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CorrectParallel lines as well as intersecting lines must be coplanar (in Euclidean geometry not quite sure about hyperbolic geometry...).Lines in space which neither are coplanar nor intersecting are called "skew"
The locus in a plane is two more intersecting lines, perpendicular to each other (and of course half-way between the given lines.
draw two intersecting lines that are not perpendicular
Intersecting lines are two lines that cross in a coordinate plane. So, yes, intersecting lines cross.
Two lines are intersecting if they have exactly one point in common.