Plane Geometry - Bruce E. Shapiro

282 SECTION 48. PARALLEL LINES IN HYPERBOLIC GEOMETRYTheorem 48.16 If l ‖ m are parallel lines that admit a common perpendicular,the for every d 0 > 0 there exists a point P ∈ m such that d(P, l) > d 0 ;and P may be chosen to line on either side of the common perpendicular.Proof. (omitted)Theorem 48.17 (Transitivity of Limiting Parallels) If l is asymptoticallyparallel to m in the direction −→ AB and l is asymptotically parallel ton in the direction −→ AB, then either m = n or m is asymptotically parallelto n.Proof. (omitted)Theorem 48.18 If l ‖ m then either l and m admit a common perpendicularor l and m are asymptotically parallel.We have already shown that the critical function is decreasing. In fact, itis strictly decreasing.Theorem 48.19 The critical function κ(x) is strictly decreasing.Proof. (See Venema.)Theorem 48.20 If l and m are asymptotically parallel lines then for everyɛ > 0 there exists a point T ∈ m such that d(T, l) < ɛ.Proof. (See Venema.)Theorem 48.21 limx→∞ κ(x) = 0Proof. (See Venema.)Theorem 48.22 limx→0 + κ(x) = 90Proof. (See Venema.)Theorem 48.23 The critical funnction κ(x) is onto.Proof. (Exercise.)Theorem 48.24 The critical functionis continuous.Proof. (Exercise.)κ(x) : (0, ∞) ↦→ (0, 90)« CC BY-NC-ND 3.0. Revised: 18 Nov 2012