Equivalent to Proclus' Axiom and Playfair's Axiom, its rejection led to Bolyai, Lobachevsky, and Riemann's development of hyperbolic and elliptic non-Euclidean geometries.
It follows from this result that any three non-colinear points lie on a circle, and that the summit angles of a Saccheri quadrilateral are right angles. Beltrami proved its independence, and Klügel demonstrated the flaws in 28 proofs of this statement. Equivalent to Proclus' Axiom and Playfair's Axiom, its rejection led to Bolyai, Lobachevsky, and Riemann's development of hyperbolic and elliptic non-Euclidean geometries. In Euclid's Elements, it is the fifth postulate. For 10 points, name this true statement asserting the uniqueness of a line that never intersects another line.
It follows from this result that any three non-colinear points lie on a circle, and that the summit angles of a Saccheri quadrilateral are right angles.
{parallel} postulate [accept {Euclid's fifth} postulate before mentioned; accept " axiom " for " postulate "]