Section 4.2 Postulates
¶We now finally give an informal (and slightly incomplete) list of postulates for neutral geometry, adapted for two dimensions from those of the School Mathematics Study Group (SMSG), and excluding for now postulates about area.
Postulate 4.2.1. Incidence Postulate.
Two distinct points determine a unique line, and there exist three non-collinear points.
Postulate 4.2.2. Distance Postulate.
Every pair of distinct points determines a unique positive number denoting the distance between them.
Postulate 4.2.3. Ruler Postulate.
The points of a line are in one-to-one correspondence with the real numbers, with the absolute value of the difference giving the distance between points.
Postulate 4.2.4. Plane Separation Postulate.
Every line separates the plane into two disjoint sets of points such that any line connecting points on opposite sides must cross the given line.
Postulate 4.2.5. Angle Postulates.
Angles can be measured, resulting in a real number between \(0^\circ\) and \(180^\circ\text{,}\) inclusive, and all values are realized at every point.
Postulate 4.2.6. Angle Addition Postulate.
If \(D\) is in the interior of \(\angle BAC\text{,}\) then \(\angle BAC=\angle BAD+\angle DAC\text{.}\)
Postulate 4.2.7. Supplement Postulate.
Supplementary angles (those along a line) sum to \(180^\circ\text{.}\)
Postulate 4.2.8. SAS Postulate.
Two triangles are congruent if two corresponding sides and the angle between them are congruent.
For the purposes of this book, Euclidean geometry is the two-dimensional geometry satisfying the above neutral postulates, together with the Euclidean parallel postulate.
Postulate 4.2.9. Euclidean Parallel Postulate.
There exists a unique line parallel to a given line through a given point not on the given line.
As we will see, hyperbolic geometry is neutral geometry together with the hyperbolic parallel postulate (infinitely many parallel lines), and elliptic geometry has no parallel lines but also requires changes to other neutral postulates.