- AProof by contradiction
- BProof by example
- CDirect proof
- DProof by measurement
Step-by-Step Solution
Assuming the opposite and deriving a contradiction is 'proof by contradiction' (reductio ad absurdum).
Answer: A) Proof by contradiction
A proof method where you assume the opposite of what you want to prove and reach a contradiction is called ____.
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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