Prove rigorously that for any hyperbolic line l H2 and any p

Prove rigorously that for any hyperbolic line l H^2 and any point p NotElement l, there exists at least two hyperbolic lines m_1 and m_2 parallel to l through p.

Solution

We can assume all axioms of neutral geometry, so we can use the following theorem: Theorem Given a line l and a point P / l, let Q denote the foot of the perpendicular dropped from P to Q. Then there exist two rays P~R and P~S on opposite sides of P~Q such that

(a) The ray P~R and P~S do not intersect l.

(b) A ray P~X intersects line l if and only if P~X is between P~R and PS.

(c) QPR \' QPS.

The measurement axioms for angles imply that for every real number x [0, 180] there exists a point X on one side of the line through P and Q such that the measure of the angle XPQ equals x. Now, let Y = {x [0, 180degree ]/ the ray P~X intersects l}.Note that Y is nonempty and bounded. Therefore, the set has a supremum supY = s. Then there exists a point S on one side of PQ such that m(SPQ) = s and that the ray P~S does not intersect line l. In fact, that in Euclidean geometry the sets supremum will be 90 degree and in Hyperbolic geometry the supremum of the set is less than 90 degree .

In hyperbolic geometry the measure of this angle is called the angle of parallelism of l at P and the rays PR and PS the limiting parallel rays for P and l.

Hence Proved