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Theorem: A Proven Statement. Postulate: An Accepted Statement without Proof. They mean similar things. A postulate is an unproven statement that is considered to be true; however a theorem is simply a statement that may be true or false, but only considered to be true if it has been proven.
Well, this will depend on the length of the sides of the triangle for what postulate or theorem you will be using.
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the logical structure of the formulation of the CAP is on the form "p implies q", or "If p, then q". In symbols: p => q with p being the statement "l and l' are lines cut by a transversal t in such a way that two corresponding angles are congruent" and q the statement "l is parallel to l'" It's corollarys are also on this form, obviously with other p and q. Not sure if this is what you were looking for.
The Criminal Code of Canada postulates criminal activity and justice in general. One postulate involves that of sentencing. One can find more information about this topic from Wikipedia.