Hyperbolic Geometry Axiom • Hyperbolic Axiom (HA): In hyperbolic geometry there exist a line l and point P not on l such that at least two distinct lines parallel to l pass through P. – Lemma 6. 1: Rectangles do not exist.
Hyperbolic Geometry UHT • Universal Hyperbolic Theorem (UHT): In hyperbolic geometry, for every line l and every point P not on l there pass through P at least two distinct parallels to l. – Cor: In hyperbolic geometry, for every line l and every point P not on l, there are infinitely many parallels to l through P.
Hyperbolic Geometry Theorems 1 - 3 • Thm 6. 1: In hyperbolic geometry, all triangles have angle sum less than 180. – Cor: In hyperbolic geometry, all convex quadrilaterals have angle sum less than 360. • Thm 6. 2: In hyperbolic geometry, if two triangles are similar, they are congruent. • Thm 6. 3: In hyperbolic geometry, if l and l' are any distinct parallel lines, then any set of points on l equidistant from l' has at most two points in it.
Hyperbolic Geometry Theorem 4 • Thm 6. 4: In hyperbolic geometry, if l and l' are parallel lines for which there exists a pair of points A and B on l equidistant from l', then l and l' have a common perpendicular segment that is also the shortest segment between l and l' – L 2: The segment joining the midpoints of the base and summit of a Saccheri quadrilateral is perpendicular to both the base and the summit, and this segment is shorter than the sides.
Hyperbolic Geometry Theorem 5 • Thm 6. 5: In hyperbolic geometry, if lines l and l' have a common perpendicular segment MM' then they are parallel and MM' is unique. Moreover, if A and B are any points on l such that M is the midpoint of segment AB, then A and B are equidistant from l'.
Hyperbolic Geometry Theorem 6 • Thm 6. 6: For every line l and every point P not on l, let Q be the foot of the perpendicular from P to l. Then there are two unique nonopposite rays. and on opposite sides of that do not meet l and have the property that a ray emanating from P meets l if and only if it is between and. . Moreover these limiting rays are situated symmetrically about in the sense that XPQ X'PQ. (See following figure. )
Theorem 6 Figure