Mathematicians tried to prove the parallel postulate, with no success. Then they tried to prove it using proof by contradiction. This technique assumes the statement is false, and tries to prove the opposite is true. When a rigorous proof arrives at an obvious contradiction, yet steps of the proof were geometrically valid, then the only recourse is to decide that the initial assumption was false -- thus proving that the original statement was correct.

So what is the opposite, or negation, of the parallel postulate? Well, the opposite of one parallel line would be no parallel lines or more than one parallel line. We will consider these two cases separately.

Case 1: Given a line and a point not on the line, there are NO lines through the point parallel to the given line

Case 2: Given a line and a point not on the line, there are infinitely MANY lines through the point parallel to the given line