Proposition 36
Parallelograms which are on equal bases and in the same parallels equal one another.

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

I say that the parallelogram ABCD equals EFGH.

Join line BE and line CH.

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

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

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

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

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