In 2D geometry, that is the definition of parallel lines. Two non-intersecting lines are indeed parallel.
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Here are some: They can be (1) coincident, (2) intersecting at one point, (3) coplanar but non-intersecting (ie parallel), (4) non-coplanar, non-parallel and non-intersecting (eg paths of a rail line and of an elevated road going over it - from above, they may appear to intersect but they do not), (5) intersecting at two points (eg longitudes on the earth). etc.
A rhombus, two sets of parallel lines intersecting,
Yes, squares do have intersecting lines. Infact they have 2 diagnol intersecting 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)
More info needed. Are the 2 lines parallel, perpendicular, or? are the angles that you are interested in on opposite sides of the intersecting line or the same side. The intersecting line is called a transversal. If the original lines are parallel, angles between the 2 lines on opposite sides of the transversal are called alternate interior angles, etc.