Is triangle congruence SAS an axiom?

Question

I was wondering if there is a way to prove SAS in triangle congruence with Euclidean axioms.

Thank you for your help!

Answer 1

So I actually spend a good bit of time trying to think through this a few years back. This doesn't fully answer the question the way you will want it to, but it won't fit in a comment.

Firstly since SAS is true, (and relatively simple so Godel is unlikely to show up here) it has to follow from Euclid's axioms (or Hamilton's but they are equivalent) Secondly, using some trig (law of sines and cosines stuff) you can show that SAS and SS and AAS are all equivalent (as in if one holds all the rest do by CPCTC). Then you can use some coordinate geometry to show that SSS (which is SS) must work. You can do this by noting that rotations are isometries (preserve distance) and so do translations. So then you move the triangles such that they overlap at all points and then they must be congruent because every part of them overlaps, the sides must be the same length and so do the angles. This isn't entirely rigorous, but it made me personally feel a bit better about the various triangle proofs. It is entirely possible that full rigor for something like this is very complex. Hope the above is interesting/useful even if not entirely complete. I could go back and look for my notes on how I did the specifics of the above, but I think it was mostly trig and coordinate geometry reasoning.

Answer 2

The answer to this question is a bit complicated. It's not a straight yes or no.

Euclid claims to prove side-angle-side congruence in his Proposition 4, Book 1. He does this by "applying" one triangle to the other. Essentially, this means he moves one triangle until it coincides with the other.

Euclid's proof is universally regarded as problematic from a logical standpoint, because there are absolutely no axioms in the Elements that tell you when, or in what ways, a figure can be "applied" to another.

There are modern axiom systems that are logically satisfactory, and they fill the gaps in Euclid in different ways. Some systems have axioms involving rigid motions, and in some cases it may be possible to give a proof similar to Euclid's.

The best-known modern axiom system intended to replace Euclid's, while staying close to his in spirit, is the one given by Hilbert in 1899 in Grundlagen der Geometrie. Hilbert assumes as an axiom something slightly weaker than the full SAS property. Namely, Axiom IV-6 states that if two triangles have two corresponding sides and the angle between them equal, then they also have their remaining angles equal. From this, and his other axioms, Hilbert then proves that the remaining side must also be equal.

So in Hilbert's case, two-thirds of the conclusion of the SAS property is assumed as an axiom, and the remaining one-third is proved.

Answer 3

This research paper talks about this: http://www.iosrjournals.org/iosr-jm/papers/Vol10-issue1/Version-3/D010132633.pdf

To sum it up: it is a theorem.

This research has shown that the elusive SAS (side angle side) theorem of congruence of triangles can be proved analytically. Indeed same goes for the other congruence theorems of triangles by applying, mutatis mutandis, the data for each theorem on the cross section of a double cone with the appropriate construction.