Parallel Lines

In Euclidean geometry, parallel lines are coplanar lines that do not intersect. In absolute geometry, there is at least one line that passes through a point not lying on a given line and that does not intersect the given line. Only one such line exists in Euclidean geometry (Euclid’s fifth, or parallel postulate). In Lobachevskian geometry, a plane contains an infinite set

Figure 1

of lines that pass through a point C not lying on a given line AB and that do not intersect AB (see Figure 1). Only two of these lines are said to be parallel to AB. The line CE is said to be parallel to AB in the direction from A to B if (1) B and E are on the same side of AC, (2) CE does not intersect AB, and (3) every ray within the angle ACE intersects AB. The line CF parallel to AB and directed from B to A is analogously defined.