parallelogram equality
======================

.. index:: proof, parallelograms

.. image:: elem.1.prop.36.png
   :align: right
   :width: 300px

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

===

Let `ABCD`, `EFGH` be parallelograms which are on equal bases `BC`, `FG` and in the same parallels `AH`, `BG`;  I say that the parallelogram `ABCD` is equal to `EFGH`.

For let `BE`, `CH` be joined.

Then, since `BC` is equal to `FG` while `FG` is equal to `EH`, 

- `BC` is also equal to `EH`. [I.c.n.1]

But they are also parallel.

And `EB`, `HC` join them; but straight lines joining equal and parallel straight lines (at the extremities which are) in the same directions (respectively) are equal and parallel. [I.33]

Therefore `EBCH` is a parallelogram. [I.34]

And it is equal to `ABCD`; for it has the same base `BC` with it, and is in the same parallels `BC`, `AH` with it. [I.35]

For the same reason also `EFGH` is equal to the same `EBCH`; [I.35] so that the parallelogram `ABCD` is also equal to `EFGH`. [I.c.n.1]

Therefore etc. 

- Q. E. D.

## References


[I.33]: /elem.1.33 "Book 1 - Proposition 33"
[I.34]: /elem.1.34 "Book 1 - Proposition 34"
[I.35]: /elem.1.35 "Book 1 - Proposition 35"
[I.c.n.1]: /elem.1.c.n.1 "Book 1 - Common Notion 1"