Assertion (A): Euclid's fifth postulate is independent of the first four. Reason (R): Replacing it gives a valid non-Euclidean geometry. 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

The existence of consistent non-Euclidean geometries (where Postulate 5 is replaced) proves Postulate 5 cannot be derived from the other four — so it is independent.

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

Key idea: The fifth postulate and its legacy

Postulate 5 stood out — it was much longer and less obvious than the other four. For 2000 years, mathematicians tried to prove it from the others. They all failed.

In the 19th century, mathematicians like **Lobachevsky**, **Bolyai**, and **Riemann** showed why: if you replace Postulate 5 with a different assumption, you get a different but perfectly valid geometry — **non-Euclidean geometry**. These are real and useful (Einstein's General Relativity uses Riemannian geometry).

An equivalent of Postulate 5 — easier to state — is **Playfair's axiom**: 'Through a point not on a line, exactly one line parallel to the given line can be drawn.'

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