In equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional; and equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal.

Let *AB* and *BC* be equal and equiangular parallelograms having the angles at *B* equal, and let *DB* and *BE* be placed in a straight line. Therefore *FB* and *BG* are also in a straight line.

I say that, in *AB* and *BC,* the sides about the equal angles are reciprocally proportional, that is to say, *DB* is to *BE* as *BG* is to *BF.*

Complete the parallelogram *FE.*

Then since the parallelogram *AB* equals the parallelogram *BC,* and *FE* is another parallelogram, therefore *AB* is to *FE* as *BC* is to *FE.*

But *AB* is to *FE* as *DB* is to *BE,* and *BC* is to *FE* as *BG* is to *BF.* Therefore *DB* is to *BE* as *BG* is to *BF.*

Therefore in the parallelograms *AB* and *BC* the sides about the equal angles are reciprocally proportional.

Next, let *DB* be to *BE* as *BG* is to *BF.*

I say that the parallelogram *AB* equals the parallelogram *BC.*

Since *DB* is to *BE* as *BG* is to *BF,* while *DB* is to *BE* as the parallelogram *AB* is to the parallelogram *FE,* and, *BG* is to *BF* as the parallelogram *BC* is to the parallelogram *FE,* therefore also *AB* is to *FE* as *BC* is to *FE.*

Therefore the parallelogram *AB* equals the parallelogram *BC.*

Therefore, *in equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional; and equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal.*

Q.E.D.