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→The fifth postulate
That has never seemed as "self-evident" as the others, and in fact for centuries mathematicians thought it could be proved and tried to produce proofs. Eventually, mathematicians realized that this was impossible. If you removed the parallel postulate, you ended up with a perfectly logical, consistent system, a ''non-Euclidean geometry'' that simply happened to describe a kind of geometry ''different'' from plane geometry.
That has never seemed as "self-evident" as the others, and in fact for centuries mathematicians thought it could be proved and tried to produce proofs. Eventually, mathematicians realized that this was impossible. If you removed the parallel postulate, you ended up with a perfectly logical, consistent system, a ''non-Euclidean geometry'' that simply happened to describe a kind of geometry ''different'' from plane geometry.
===Equivalent statments of the fifth postulate===
Without the fifth postulate many familar geometric ideas are no longer true, these include,
* The sum of the interior degrees of a triangle are two right angle (180 degrees)
[[Category:Mathematics]]
[[Category:Mathematics]]
[[category:geometry]]
[[category:geometry]]