Proposition 27

If a straight line falling on two straight lines makes the alternate angles equal to one another, then the straight lines are parallel to one another.

Let the straight line EF falling on the two straight lines line AB and line CD make the alternate angles angle AEF and angle EFD equal to one another.
I say that line AB is parallel to line CD.

If not, line AB and line CD when produced meet either in the direction of point B and point D or towards point A and point C.
Let them be produced and meet, in the direction of point B and point D, at point G.

Then, in the triangle GEF, the exterior angle AEF equals the interior and opposite angle EFG, which is impossible.
Therefore line AB and line CD when produced do not meet in the direction of point B and point D.
Similarly it can be proved that neither do they meet towards point A and point C.

But straight lines which do not meet in either direction are parallel. Therefore line AB is parallel to line CD.
Therefore if a straight line falling on two straight lines makes the alternate angles equal to one another, then the straight lines are parallel to one another.

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