David: << The book "Hilbert's Third Problem" by Vladimir Boltyanskii 1978 is the definitive reference.
Me: << I have that book and it's very nice, written at a level most upper division math undergrads could understand, though I don't think he tries to prove Sydler's sufficiency proof.
David: << Dan, are you perhaps thinking of a different Boltyanskii book "Equivalent and Equidecomposable Figures"? This is the book which I used for my website exhibit. (I still have hopes someone will look at it). http://mathsite.math.berkeley.edu/polygon/polygonGalleryIntro.html The book "Hilbert's Third Problem" is non-elementary and it's devoted mostly to proving Sydler's converse.
Oops, that must be the book I have (in the sense that it's in storage with most of my other math books somewhere in Chico, Calif.). --Dan