Question

Is it possible to represent a plane with two parallel lines?

Answer 1

Lines are parallel if they lie in the same plane and they don't intersect. In other geometries, there may be no parallel lines, lines may not have a common point but they may have a common limit point at infinity, or they may just not intersect.

You might be thinking about Projective Geometry where a "point at infinity" is added to every family of parallel lines and the set of all points at infinity is called the "line at infinity".

Addendum

Typically, in formal geometry, points, lines, and planes are not defined. But postulates define their "baseline" behavior. The postulate that every geometry seems to agree on is the one that states

L1: Given two distinct points in a plane, there is exactly one line in that plane that contains them.

The "dual" of that postulate is

DL1: Given two distinct lines in a plane, there is exactly one point in that plane that belongs to both lines.

Since Euclidean geometry contains parallel lines, DL2 is false. But Projective geometry accepts DL2 as a postulate. The big question is, "Does there exists a geometry that satisfies the postulates of Projective geometry?" Yes there does.

The creation of such a geometry is really quite clever. You start with a Euclidean plane and you add points to it as follows. Pick any line in the plane. To that line and all lines parallel to it, you add one extra point, a point at infinity. This is a set thing. We are treating a Euclidean line, l" roleSome lines will still intersect. Those that intersected at a point on the line that was removed will now be parallel.

Mark as Brainliest!