Proposition VI.14 — In equal and equiangular parallelograms the sides about the …

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.

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Depends on (2)

VI.1Proposition VI.1Triangles and parallelograms which are under the same height are to one another as their bases.
V.11Proposition V.11Ratios which are the same with the same ratio are also the same with one another.

Required by (dependents) (4)

VI.15Proposition VI.15In equal triangles which have one angle equal to one angle the sides about the equal angles are reciprocally…
VI.16Proposition VI.16If four straight lines be proportional, the rectangle contained by the extremes is equal to the rectangle contained by…
XI.34Proposition XI.34In equal parallelepipedal solids the bases are reciprocally proportional to the heights; and those parallelepipedal…
XII.9Proposition XII.9In equal pyramids which have triangular bases the bases are reciprocally proportional to the heights; and those…

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