Parallelograms which are one 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
I say that ABCD is equal to the parallelogram EBCF.
For, since ABCD is a
AD is equal to BC. [I. 34]
But AB is also equal to
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
therefore the trapazium ABGD which remains is equal to the trapezium EGCF which remains. [C.N. 3]
Let the triangle GBC
be added to each;
therefore the thole parallelogram ABCD is equal to the whole parallelogram EBCF. [C.N. 2]
Therefore, etc. Q.E.D.
Book I: Euclid, Book I (ed. Sir Thomas L. Heath 1st Edition, 1908)
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