Question

Distinguish between the following statements. Which statement is the best example of a theorem?

Statement #1: All triangles have three sides and three angles.

Statement #2: The interior angles of any triangle sum to 180 degrees.

Statement #3: a+b=b+a

Answer 1

Let's analyze the three statements to determine which best exemplifies a theorem.

**Statement #1:** "All triangles have three sides and three angles."
This statement is more of a definition than a theorem. It describes the properties that define triangles rather than asserting a mathematical truth derived from a set of axioms or previously established statements.

**Statement #2:** "The interior angles of any triangle sum to 180 degrees."
This is a formal mathematical proposition that can be proven based on the properties of Euclidean geometry. It makes a specific claim about all triangles that can be tested and validated using logical reasoning or geometric proofs.

**Statement #3:** "a + b = b + a."
This is a statement of the commutative property of addition, which is accepted as a mathematical axiom or fundamental property rather than a theorem derived from other theorems or axioms.

**Conclusion:** The best example of a theorem among these statements is **Statement #2**: "The interior angles of any triangle sum to 180 degrees." It is a valid mathematical assertion that can be proven within the framework of geometry, making it a theorem.