Robert Kern Curtis
02/08/04 If two sides of a triangle are congruent, then the angles opposite these sides are congruent.

Restatement:
Given triangle ABC. If segment AB is congruent to segment AC, then angle B is congruent to angle C.

Proof:
Consider the correspondence ABC <--> ACB
between triangle ABC and itself. Under this correspondence, we see that segment AB <--> segment AC,
segment AC <--> segment AB,
angle A <--> angle A.
Since this is an SAS correspondence, it follows by SAS that triangle ABC is congruent to triangle ACB, that is, the correspondence ABC <--> ACB is a congruence.
By the definition of a congruence between triangles all pairs of corresponding parts are congruent. Therefore angle B is congruent to angle C because these angles are corresponding parts.
We now show how the above proof looks in two-column form. The same restatement is used.
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Statements                                       Reasons

1.  segment AB is congruent to segment AC        Given.

segment AC is congruent to segment AB

2.  angle A is congruent to angle A               Identity congruence.

3.  triangle ABC is congruent to triangle ACB     Steps 1 & 2 and SAS.

4.  angle B is congruent to angle C               C.P.C.T.C.

QED

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Notes: Isosceles and equilateral triangles

A. The Isosceles Triangle Theorem. The isosceles triangle theorem is a striking example of the use of a particular correspondence to establish a congruence. We merely show that an isoceles is congruent to itself under a correspondence which interchanges the vertices at the ends of the base. To establish a correspondence between a triangle and itself is at this stage a rather subtle maneuver, and it is advisable to discuss the theorem in detail in class. We suggest that you prove it, after preliminary discussion, in two-column form....

B. The "traditional" proof of the Isosceles Triangle Theorem. Although a good proof of the Isosceles Triangle Theorem was known in antiquity, a rather unsatisfactory proof became conventional. This "conventional" proof says to bisect angle BAC, letting D be the point at which the bisecting ray AF intersects the base, and then show that triangle ADB and triangle ADC are congruent. This "conventional" proof is longer than the proof in the text, and, moreover, it is incomplete. From a rigorous point of view it is necessary to show that ray AF intersects segment BC. this does follow from the Crossbar Theorem ... but this latter theorem is extremely difficult, and its difficulties are foreign to the problem before us. The proof of the Isosceles Triangle Theorem in the text is simple, in keeping with the simplicity of the theorem itself, and is free from logical gaps.

C. Historical note on the Isosceles Triangle Theorem. Euclid's own proof of the Isosceles Triangle Theorem ... is rather difficult. (Euclid's proof was a stumbling block to some students in the Middle Ages, and the theorem consequently acquires the name pons asinorum or "The Bridge of Asses.") The proof given in the text is due, essentially, to Pappus, although Pappus naturally did not use the sort of formulation for the congruence postulate that we have been using here. Not many years ago--or so the story goes--an electronic computing machine was programmed to look for proofs of elementary geometric theorems. When the pons asinorum theorem was fed into the machine, it promptly printed Pappus' proof on the tape. This is said to have been a surprise to the people who had coded the problem; Pappus' proof was new to them. What had happened, of course, was that the SAS postulated had been coded in some form such as this:

"If (1) A, B, and C are noncollinear; (2) D, E, and F are noncollinear; (3) segment AB is congruent to segment DE; (4) segment BC is congruent to segment EF; and (5) angle ABC is congruent to angle DEF, then (6) segment AC is congruent to segment DF; (7) angle ACB is congruent to angle DFE; and (8) angle BAC is congruent to angle EDF."

This is the sort of austere language in which people commonly talk to vacuum tubes and transistors; you can't indoctrinate them with vague preconceptions and prejudices; and so, if you want them to get the idea that the triangles in the SAS postulate are supposed to be different, you have to say so explicitly. It did not occur to anybody to do this, and so the machine proeeded, in its simple-minded way, to produce the simplest and most elegant proof.

Notes from pages 46 and 47 of Geometry Teachers' Manual prepared by Gerhard Wichura. Addison-Wesley c.d., 1964.

Text: Edwin E. Moise and Floyd L. Downs, Jr., Geometry Allison-Wesley Publishing Company, Inc., c.d., 1964.