If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another.
一条直线落在两条直线上，若内错角相等，则这两条直线平行。
直线 AB 与 CD 同被横截线 EF 截，分别交于 E、F；若内错角 AEF=EFD 则 AB∥CD；G 是假设两线相交所得的虚构交点。
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For let the straight line EF falling on the two straight lines AB, CD make the alternate angles AEF, EFD equal to one another; I say that AB is parallel to CD. For, if not, AB, CD when produced will meet either in the direction of B, D or towards A, C.
一条直线落在两条直线上，且内错角相等。
Let them be produced and meet, in the direction of B, D, at G. Then, in the triangle GEF, the exterior angle AEF is equal to the interior and opposite angle EFG: which is impossible.
若两直线相交，则形成一个三角形。
[I. 16] Therefore AB, CD when produced will not meet in the direction of B, D. Similarly it can be proved that neither will they meet towards A, C.
这个三角形的外角会等于一个内对角，违反 euclid-elements/book1-prop-016。
But straight lines which do not meet in either direction are parallel; [Def. 23] therefore AB is parallel to CD. Therefore etc.
所以两直线不相交，即平行。