the parallel postulate
Euclid's first four postulates are short and self-evident — you can draw a line between two points, extend it, draw a circle, all right angles are equal. The fifth is conspicuously different: long, intricate, and not obvious at all. In Euclid's words, if a line crossing two others makes the two interior angles on one side add to less than two right angles (less than 180 degrees), then those two lines, extended far enough on that side, must eventually meet. For two thousand years mathematicians felt this was really a theorem in disguise and tried to prove it from the other four. Every attempt failed — and the failure turned out to be the most fruitful in the history of geometry.
Read the postulate as a condition for meeting. Imagine two roads cut by a third. If the angles on one side fall just short of a straight 180 degrees, the roads are leaning toward each other on that side, and the postulate guarantees they cross there. The contrapositive is the picture most people carry: if the interior angles sum to exactly two right angles, the lines never meet — they are parallel. The deep content is the word 'eventually', because the meeting point can lie arbitrarily far away, beyond any diagram, which is exactly why the postulate cannot be checked by inspection and why it is not obvious. It is doing real work that the other postulates cannot do.
The resolution, reached in the 1820s and 1830s by Lobachevsky, Bolyai, and Gauss, is that the fifth postulate is independent of the other four — it can be neither proved nor disproved from them. So it is not a hidden theorem and it is not false; it is a genuine choice. Accept it and you get Euclidean (flat) geometry; replace it and you get consistent non-Euclidean geometries, just as logically sound. The honest statement is the careful one: the parallel postulate was not refuted by non-Euclidean geometry; it was revealed to be one consistent option among several, with the flat plane being the special case our everyday intuition is tuned to.
A transversal crosses two lines making interior angles of 88 and 89 degrees on one side; their sum, 177, is under 180. The parallel postulate says the two lines must meet somewhere on that side — though the meeting point could be kilometres past the edge of the page. Bump both angles up to 90 and 90: the sum is exactly 180, and the lines are parallel, never meeting.
Interior angles under 180 force a meeting; exactly 180 gives parallels — but 'eventually' can be very far.
Common misconception: that non-Euclidean geometry 'proved the parallel postulate false'. It did the opposite — it proved the postulate independent, so geometries with it and without it are both consistent. Euclid was not wrong, only one of several options.