Plane Geometry - Bruce E. Shapiro

Section 48Parallel Lines inHyperbolic GeometryDefinition 48.1 (Intersecting Set) Let l be a line, P ∉ l a point, Athe foot of the perpendicular dropped from P to l, and B ∈ l any otherpoint in l. Then for each θ ∈ (0, 90] there is some point D(θ) such thata µ(AP Dθ) = θ. The critical set K is the set of all θ such that −−−−→ P D(θ)intersects −→ AB:K = {θ| −−−−→ P D(θ) ∩ −→ AB ≠ ∅}Since K is a bounded set of real numbers (it is bounded because 90 is anupper bound) it has a least upper bound.Definition 48.2 (Critical Number) The critical number θ C of P for−→AB is the leqst upper bound of the critical set K:θ C = lub KThe critical number defines an angle such that any rays that make a largerangle with P (on the same side of ←→ AP as B) do not intersect ←→ AB and henceare parallel to ←→ AB. The angle with measure given by the critical number iscalled the angle of parallelism. Rays that make a smaller angle with −→ P Aare not parallel to ←→ AB and intersect it. There are two angles of parallelism,one on each side of P A; these two angles are congruent.Definition 48.3 (Angle of Parallelism) Let D be a point of the sameside of ←→ P A as B such that µ(∠P AD) = θ C . Then angle ∠P AD is called275