One, if they're in a row.
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2 points
the parallel lines never intercept so they do not have any common points.
In Euclidian or plane geometry, there can be only one line through two fixed points. Lines cannot actually be drawn; if you see it it is not a geometric line. If the points are on a curved surface as in a geometry that is non-Euclidian, then there can be infinitely many lines connecting two points.
Rotational symmetry is based on points of rotation not lines
2 lines, I believe.