use these multiple parallels, all that billions of lines would be parallel to the first line and not parallels between theirs.
By reciprocity in all or many of the points of the first line would exist many parallel lines parallel to any point of the millions parallels lines to the first, but not parallels between theirs.
According to this we cannot say that a parallel from a parallel are also both parallels. By that probably the geometry would need to go out of the maths.
But there are more:
According to hyperbolic geometry (the acute case) and elliptic geometry (the obtuse case) the sum of angles of a triangle can to be more or less of 180º : "The sum of the measures of the angles of any triangle is less than 180° if the geometry is hyperbolic, equal to 180° if the geometry is Euclidean, and greater than 180° if the geometry is elliptic" according to http://en.wikipedia.org/wiki/Non-Euclidean_geometry#History
By that they give an example of a triangle formed by the equator of the Earth and meridians that with 2 angles of 90º have another angle in the pole and so add more of 180º.
According to this :
1 - Know parallel lines can not be parallels, because according to this 2 lines with 90º angle (parallels) make a triangle. So to the before note of that in a point can to be more or a parallel also can affirm that a parallel also can not to be a parallel.
2 - A semi-sphere is not a triangle, in the example the line of the equator is equidistant to the pole angle and by that is a line but is not straight.
Really in semi-spheres, cones and semi-cones there is near a triangle (it has 2 angles of 90º and 1 more), but semi-spheres and semi-cones are not triangles and the line between the 2 angles of 90º is equidistant to the other angle.
If I could make a triangle with lines not straight I could make triangles from 0º to 360x3º (less the minimum angle we consider x 3), but a triangle has 3 straight lines and 3 angles.
Really in hyperbolic and elliptic geometries the triangles really are also of 180º. How? easy: make a triangle in a sheet of paper and now you can curve the paper in hyperbolic to see the result of a hyperbolic triangle, ... also you can bend the paper in the form you like to understand how would be a triangle in any other geometry environment.
Also for parallels in a hyperbolic geometry would seem a parallel different from a elliptic geometry but an space cannot to be at same time hyperbolic and elliptic and like in the case of triangle the result is to make 1 or more parallel in a sheet of paper and bend the paper if you bend the paper the position of the parallel and point change thinking in a 3d space because the space has changed. By that really is 1 only point or parallel but like in the case of sheet of paper bended the parallel and point has changed their position. You cannot thing in a 3d space in that change the parallel because if you change the geometry also the point change of position. Proof with thes heet of paper to see that positions of the points are differents but by the same point only is 1 parallel.
Also in sphere and cones, ... yo can use the triangle of the sheet of paper to see the form of a triangle in this forms.
If this they say a train that travel by parallel railroad could not travel because the railroad would change the separation.
Remember that a triangle can to be made in 2d and at least in 3d because in 3d 3 points make a plane.
I don't understand very well how mathematicians have admitted this impossible near of 100 years because is worse that a bad novel fiction. The classical scientists create in Earth paralells and meridians for localization in Earth because the meridians are not parallels, they understood well the maths principes but actual scientists have mistake all this information.
If this Non-Euclidean geometry would true:
1 - would not exist the parallels because many parallels in 1 point is same that not exist parallels because have 1 common point, also the triangle with parallel lines that have a common angle.
2 - Geometry would to go out of maths because to say that a triangle can to have more an less of 180º is so math that to say that 2+2 can to be 4 and more and less of 4.
3 - trains could not work because would not exist parallel lines.