Plane Geometry - Bruce E. Shapiro

280 SECTION 48. PARALLEL LINES IN HYPERBOLIC GEOMETRYFigure 48.6: The angle of parallelism is acute (theorem 48.7).Proof. (Exercise.)Theorem 48.9 −−→ P D| −→ ←→ ←→AB =⇒ P D ‖ ABProof. (Exercise.)Definition 48.10 If −−→ P D| −→←→ ←→AB then the lines P D and AB are called asymptoticallyparallel in the direction −→ AB.Theorem 48.11 (Symmetry of Limiting Parallels) If −−→ P D| −→ AB, Q is thefoot of the perpendicular from A to −−→ P D, and D ′ ∈ −−→ P D such that P ∗Q∗D ′ ,then −→ AB|−−→ QD ′ .Proof. (Exercise.)Corollary 48.12 (Symmetry of Asymptotic Parallelism) If l is asymptoticallyparallel to m then m is asymptotically parallel to l.Proof. (Exercise.)Theorem 48.13 If l and m admit a common perpendicular, then they arenot asymptotically parallel.Proof. Let l and m be two lines that admit a common perpendicular, whichwe will call t. Let the intersections of t with l and m be S and R.Since m is perpendicular to RS, no subray of M starting at R can beasymptotically parallel to l, because the angle of parallelism must be acute.Let P ∈ m such that P ≠ R. Drop a perpendicular from P to its footA ∈ l.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012