Assertion (A): Two distinct lines in a plane can intersect in at most one point. Reason (R): Through two distinct points, exactly one line can be drawn. Which is correct?

Class 9Introduction to Euclid's GeometryGeometry — Foundations

ABoth A and R are true, and R is the correct explanation of A.
BBoth A and R are true, but R is not the correct explanation of A.
CA is true but R is false.
DA is false but R is true.

Step-by-Step Solution

If two lines met at two points, both lines would pass through those two points, but only one line can — so they would coincide. R directly explains A.

Answer: A) Both A and R are true, and R is the correct explanation of A.

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.

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