If a straight line falling on two other straight lines, make the alternate angles equal to one another, the two straight lines shall be parallel to one another.
Let the straight line EF, which falls on the two straight lines AB, CD, make the alternate angles AEF, EFD equal to one another: AB shall be parallel to CD.
For if not, AB and CD, being produced, will meet either towards B, D or towards A, C. Let them be pro- duced and meet towards B, D at the point G.
Therefore GEF is a triangle, and its exterior angle AEF
is greater than the interior opposite angle EFG; [I. 16.
But the angle AEF is also equal to the angle EFG; [Hyp.
which is impossible.
Therefore AB and CD being produced, do not meet to- wards B, D.
In the same manner, it may be shewn that they do not
meet towards A, C.
But those straight lines which being produced ever so far both ways do not meet, are parallel. [Definition 35.
Therefore AB is parallel to CD.
Wherefore, if a straight line &c. q.e.d.
If a straight line falling on two other straight lines, make the exterior angle equal to the interior and opposite angle on the same side of the line, or make the interior angles on the same side together equal to two right angles, the two straight lines shall be parallel to one another.