I'm going through a text, Elementary Geometry from an Advanced Standpoint, by Edwin E. Moise that gives an axiomatic development of geometry. In the first few chapters, the author covers
- incidence axioms for points, lines, and planes;
- a ruler postulate for line-segment measure;
- betweenness for points on a line;
- a plane separation axiom for dividing planes into two convex half planes;
- angle measure postulates.
(Granted, this is a rough outline of the axioms, rather than an explicit list.) The author then presents standard axioms for congruency of triangles, such as the side-angle-side postulate---if two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. My question is whether this triangle congruency axiom follows from the axioms listed above, or whether it's independent from them. For instance, if the law of cosines follows from the above axioms, then the triangle congruency axioms should not be necessary. On the other hand, it is hard to imagine geometric models of the axioms listed above in which the triangle congruency axioms fail.
So, does it seem more likely that such models really exist or that you can prove the side-angle-side axiom?