Side-Angle-Side Congruence and the Parallel Postulate - MathOverflow

Side-Angle-Side Congruence and the Parallel Postulate

2009-11-12T21:41:29Z

Hallo!

Is there a link between the side-angle-side congruence of triangles and the parallel postulate? Specifically, does it follow from Euclid's first four axioms alone? In fact, does it even follow from all five?

Answer by Michael Lugo for Side-Angle-Side Congruence and the Parallel Postulate

2009-11-12T21:51:55Z

Side-angle-side congruence is the fourth proposition of the Elements. Euclid doesn't use the fifth postulate until I.29, which is the equality of alternate interior angles.

Answer by Hugh Thomas for Side-Angle-Side Congruence and the Parallel Postulate

2009-11-12T22:44:39Z

You don't need the parallel postulate, but you are right to be a bit worried about triangle congruences following from Euclid's axioms. The link provided by Michael Lugo explains the issue: Euclid's proof uses "superposition" but his axioms do not allow him to draw conclusions on this basis.

Answer by Kristal Cantwell for Side-Angle-Side Congruence and the Parallel Postulate

2009-11-13T00:33:44Z

The first 28 propositions of Euclid's geometry use the first four axioms and are theorems in both hyperbolic(many parallels through a point parallel to a given line) and euclidean geometry(one parallel through a point parallel to a given line). see the following:

http://en.wikipedia.org/wiki/Absolute_geometry

The first 15 propositions of Euclid hold in Elliptic geometry(no parallels) see:

http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI16.html

So since side angle side is proposition 4 it can hold in all three systems. But as mentioned elsewhere it is one of Hilbert's axioms that were added as part of the process of formalizing Euclid's geometry. Hilbert's book The foundations of geometry is available here:

http://www.gutenberg.org/files/17384/17384-pdf.pdf

Answer by Noah Snyder for Side-Angle-Side Congruence and the Parallel Postulate

2009-11-13T07:50:13Z

My understanding was that SAS does not follow from the Euclid's postulates and should instead be considered an axiom itself. See Hilbert's axioms for more detail.

Answer by Jaikrishnan for Side-Angle-Side Congruence and the Parallel Postulate

2009-11-13T12:44:05Z

The proof given by Euclid is incorrect. Interestingly, I read somewhere that by using techniques used by Euclid in his Elements, it is possible to prove that every triangle is equilateral!

Answer by some guy on the street for Side-Angle-Side Congruence and the Parallel Postulate

2009-11-30T01:45:44Z

On reflection, SAS tells me that Euclidean geometry has a strong semi-local homogeneity, in that every neighborhood of every point is isotropically isomorphic with some neighborhood of any other point --- once you find a good way to say "neighborhood", that is.

The parallel postulate, on the other hand, can be used to construct canonical isomorphisms of point(ed)-neighborhoods --- by parallel translation of course; but since the constructed isomorphisms are all parallel in a good sense, we don't get the isotropy structure without SAS. (edit/add:ed): in the other direction, SAS doesn't give any canonical isomorphisms, which is just as well because hyperbolic and elliptical space both have SAS, but not the parallel postulate. (end edit)

The related postulate that Euclid states properly --- that all right angles are equal --- only gives a pointwise isotropy; it doesn't help much for segments subtended by respectively equal segments at equal angles.

Answer by Gabriel Benamy for Side-Angle-Side Congruence and the Parallel Postulate

2009-12-03T19:00:36Z

Ironically, I had intentions of working the other way around. As Hugh Thomas said, Euclid used a superposition of one figure onto another to demonstrate SAS congruence. My idea was to use a superposition of one figure onto another to demonstrate the parallel postulate, as well, but it never seemed to work well enough. However, regarding what Kristal Cantwell says, using the superposition "axiom" is contrary to hyperbolic geometry, and thus should be able to restrict it to our geometry.