公理和定理有什么不同？

公理(Axiom) is some theory that no proof can be given.
It is arrived by observation, or some experiments, or common sense.
eg "vertically opposite angles are equal", "corresponding angles of parallel lines are equal", "sum of all angles at a point is 360 degrees", "sum of all adjacent angles on a straight line is 180 degrees", etc. they are all axioms.

定理(Theorem) is theory formed based on axioms or other theorems that based on axioms. It must have a proof.
eg "alternate angles of parallel lines" are equal is based on "vertically opposite angles are equal" and "corresponding angles of parallel lines are equal".