**Euclid's Elements: Book I: Proposition 35**

**Proposition 35: Parallelograms which are on the same base and in the same parallels are equal to one another.**
Let ABCD, EBCF be parallelograms on the same base BC and in the same parallels AF, BC;

I say that ABCD is equal to the parallelogram EBCF.

For, since ABCD is a parallelogram, AD is equal to BC. I. 34

For the same reason also EF is equal to BC, so that AD is also equal to EF; C.N. 1 and DE is common; therefore the whole AE is equal to the whole DF. C.N. 2

But AB is also equal to DC; I. 34 therefore the two sides EA, AB are equal to the two sides FD, DC respectively, and the angle FDC is equal to the angle EAB, the exterior to the interior; I. 29

Therefore the base EB is equal to the base FC, and the triangle EAB will be equal to the triangle FDC. I. 4

Let DGE be subtracted from each; therefore the trapezium ABGD which remains is equal to the trapezium EGCF which remains. C.N. 3

Let the triangle GBC be added to each; therefore the whole parallelogram ABCD is equal to the whole parallelogram EBCF. C.N. 2

Therefore etc.

Q.E.D.