Proposition 36.
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. [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. [
C.N. 1]
Therefore etc. Q. E. D.