Euclid's Elements - Book 1 - Proposition 27
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.
In other words, given two lines and a transversal, if the alternate interior angles are equal to each other, the two lines are parallel.
Construction:
1. Draw two straight lines AB
2. and CD.
3. Draw a third line such that it intersects lines AB and CD at points E and F.
Given angle AEF and angle EFD are equal, prove that the lines AB and CD are parallel.
Proof by contradiction:
4. Assume that lines intersect at point G.
Then angle AEF is an exterior angle to the triangle EFG, which means that angle AEF is larger than angle EFG (I.16).
Similarly it can be proved that the lines will not meet towards A and C.
Straight lines which do not meet in either direction are parallel; therefore, AB is parallel to CD.
Q.E.D.