# Need for triangle congruency axioms

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?

-
+1 for reading Moise. –  lhf Oct 19 '11 at 1:39

## 2 Answers

Look at: http://en.wikipedia.org/wiki/Taxicab_geometry which discusses the taxicab plane. Eugene Krause has written a lovely book about this geometry: Taxicab Geometry, now available through Dover Press.

-
That's a nice example (which I certainly wouldn't have thought of on my own). Thanks. –  Rus May Oct 19 '11 at 11:10

Moise devotes section 4 of Chapter 6 to your question. He gives an example of "a structure...satisfying all of the postulates of metric geometry" but with a distance function differing slightly from the usual one, and showing that side-angle-side fails in this structure.

-
Moise has a strange presentation of measure for line segments (for instance, he doesn't assume the triangle inequality) that common intuition (at least my intuition) doesn't mesh with. I suppose the fact that SAS doesn't follow from the previous axioms is an example of a bizarre model for a weak axiom system. I'll have to work on convincing myself that the model that Moise presents at the end of chapter 6 really foots the bill. –  Rus May Oct 19 '11 at 11:39
Well, one has to do something that doesn't mesh with intuition in order to come up with a geometry where SAS fails, don't you think? –  Gerry Myerson Oct 19 '11 at 22:20