Final answer:
An axiom is a statement believed to be true without requiring proof, while a theorem is a statement that has been proven to be true based on previously established axioms or theorems.
Step-by-step explanation:
An axiom is a statement believed to be true without requiring proof, while a theorem is a statement that has been proven to be true based on previously established axioms or theorems.
For example, in geometry, Euclid's fifth postulate, also known as the parallel postulate, is an axiom that states that if two lines are intersected by a transversal and the interior angles on the same side of the transversal sum to less than 180 degrees, then the lines will eventually intersect. On the other hand, the Pythagorean theorem is a theorem that has been proven using axioms and other theorems, and it states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
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