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I have seen a proof of SAS congruence rule using rigid transformations. So, why isn't it a theorem. Also, are there any other proofs of it using the existing postulates? I am very confused regarding this matter.

I was able to prove that if two triangles two corresponding sides and the angle contained by those sides equal, then the triangles are congruent, using the postulates that describe parallel lines and some construction. So, does that make it a theorem? I highly doubt that I used any other assumptions or followed any incorrect steps.

Please clear my above confusion.

Bill Dubuque
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VizDracViz
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    Welcome to the site! SAS congruence is indeed a postulate. To evaluate your proof, we need to see your complete work though. – Vasili Jun 29 '22 at 13:57
  • “Why isn’t it a theorem?” Please clarify this point. Many theorems in maths also go by the name of a lemma, rule, proposition, principle, etc. this may simply be an issue of naming convention – FShrike Jun 29 '22 at 13:57
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    You need to specify which axiom system you are using. The "proof" given in the Elements is widely regarded as incomplete, as it relies on Superposition (not described in the postulates). Hilbert's treatment of Euclid's Geometry makes SAS into a postulate and thereby avoids the gap. – lulu Jun 29 '22 at 13:59
  • See the post Congruence of angles axiom as weel as The Axioms of Geometry (1929) – Mauro ALLEGRANZA Jun 29 '22 at 14:47
  • See also Need for triangle congruency axioms. – Mauro ALLEGRANZA Jun 29 '22 at 14:49
  • Also, it is possible to prove the other two congruency principles if you take one of them as postulate. – Vasili Jun 29 '22 at 15:09
  • What are “rigid transformations”? What are the postulates that govern them? I can almost guarantee that one of the postulates governing rigid transformations is SAS congruence. – Mark Saving Jun 29 '22 at 16:56
  • @FShrike, I am asking why it is not a "theorem" as a proposition or a rule is something obvious that can be accepted without a proof, something universally acceptable. But if a valid proof exists based on already existing axioms, then why is it not called a theorem? Also, I am using Euclidean geometry and Euclid's postulates. – VizDracViz Jun 30 '22 at 09:42
  • @Vasili I will upload it as soon as possible. – VizDracViz Jun 30 '22 at 09:51
  • ...Hence my comment. Many theorems with proofs are still called rules or principles – FShrike Jun 30 '22 at 09:58
  • @FShrike, are the terms 'rule' and 'axiom' used interchangeably? – VizDracViz Jul 04 '22 at 05:37
  • @VizDracViz Not in my experience, and they shouldn’t be. It’s just that famous theorems tends to be called by different names or different naming conventions. I’m not talking about the geometry anymore - I couldn’t comment on the axiomatic status of SAS - just making a general point that “principles”, “rules”, “propositions” and “theorems” can be used to mean the same thing (not always) – FShrike Jul 04 '22 at 05:49
  • I am still very confused. I have also looked up the meanings of all these terms(axioms, postulates, propositions, rules, etc.)but havent been able to clarify the difference in all of these. – VizDracViz Jul 09 '22 at 07:48

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