This is not an ancient proposition, although the Greeks were well aware of the principle. It may be stated so as to include the case of the sides being perpendicular, each to each, but this is better left as an exercise. It is possible, by some circumlocution, to so state the theorem as to tell in what cases the angles are equal and in what cases supplementary. It cannot be tersely stated, however, and it seems better to leave this point as a subject for questioning by the teacher.
Theorem. The opposite sides of a parallelogram are equal.
Theorem. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram.
This proposition is a very simple test for a parallelogram. It is the principle involved in the case of the common folding parallel ruler, an instrument that has long been recognized as one of the valuable tools of practical geometry. It will be of some interest to teachers to see one of the early forms of this parallel ruler, as shown in the illustration.[63] If such an instrument is not available in the school, one suitable for illustrative purposes can easily be made from cardboard.
Parallel Ruler of the Seventeenth Century
San Giovanni's "Seconda squara mobile," Vicenza, 1686
A somewhat more complicated form of this instrument may also be made by pupils in manual training, as is shown in this illustration from Bion's great treatise. The principle involved may be taken up in class, even if the instrument is not used. It is evident that, unless the workmanship is unusually good, this form of parallel ruler is not as accurate as the common one illustrated above. The principle is sometimes used in iron gates.
Parallel Ruler of the Eighteenth Century
N. Bion's "Traité de la construction ... des instrumens de mathématique," The Hague, 1723