Congruence Of Triangles

© Copyright 1997, Jim Loy

In Plane Geometry, two objects are congruent if all of their corresponding parts are congruent. In the first diagram, the two triangles have two sides which are congruent, and the angle between these sides is also congruent. Euclid proved that they are congruent triangles (Theorem I.4, called "Side-Angle-Side" of SAS). But, he was not happy with the proof, as he avoided similar proofs in other situations. The way he proved it, is to move one triangle until it is superimposed on the other triangle. Such a trick (superposition: placing one triangle on top of another, to see if they are congruent), is not considered legal, now. It involves some complicated assumptions. So, now this (SAS) is given as an assumption (postulate). It is the Side-Angle-Side Postulate.

Euclid proved his Side-Side-Side (SSS) Theorem (I.8) and his Angle-Side-Angle (ASA, diagram at the right) Theorem (I.26) in a similar way. In SSS, if a triangle has all three sides conguent to the corresponding sides of a second triangle, then they are congruent. In ASA, if a triangle has two angles and the side in between the angles congruent to the corresponding parts of a second triangle, then they are congruent. These too, are now assumed as postulates.

It is easy to construct a triangle given the lengths of the three sides (diagram on the left), or two sides and the included angle, or two angles and the included side. Given the ease of these constructions (especially SSS), it seems strange that the corresponding congruence "theorems" cannot be proven, except by slightly shady means (superposition).

All three of these congruence postulates are equivalent. You can assume any one of them, and prove the other two from there. See Congruence Of Triangles, Part II for a proof of this.

Euclid also proved an Angle-Angle-Side (AAS) as a corollary (a minor theorem derived directly from the parent theorem) to Theorem I.26. The situation SSA (or ASS) is not a theorem or postulate, as it is often ambiguous, there are often two different non-congruent triangles with two congruent sides and a congruent angle at the end of one of the sides.

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Note: In the above, I used the term "congruent" instead of "equal" when comparing sides and angles. Numbers are equal. Line segments (sides) and angles are congruent. Calling them "equal" is a sloppy way of saying that their measures (lengths or sizes) are equal. Nevertheless, I do use this more informal terminology in some of my articles.

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Addendum:

In answer to email about proving these congruence theorems/assumptions, I wrote this:

The issue is this (using SAS as an example). If you take your angle and two segments, and move it to somewhere else, does the triangle that you get, by drawing the third side of your moved object, change shape (is it congruent to the original triangle)? We can say that it doesn't change shape (that would seem to be right), but we can't prove it. Moving things is a complicated process; do we need postulates that say what changes and what does not change when we move a geometric figure? It is simpler to make congruence assumptions. Then we know that a triangle over here is congruent to a triangle over there. And motion never enters into it.