 Side Angle Side Proof
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Steve Conner
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Registered Member #30
Joined: Fri Feb 03 2006, 10:52AM
Location: Glasgow, Scotland
Posts: 6706
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OK, I admit that I didn't know what kind of a "theorem" SAS was or how to prove it, so I did some more research on this, and there does seem to be a fair bit of confusion.
In US schools they teach that the Side-Angle-Side condition and its fellow congruence/similarity relations are postulates in their own right: students are just to accept them without questioning. But this isn't universally agreed. (Only dull students accept things without questioning, anyway.  )
Some people think that Side-Angle-Side is based on Euclid's fourth postulate, because that is also a kind of congruence condition. Others think it was an axiom introduced by Hilbert in his "Foundations Of Geometry". 
That thread links to an e-book version of "Foundations Of Geometry" and Hilbert's proof of SAS is on pages 10-11. I'm not the world's greatest mathematician, but I'm pretty sure he did it without Pythagoras' theorem.
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