If the two interior angles on the same side add to less than 180°, the drawn lines will, if they continued, meet. Now draw a transversal (line crossing both of those first two lines). Move away a few centimeters from it and draw another 10 cm line. Take a sheet of paper, pencil, and straightedge. The sum of both same-side interior angles is less than 180°, so Euclid is saying the lines represented by the first two spaghetti strands will, if extended, eventually meet. Look at the same-side interior angles toward the close ends of spaghetti. You see you have created eight angles at the two intersections. Take two strands and arrange them a bit apart from each other but leaning toward each other. The fastest way to understand the Parallel Postulate is to set up some line segments. 'If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.' Parallel Postulate Example What is the Parallel Postulate?Īfter Euclid knocked out four postulates (the foundation for absolute geometry), he waited before springing his fifth postulate, which in an English translation by Thomas Heath states: Same-side interior angles are the two angles on the same side of the transversal. The interior angles are between the two other lines exterior angles are outside the two other lines. Interior angles are the angles formed when a transversal crosses two other lines. Contrast a postulate with a theorem, which is shown to be true by using proofs. A postulate is an idea (also called an axiom) that is taken to be true even without proof.

How can anyone be sure lines are parallel, if lines go on forever? You and your classmates may be new to geometry, but geometry has existed for thousands of years, and thousands of years ago, Euclid wrote down five postulates, one of which is the kernel of the Parallel Postulate.Įuclid had many great ideas, but not all could be proven.

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