One Line Crossing Two Others Creates Eight Angles. Only Two Are Different

Take two lines and draw a third crossing both of them. Each crossing produces four angles, so eight in total. At a single crossing, angles opposite each other are equal and angles next to each other add to a straight line, so only two distinct values appear at each crossing. If the two original lines are parallel, the two crossings produce the same pair of values, so across the whole figure there are only two different angles and every one of the eight is equal to one or the other.

Three pairings have names and each is equal when the lines are parallel:

• •Corresponding angles, in matching positions at each crossing
• •Alternate angles, on opposite sides of the crossing line
• •Co-interior angles, between the lines on the same side
• •The first two are equal to each other
• •The third pair adds to one hundred and eighty degrees
• •Each of these fails if the two lines are not parallel

The relationships work in both directions, which is what makes them useful rather than merely true. If the lines are parallel the angles match, and equally, if the angles match then the lines must be parallel. That converse is what allows a builder to check that two walls are parallel using only a straight edge and a protractor, and what allows a proof to establish that two lines never meet without following them forever. Most school geometry proofs about triangles and parallelograms reduce to drawing such a line and applying one of these relationships.

The single most used consequence is the proof that the angles of a triangle add to a straight line, and it is worth following because it is two lines long. Draw a line through one corner parallel to the opposite side. The two angles at that corner beside the new line are alternate angles with two of the triangle's angles, so each equals one of them. Those two, together with the triangle's third angle at that corner, sit along a straight line and therefore add to one hundred and eighty degrees. The triangle's three angles are equal to those three, which completes it.

The whole arrangement rests on a statement that troubled mathematicians for two thousand years. Euclid's fifth postulate says in effect that if the angles on one side add to less than two right angles, the lines meet on that side, and it is far less obviously true than his other postulates. Generations tried to prove it from the others and failed. In the nineteenth century it was shown that consistent geometries exist in which it is false, where the angles of a triangle do not add to a straight line and parallel lines behave quite differently, which is the mathematics that general relativity later required.

A line crossing two others creates eight angles, and if the two lines are parallel only two distinct values appear among them. Corresponding and alternate pairs are equal and co-interior pairs add to a straight line. The relationships work in reverse, which is how parallelism is proved rather than assumed. The whole structure rests on Euclid's fifth postulate.