In Euclidean space, never.
But they can in non-Euclidean geometries.
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parallel lines never intersect.
Lines that never intersect are parallel.
No, parallel lines do not intersect, and never will.
If lines intersect they CANNOT BE PARALLEL. Speak to your teacher!
Perpendicular lines intersect at a 90 degree angle. Parallel lines do not intersect, nor would they ever intersect if continued infinitely.