Points cannot be parallel to other points.
A straight line joining two points is parallel to a straight line joining another pair of points if the gradient (slope) of the two lines is the same.
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prism
Points cannot be parallel or perpendicular. So they never are.
A plane midway between the two given planes and parallel to them.
Parallel.
If the two lines are actually "on top of each other", they can have infinitely many points in common. If they are parallel, they have no points in common. If they are perpendicular, they have one point in common.