Proposition XXXV. Theorem.
[Euclid, ed. Lardner, 1855, on Google Books]
| (165) | Parallelograms on the same base (B C) and between the same parallels are equal. |
For the angles B A F and C D F and also B E A and C F D are equal (XXIX), and the sides A B and D C are also equal (XXXIV), and therefore (XXVI) the triangles B A E and C D F are equal. These being successively taken from the whole quadrilateral B A F C, leave the remainders, which are the parallelograms B D and B F, equal.
Book I: Euclid, Elements, Book I (ed. Dionysius Lardner, 11th Edition, 1855)
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This proposition in other editions:
We have in this proof departed from Euclid in order to avoid the subdivision of the proposition into cases. The equality which is expressed in this and the succeeding propositions is merely equality of area, and not of sides or angles. The mere equality of area is expressed by Legendre by the word equivalent, while the term equal is reserved for equality in all respects. We have not thought this of sufficient importance however to justify any alteration in the text.