Yes. Parallel means they they are always the same distance and they never intersect, and this could only mean on the same plane. However, the proof that it works is a little harder than one might think.
Two lines that do not intersect on the same plane are Parallel lines.
Parallel lines.
If the two lines lie in the same plane, and they do not intersect, then they are parallel. If they are not in the same plane, and do not intersect, then they are called skew lines.
They are parallel lines
Parallel
Lines in the same plane that do not intersect Lines in the same plane that do not intersect Lines in the same plane that do not intersect Lines in the same plane that do not intersect
Lines in the same plane that are not parallel are intersecting lines.
They are parallel lines
If the 2 lines lie in the same plane, and they are not parallel, then they will intersect at some point. If the 2 lines are skew lines, then they are not in the same plane, and they will not intersect (but they are Not Parallel)
Two lines that do not intersect on the same plane are Parallel lines.
If they are not parallel and do not intersect then they would not be in the same plane (not possible), but they would be called skew lines.
Parallel lines.
Parallel lines
Parallel lines. they never intersect and lie on the same plane.parallel lines
If the two lines lie in the same plane, and they do not intersect, then they are parallel. If they are not in the same plane, and do not intersect, then they are called skew lines.
Lines which are parallel. All other lines on the same plane eventually intersect.
They can be, but not necessarily. If two lines in the same plane never intersect, they are parallel. Though, two lines can exist in the same plane without being parallel. Ie, two lines that intersect at a 90 degree angle are perpendicular. Most lines in a plane aren't parallel.