Mathematics vocabulary – which option best describes the term theorem?

Introduction / Context:
Although this question appears under English, it actually tests basic mathematical vocabulary that students frequently encounter in geometry and algebra. The term theorem is central in mathematics, especially in geometry, where many exam questions ask you to recall or apply specific theorems. Knowing the precise meaning of the word theorem helps you distinguish it from related terms such as definition, axiom, and conjecture.

Given Data / Assumptions:

The question asks for the option that best describes the meaning of theorem in standard mathematics.
Some options represent other technical terms such as definition, conjecture, and axiom.
We assume a high school level understanding of logical reasoning and proof in mathematics.
The correct option must highlight proof by deductive reasoning as the key feature of a theorem.

Concept / Approach:
In mathematics, a theorem is a statement that has been logically proven to be true using deductive reasoning, starting from definitions, axioms, and previously established results. A conjecture, by contrast, is an unproven statement believed to be true based on observation or inductive reasoning. A definition explains the precise meaning of a mathematical term, and an axiom (or postulate) is a statement accepted without proof. Understanding these distinctions makes it clear which option correctly describes theorem.

Step-by-Step Solution:
Step 1: Recall the textbook meaning: a theorem is a proven mathematical statement established by logical deduction. Step 2: Read option A: A conclusion proved by deductive reasoning. This directly captures that essential idea. Step 3: Examine option B: A statement explaining the meaning of a geometric term. This describes a definition rather than a theorem. Step 4: Consider option C: A conjecture based on inductive reasoning. This is almost the opposite of a theorem because a conjecture has not yet been proved. Step 5: Look at option D: A statement that is accepted as true without proof, which is the definition of an axiom or postulate, not a theorem. Step 6: Option E mentions a formula relating two variables, which could describe many equations, some of which may be theorems, but the phrase does not capture the core idea of proof and is therefore incomplete.

Verification / Alternative check:
Think of well known examples such as Pythagoras theorem or theorems about angles in a triangle. Each of these is not simply a definition or an assumption; instead they have detailed proofs that show step by step why the statements must be true for all valid figures. This is the hallmark of a theorem. In contrast, axioms like parallel lines never meet are taken as starting points, and definitions like radius of a circle explain terms. Conjectures such as Goldbach conjecture remain unproved. The only option that clearly emphasises proof by deductive reasoning is option A.

Why Other Options Are Wrong:
Option B describes a definition, which is used to introduce terms like triangle or perpendicular but does not itself require proof.
Option C explicitly states conjecture based on inductive reasoning, which refers to patterns that seem true from examples but are not yet proved for all cases.
Option D outlines an axiom or postulate, which is assumed to be true and used as a starting point for proofs, but is not itself a theorem.
Option E talks about a formula relating variables, which might be derived from a theorem but is not the definition of theorem itself.

Common Pitfalls:
Students sometimes confuse axiom and theorem because both appear in proofs. A helpful way to remember the difference is that axioms are starting points without proof, while theorems are end points with proof. Conjectures sit in between, waiting to be proved or disproved. In multiple choice questions, always look for the phrase proved by deductive reasoning when the term theorem is involved.

Final Answer:
The term theorem is best described as a conclusion proved by deductive reasoning.