- AAxiom (or postulate)
- BTheorem
- CCorollary
- DConjecture
Step-by-Step Solution
Axioms and postulates are accepted without proof. Theorems are proved; conjectures are unproved guesses.
Answer: A) Axiom (or postulate)
A statement accepted as true without proof and used as a starting point for reasoning is generally called a/an ____.
Class 9Introduction to Euclid's GeometryGeometry — Foundations
Key idea: How a Euclid-style proof works
A proof builds a chain of statements, each justified by something already accepted.
**Example**: Two distinct lines cannot have more than one point in common.
**Proof**: Suppose two distinct lines ℓ and m had two points A and B in common.
- By Postulate 1, only one straight line can be drawn through A and B.
- So ℓ and m must be the same line — contradicting that they are distinct.
- Hence our supposition is false. ∎
This is **proof by contradiction** (reductio ad absurdum). Assume the opposite of what you want, derive an impossible conclusion, conclude the original must be true.
**Example**: Two distinct lines cannot have more than one point in common.
**Proof**: Suppose two distinct lines ℓ and m had two points A and B in common.
- By Postulate 1, only one straight line can be drawn through A and B.
- So ℓ and m must be the same line — contradicting that they are distinct.
- Hence our supposition is false. ∎
This is **proof by contradiction** (reductio ad absurdum). Assume the opposite of what you want, derive an impossible conclusion, conclude the original must be true.
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