Equal triangles which are on the same base and on the same side are also in the same parallels.

Let *ABC* and *DBC* be equal triangles which are on the same base *BC* and on the same side of it. Join *AD.*

I say that *AD* is parallel to *BC.*

If not, draw *AE* through the point *A* parallel to the straight line *BC,* and join *EC.*

Therefore the triangle *ABC* equals the triangle *EBC,* for it is on the same base *BC* with it and in the same parallels.

But *ABC* equals *DBC,* therefore *DBC* also equals *EBC,* the greater equals the less, which is impossible.

Therefore *AE* is not parallel to *BC.*

Similarly we can prove that neither is any other straight line except *AD,* therefore *AD* is parallel to *BC.*

Therefore *equal triangles which are on the same base and on the same side are also in the same parallels.*

Q.E.D.