Final answer:
An axiom is a statement believed to be true within defined parameters, whereas a theorem is a statement that has been proven to be true using logical reasoning.
Step-by-step explanation:
An axiom is a statement believed to be true within defined parameters or by observation. It is a basic assumption or principle that is accepted without proof. For example, in geometry, the axiom that states 'the sum of the angles in a triangle is 180 degrees' is considered true without needing to be proved.
A theorem, on the other hand, is a statement that has been proven to be true using logical reasoning based on axioms, definitions, and previously proven theorems. The process of proving a theorem involves constructing a logical argument to support its truth. For instance, the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, is a well-known theorem.
In summary, axioms are assumed to be true, while theorems are proven to be true through logical reasoning. Axioms serve as the foundation upon which theorems are built, and theorems help extend our understanding of mathematical concepts.
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