Plane Geometry - Bruce E. Shapiro

SECTION 48. PARALLEL LINES IN HYPERBOLIC GEOMETRY 277Figure 48.2: Proof of theorem 48.4.Figure 48.3: There are two critical angles, one on each side of the perpendiculardropped from P to l.Define β = µ(∠AP T ).Since P −−→ D β ∩ −→ AB ≠ ∅, then β ∈ K. But β > θ because A ∗ U ∗ T . Henceβ ≥ θ > θ C which contradicts the RAA. Hence θ ∉ K.Theorem 48.5 The critical number θ C depends only on the distance d(P, l).Proof. Let l be a line, let P ∉ l be a point, let A be the foot of theperpendicular from P to l, and let B ≠ A be another point in l.Let l ′ be a second line, let P ′ ∉ l ′ , let A ′ be the foot of the perpendicularfrom P ′ to l ′ such that P ′ A ′ = P A, and let B ′ ≠ A ′ be another point in l ′ .Let the intersecting sets of the two points and their respective lines be Kand K ′ , and suppose that θ ∈ K.Then there is a point D(θ) such that −−−−→ P D(θ) intersects ←→ AB at some pointT .Choose T ′ ∈ −−→ A ′ B ′ so that A ′ T ′ = AT .By SAS, △P AT ∼ = △P ′ A ′ T ′ . Therefore there is a point D ′ (θ) such that−−−−−→P ′ D ′ (θ) intersects ←−→ A ′ T ′ (pick any point on ←−→ A ′ T ′ .Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.