3.3 Polygons in the hyperbolic plane
Theorem: The summit angles of a Saccheri quadrilateral are acute.
Theorem: The sum of the angles of a hyperbolic quadrilateral is less than 180 degrees.
Definition: The defect of a hyperbolic triangle with degree angle sum x is 180 – x
degrees (is   x radians if the angle sum is x radians).
Theorem: The sum of the acute angles of a right triangle is less than 90 degrees.
Theorem: The sum of the interior angles of a convex hyperbolic polygon is less than
(n -2)180 degrees.
Theorem: The sum of the angles of a Saccheri quadrilateral is less than 360 degrees.
Theorem: Rectangles do not exist in hyperbolic space.
Theorem: ABC is a hyperbolic right triangle with angle C the right angle and sides a, b,
and c (in the usual notation). Then cosh( c)  cosh( a) cosh( b).
Definition: The shortest distance between two parallel lines is measured along a
common perpendicular between the lines.
Theorem: Parallel lines cannot have more than one common perpendicular.
Theorem: Parallel lines are not everywhere equidistant.
Theorem: Intersecting lines cannot have a common perpendicular.
Theorem: The fourth angle of a Lambert quadrilateral is acute.
Area Axiom 1: The area of any portion of the plane must be non-negative.
Area Axiom 2: The area of congruent portions of the plane must be the same.
Area Axiom 3: The area of the union of disjoint regions of the plane must equal the sum
of the areas of the regions.
Theorem: The area of a hyperbolic triangle with degree angle sum x is 180 – x degrees
(is   x radians if the angle sum is x radians).
Theorem: If two hyperbolic triangles have the same angle sum, they have the same area.
Note: The ‘largest’ triangles in hyperbolic space are those in which all three vertices are
omega points.
Exercises
1.
In a hyperbolic right triangle ABC, sides a, b, and c are opposite angles A, B, and
C, respectively. Angle C is a right angle. If c = 4, and b = 3, find a.
2.
What considerations determine whether a given set of perpendicular lines may be
used to create a Lambert quadrilateral.
3.
What range of values are possible for the sum of the interior angles of a Saccheri
quadrilateral? For a Lambert quadrilateral?
4.
Investigate whether the segment joining the midpoints of the sides of a Saccheri
quadrilateral is perpendicular to the sides.
5.
Investigate whether the following Euclidean shapes exist in hyperbolic space.
 Parallelogram
 Rhombus
6.
Prove: Sensed-parallel lines are not everywhere equidistant.
7.
Prove: The sides of a Lambert quadrilateral forming the acute angle are longer
than the sides opposite the acute angle.