Definition
The parallel postulate is Euclid's fifth postulate, often stated as: through a point not on a given line, there is exactly one line parallel to the given line (the "Playfair form"). Euclid's original phrasing is more elaborate. The postulate seemed less self-evident than Euclid's other four, and mathematicians spent 2000 years trying to prove it from the rest.
Why it matters
How it works
Many alternative formulations are equivalent to the standard one: "the angles of every triangle sum to 180°"; "two parallel lines are everywhere equidistant"; "there exist similar but non-congruent triangles." To show the postulate is independent of Euclid's other four, construct models that satisfy the other four but make different assignments to parallels — the Poincaré disk model (hyperbolic, multiple parallels) and the sphere with antipodal points identified (elliptic, no parallels) both work.