Parallelograms which are on equal bases and in the same parallels equal one another.

Let *ABCD* and *EFGH* be parallelograms which are on the equal bases *BC* and *FG* and in the same parallels *AH* and *BG.*

I say that the parallelogram *ABCD* equals *EFGH.*

Join *BE* and *CH.*

Since *BC* equals *FG* and *FG* equals *EH,* therefore *BC* equals *EH.*

But they are also parallel, and *EB* and *HC* join them. But straight lines joining equal and parallel straight lines in the same directions are equal and parallel, therefore *EBCH* is a parallelogram.

And it equals *ABCD,* for it has the same base *BC* with it and is in the same parallels *BC* and *AH* with it.

For the same reason also *EFGH* equals the same *EBCH,* so that the parallelogram *ABCD* also equals *EFGH.*

Therefore *parallelograms which are on equal bases and in the same parallels equal one another.*

Q.E.D.