- APlayfair's axiom
- BPythagoras' theorem
- CThales' theorem
- DEuclid's lemma
Step-by-Step Solution
This statement is Playfair's axiom, a well-known equivalent form of Euclid's fifth (parallel) postulate.
Answer: A) Playfair's axiom
The equivalent of Euclid's fifth postulate that says 'through a point not on a given line, exactly one parallel can be drawn' is called ____.
Class 9Introduction to Euclid's GeometryGeometry — Foundations
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.'
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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