Hyperbolic Geometry:

Hyperbolic Parallel Postulate

If P is a point not on the line AB and if Q is the foot of the perpendicular from P to AB,there are 2 rays PX and PY from P, not in the same line and not intersecting AB such that any ray PZ from P lying within <XPY containing Q intersects AB. (Eves pg 340. He notes that this is stronger than is really necessary, but easier to work with. See Cederberg for a stricter version.)

The second figure illustrates this within the Poincaré Disk model

In the illustrations accompanying the following definitions and Theorems I have sometimes used diagrams similar to the left hand representation and sometimes using the Poincaré model. It should be clear which is being used!

Theorems

Theorem

Any line through P not passing within <XPY does not intersect AB.

Parallel and Ultraparallel

The lines PX and PY are called the parallels through P to line AB. The directed line PX is parallel to the directed line AB (PX is the right-sensed parallel through P to use another terminology).
Lines through P not passing within <XPY are called ultraparallels.

Pasch's Axiom(s)

 

Theorem

If Q is the foot of the perpendicular from P on line AB and if PX and PY are the 2 parallels through P to AB, then <XPQ = <YPQ and are acute. Cabri Java Illustration

Theorem

If a line l is right(left)-parallel through a point P to a line m, then it is at each of its points the right(left)-parallel to m.
(If PX is parallel to AB and R is another point on PX so that P and R are on the same side of X, then RX is parallel to AB (where all lines such as PX are to be understood to mean directed line PX etc). (Note: 2 cases)) Cabri Java Illustration

Theorem

If a line l is right(left)-parallel to line m, then m is right(left)-parallel to l.
(If CD is parallel to AB, then AB is parallel to CD (directed lines again)). CabriJava Figure

Theorem

If line l is right(left)-parallel to m and m is right(left)-parallel to n, then l is right(left)-parallel to n.
(If AB and CD are parallel to EF, then AB is parallel to CD (directed lines)).

Limit or Asymptotic triangle

A figure formed by 2 parallel rays and a segment joining their origins is called a limit or asymptotic triangle.

Two Theorems about asymptotic triangles

A line containing a vertex of an asymptotic triangle and a point interior to the triangle will intersect the opposite side of the triangle. This is also true if the "vertex" is taken to be the ideal point where the parallel rays "intersect".

If a line intersects one side of an asymptotic triangle then it intersects one of the other sides. This is true whether the side is the finite side or either of the infinite parallel rays forming the asymptotic triangle

Theorem

An exterior angle of an asymptotic triangle is greater than the opposite interior angle.(cf Euclid Prop 16)
(Proof by contradiction: 2 cases - less than and equal to - the latter case gives, as a corollary the Theorem that if the alternate angles are equal the lines are ultraparallel.).

Theorem

If the finite sides if two asymptotic triangles are equal and an angle of one is equal to an angle of the other, then the other angles are equal. (A congruence criterion for asymptotic triangles)
(contradiction again)

Theorem

If the 2 angles of two asymptotic triangles are equal then the finite sides are equal. (another congruence criterion)
(contradiction proof)

Theorem

If in an asymptotic triangle its two angles are equal to each other, and in another asymptotic triangle its two angles are equal to each other, and if the finite sides of the two asymptotic triangles are equal, then all the angles are equal.
(construction and congruence)

Theorem

The angle of parallelism at P for a line AB decreases as PQ increases.

Saccheri Quadrilaterals

Saccheri Quadrilateral

A quadrilateral ABCD with <A and <B right angles and AD=BC is called a Saccheri quadrilateral. AB is called the base, DC the summit and <D and <C the summit angles. Illustration

Theorem

The summit angles of a Saccheri quadrilateral are equal and acute.
(Proof outline and illustration)

Theorem

The line joining the midpoint of the base and summit of a Saccheri quadrilateral is perpendicular to both.
(Congruent triangles)

Theorem

Two Saccheri quadrilaterals are congruent if they have equal summits and equal summit angles.

Theorem

The sum of the angles of any triangle is less than 2 right angles.
(Proof outline and illustration)

Three immediate and important corollaries

1. The sum of the angles of a quadrilateral is less than 4 right angles.

2. Two lines cannot have more than one common perpendicular.

3. There do not exist equidistant lines.

Defect

The difference between the sum of the angles of a triangle and 2 right angles is called its defect.

Theorem

If the angles of a triangle are equal to the angles of another then the two triangles are congruent. (There are no similar non-congruent figures)

Coursework Task