Definition: Def. 11.10: Equal Solid Figures
(Definition 10 from Book 11 of Euclid's “Elements”)
But equal and similar solid figures are those (solid figures) contained by similar planes equal in number and in magnitude (which are similarly arranged).
Modern Formulation
Please refer to the modern definition of congruence, which is valid for Euclidean spaces of any dimension and for all geometrical objects contained in them.
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Proofs: 1 2 3 4
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016