The Teaching of Geometry

by David Eugene Smith

Two triangles are congruent if two sides and the included angle of the one are equal respectively to two sides and the included angle of the other.

Two triangles are congruent if two angles and the included side of the one are equal respectively to two angles and the included side of the other.

In general, to every proposition involving points and lines there is a reciprocal proposition involving lines and points respectively that is often true,—indeed, that is always true in a certain line of propositions. This relation is known as the Principle of Reciprocity or of Duality. Instead of points and lines we have here angles (suggested by the vertex points) and lines. It is interesting to a class to have attention called to such relations, but it is not of sufficient importance in elementary geometry to justify more than a reference here and there. There are other dual features that are seen in geometry besides those given above.

Theorem. In an isosceles triangle the angles opposite the equal sides are equal.

This is Euclid's Proposition 5, the second of his theorems, but he adds, "and if the equal straight lines be produced further, the angles under the base will be equal to one another." Since, however, he does not use this second part, its genuineness is doubted. He would not admit the common proof of to-day of supposing the vertical angle bisected, because the problem about bisecting an angle does not precede this proposition, and therefore his proof is much more involved than ours. He makes CX = CY, and proves ⧌XBC and YAC congruent,[62] and also ⧌XBA and YAB congruent. Then from ∠YAC he takes ∠YAB, leaving ∠BAC, and so on the other side, leaving ∠CBA, these therefore being equal.

This proposition has long been called the pons asinorum, or bridge of asses, but no one knows where or when the name arose. It is usually stated that it came from the fact that fools could not cross this bridge, and it is a fact that in the Middle Ages this was often the limit of the student's progress in geometry. It has however been suggested that the name came from Euclid's figure, which resembles the simplest type of a wooden truss bridge. The name is applied by the French to the Pythagorean Theorem.

Proclus attributes the discovery of this proposition to Thales. He also says that Pappus (third century A.D.), a Greek commentator on Euclid, proved the proposition as follows:

Let ABC be the triangle, with AB = AC. Conceive of this as two triangles; then AB = AC, AC = AB, and ∠A is common; hence the ⧌ABC and ACB are congruent, and ∠B of the one equals ∠C of the other.

This is a better plan than that followed by some textbook writers of imagining ⧍ABC taken up and laid down on itself. Even to lay it down on its "trace" is more objectionable than the plan of Pappus.