Limiting parallel
In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line formula_1 through a point formula_2 not on line formula_3; however, in the plane, two parallels may be closer to formula_1 than all others (one in each direction of formula_3). Thus it is useful to make a new definition concerning parallels in neutral geometry. If there are closest parallels to a given line they are known as the limiting parallel, asymptotic parallel or horoparallel (horo from — border). For rays, the relation of limiting parallel is an equivalence relation, which includes  the equivalence relation of being coterminal. If, in a hyperbolic triangle, the pairs of sides are limiting parallel, then the triangle is an ideal triangle. Definition. A ray formula_6 is a limiting parallel to a ray formula_7 if they are coterminal or if they lie on distinct lines not equal to the line formula_8, they do not meet, and every ray in the interior of the angle formula_9 meets the ray formula_7. Properties. Distinct lines carrying limiting parallel rays do not meet. Proof. Suppose that the lines carrying distinct parallel rays met. By definition they cannot meet on the  side of formula_8 which either formula_12 is on. Then they must meet on the side of formula_8 opposite to formula_12, call this point formula_15. Thus formula_16. Contradiction.