Comparing Absolute (Neutral) Geometry to Other Geometries
Absolute (Neutral) Geometry is the geometric system that is based on Euclid’s first four postulates. Because Euclid’s first four postulates are true in Euclidean
Geometry and Hyperbolic Geometry as well, everything that is true in Absolute
(Neutral) Geometry is also true in Euclidean Geometry and in Hyperbolic Geometry.
However, sometimes, more specific or restricted versions of some relationships may also be true in Euclidean Geometry or Hyperbolic Geometry because of the addition of the Euclidean or Hyperbolic parallel postulates. Below are listed some of the commonly used theorems from each geometry. Please add to this list, as appropriate!
Statements that are true in Absolute Geometry (and in Euclidean and Hyperbolic
Geometries):
1.
If line L intersects line M, then their intersection is exactly one point.
2.
Postulate of Pasch: If A, B and C are distinct, noncollinear points and line l intersects segment AB, then l also intersects either segment AC or segment BC
(or both).
3.
Segment construction and Angle construction theorems
4.
Straight angle sum is 180 degrees.
5.
Vertical angles are congruent.
6.
The set of all points equidistant from two fixed points A and B is the perpendicular bisector of segment AB.
7.
Given a line m and a point P, then there exists a unique line l through P that is perpendicular to line m.
8.
Exterior angle inequality theorem – In a triangle, the measure of an exterior angle is greater than the measure of either of the two remote interior angles.
9.
Isosceles triangle theorems (altitude=median=angle bisector; base angles congruent and converse)
10.
Triangle angle sum is ≥ 180
11.
Parallel Line Theorem - If two lines are cut by a transversal such that alternate interior angles are congruent, then the lines are parallel. (Converse not true)
12.
Triangle congruence thms: SAS, SSS, ASA, AAS (not true in spherical), HL
13.
Scalene Inequality: If one side of a triangle has greater length than another side, then the angle opposite the longer side has the greater angle measure, and, conversely, the side opposite an angle having greater (angle) measure, is the longer side.
14.
Triangle Inequality: The length of any side of a triangle is always less than the sum of the other two side lengths.
15.
Median Inequality: Suppose that AM is the median to side BC of △ ABC. Then,
AM < ½ (AB + AC)
16.
SAS Inequality: If in △ ABC and △ XYZ we have AB = XY, AC = XZ, but m  A > m  X, then BC > Y Z, and conversely if BC > Y Z, then m  A > m  X.
17.
Quadrilateral Congruence Theorems (SASAS, SASAA, ASASA, SASSS)
18.
Saccheri Quadrilateral Theorems

a.
The summit angles of a SQ are congruent and ≤90. b.
The diagonals of a SQ are congruent
19.
Tangent Theorem: A tangent to a circle forms a right angle with the circle's radius, at the point of contact of the tangent.
20.
Secant Theorem: If a line l passes through an interior point A of a circle, it is a secant of the circle and intersects that circle in precisely two points.
21.
Other circle theorems (e.g. if two chords are equidistant from the center of a circle, then they are congruent, and conversely)
Statements that are true in Euclidean Geometry:
1.
Exterior angle theorem – strong version (equality): In a triangle, the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles.
2.
Triangle angle sum: In a triangle the sum of the measures of the interior angles is
180 degrees.
3.
Things that depend on E5/triangle angle sum a.
Existence of rectangles – a quadrilateral with 4 right angles. b.
Angle sum in a polygon (n-2)(180)
4.
Parallel Line Theorems a.
If two parallel lines are cut by a transversal, then alternate interior angle measures are equal (same side interior angles are supplementary; alternate exterior angles are equal; corresponding angles are equal) b.
When two line are cut by a transversal and the alternate interior angles are equal (or same side interior angles are supplementary or corresponding angles are equal), then the two lines are parallel.
5.
Angles in circles (e.g., congruent inscribed angles) – proofs depend on angle sum of 180 in a triangle.
Statements that are true in Hyperbolic Geometry:
1.
Triangle angle sum: In a triangle the sum of the measures of the interior angles is less than 180 degrees.