if [L]_B_ = [L]_C_, then _b_ cannot be greater than _c_ without violating (2), and _b_ cannot be less than _c_ without violating (3), therefore _b_ = _c_; and if [L]_B_ > [L]_C_, then _b_ cannot equal _c_ without violating (1), and _b_ cannot be less than _c_ without violating (3), therefore _b_ > _c_; similarly, if [L]_B_ < [L]_C_, then _b_ < _c_. This Law of Converse may readily be taught to pupils, and it has several applications in geometry. THEOREM. _If two triangles have two sides of the one equal respectively to two sides of the other, but the included angle of the first triangle greater than the included angle of the second, then the third side of the first is greater than the third side of the second, and conversely._ [Illustration] In this proposition there are three possible cases: the point _Y_ may fall below _AB_, as here shown, or on _AB_, or above _AB_. As an exercise for pupils all three may be considered if desired. Following Euclid and most early writers, however, only one case really need be proved, provided that is the most difficult one, and is typical. Proclus gave the proofs of the other two cases, and it is interesting to pupils to work them out for themselves. In such work it constantly appears that every proposition suggests abundant opportunity for originality, and that the complete form of proof in a textbook is not a bar to independent thought. The Law of Converse, mentioned on page 190, may be applied to the converse case if desired. THEOREM. _Two angles whose sides are parallel, each to each, are either equal or supplementary._ 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._ [Illustration] 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. [Illustration: PARALLEL RULER OF THE SEVENTEENTH CENTURY San Giovanni's "Seconda squara mobile," Vicenza, 1686]