I think the fundamental criterions for triangles congruence are:
- SAS (Side-Angle-Side)
- ASA (Angle-Side-Angle)
- SSS (Side-Side-Side)
But some proofs like this one: https://www.youtube.com/watch?v=DArQTsH6Y1s
Use the SAA (Side-Angle-Angle) which I'm not sure if it is valid. I mean, if we assume that the sum of the measure of all angles are the same in all triangles this is indeed the ASA criterion. But otherwise is invalid.
Is that correct?
Best Answer
SAA is valid even in "neutral/absolute geometry", which takes no stand on the Parallel Postulate—or, equivalently, the Angle-Sum Theorem. (Importantly, Side-Angle-Side itself is a postulate in neutral geometry, because you have to start somewhere when comparing triangles.)
See, for instance, this 1977 The Mathematics Teacher article "Neutral and Non-Euclidean Geometry—A High School Course" (JSTOR link) by Krauss and Okolica. From the article:
BTW: Limited JSTOR access is free. (For the duration of the coronavirus pandemic, the limits have been made pretty generous: 100 articles/month!) But, conveniently, SAA is mentioned on the first page of the article, and much of its proof is visible in the article's preview image.