*BOOK I. 27, 28.*

*31*

*THEOREM*.

*If a straight line falling on two other straight lines, make the alternate angles equal to one another, the two straight lines shall he 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.

*THEOREM*.

*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.*