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.
Proof
Lay the two parallelograms so the equal angles coincide; their union
forms a third parallelogram whose diagonal includes the original
common-angle vertex. By VI.1 the ratios of areas equal the ratios of
adjacent sides; equality of the original areas forces the
reciprocal-proportion relation. Converse runs the same way.