Chapter 4

Introduction to Hyperbolic Geometry

Theorem: Converse to Parallel Postulate
 

If a transversal makes equal alternate interior angles with two lines, then the lines are parallel.

Note: This is not the same as: If two lines are parallel, then a transversal will make alternate interior angles equal.

Theorem: Maximum Angle Sum
 

The sum of the measures of the angles of any triangle cannot exceed the measure of a straight angle.

Definition: Angular Defect
 

Let a triangle have angles of measure [Maple Math]. Then [Maple Math]is called the angular defect of the triangle. We write: d(ABC) to stand for the angular defect of triangle ABC.

With this definition, the Maximum Angle Sum theorem can be restated as:  All triangles have non-negative angular defect.

Theorem: Angular Defect Propagation
 

Let ABC be a triangle and D a point on side BC of the triangle.

Let [Maple Math]be the sum of the measures of the angles in triangle ABC.

Let [Maple Math]be the sum of the measures of the angles in triangle ABD.

Let [Maple Math]be the sum of the measures of the angles in triangle ADC.

Lemma:

[Maple Math]+ measure of a straight angle = [Maple Math].

Theorem:

d(ABC)=d(ABD)+d(ADC)
Corollary:
If d(ABD) >0, then d(ABC)>0.

Corollary:
 

If d(ABC)>0, then either
d(ABD)>0
or
d(ADC)>0.

Theorem: Right Triangle Defect theorem
If there is a triangle with positive angular defect, then there is a right triangle with positive angular defect.

Theorem: Existence of Rectangles
If there is a right triangle with zero angular defect, then a rectangle can be constructed.

Theorem: Inscribed Right Triangle
If any rectangle exists, then every right triangle can be inscribed at a vertex of some rectangle so that the legs of the triangle are contained in the sides of the rectangle meeting at the vertex.

Theorem: Consistency of Angular Defect
 
Either all triangles have angular defect of zero or all triangles have positive angular defect.

Theorem: Parallel Postulate Equivalents

The following statements are logically equivalent:

If a transversal falls on parallel lines, then the alternate interior angles have equal measure.

The sum of the measure of the angles of a triangle is equal to the measure of a straight angle.

The sum of the angles of a convex quadrilateral is two straight angles.

There exist two equi-angular triangles that are not congruent.


Theorem: Parallel Nonparallel Separation
Let P be a point not on line l. Let Q be the foot of the perpendicular from P to l. Let [Maple Math]and [Maple Math]be lines through P which intersect l to the right of Q. Let [Maple Math]and [Maple Math]be lines through P that do not intersect l. Then, no line p can be drawn through P between [Maple Math]and [Maple Math]so that p does not intersect l. Also, no line t can be drawn through P between [Maple Math]and [Maple Math]so that t intersects l.

Theorem and Definition: Critical Angle or Angle of Parallelism
Let P be a point not on line l. Let Q be the foot of the perpendicular, t, from P to l. Let r be any ray from P intersecting l to the right of Q. Let [Maple Math]be the angle between t and r at P. Then there is an angle [Maple Math]called the critcal angle (or angle of parallelism) such that [Maple Math]and if r is a ray which does not intersect line l to the right of Q, then [Maple Math].