parallelogram equality

../../../../_images/elem.1.prop.36.png

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

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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.

## 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”