The Teaching of Geometry

by David Eugene Smith

therefore the converses must necessarily be true as a matter of logic; for

if ∠B = ∠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 ∠B > ∠C, then b cannot equal c without violating (1), and b cannot be less than c without violating (3), therefore b > c;

similarly, if ∠B < ∠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.

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.