Avoiding Circular Reasoning: How to Define Congruent Shapes

Question

I apologize for being overly verbose here, but the question I want to know is at the very bottom. I am going to be honest and say I have no idea how to axiomatically handle congruence in geometry and want to understand it.

I have always held SSS, SAS, and CPCTC to be axiomatic where SSS and SAS are definitions to tell whether or not two triangles are congruent. Later, I realized I should not do that here...

Why does everyone say two shapes (e.g. triangles here) are congruent iff there is an isometry between the figures? I honestly view the word "isometry" as a made up word describing SSS by a Euclidean metric function that all of the sudden discusses "types of rigid motions". Clearly, if the distances between points are the same in context of triangles, then they satisfy SSS and vice versa.

I am confused why the word isometry is discussed with rotations in particular... What is a rotation and how does that preserve distance here? How can a figure possibly be rotated in space by the means of a function this early on in geometry?

We rotate points in space in terms of sine and cosine. We derive rotational matrices in terms of sine and cosine by double angle formulas. We define sine and cosine in terms of similarity which is done by similarity and axioms using SSS and SAS. We define similarity just like congruence with a scale factor. This leads me to this question again...

Question: How are figures rotated by functions in terms of axioms in the context of isometries?

Answer 1

One way to approach it is to define "reflection" as your fundamental transformation, i.e. for a line $\ell$, there is a transformation $R_\ell$ that preserves collinearity, distance, and angle, but swaps one half-plane for the other.

Then, isometries can be defined as compositions of reflections. Specifically, a rotation about $O$ consists of a consecutively-applied pair of reflections $R_{\ell_1}$ and $R_{\ell_2}$ where $\ell_1$ and $\ell_2$ cross at $O$.

Answer 2

First, I think it may help to look at a less-trivial situation - quadrilaterals. Here SSSS is not sufficient for congruence (think chevron vs. kite), so clearly there's something more interesting going on here.

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The general language of metric spaces and isometries provides an abstract definition of congruence (and "ambient congruence" - an equivalent notion in many cases, including Euclidean geometry, but not in general). Note that this definition of congruence doesn't require us to analyze the particular congruences in any way; it's entirely "from the outside" and applies to all situations. In a presentation of Euclidean geometry which includes distance as a primitive notion we can then show that a pair of reflections is an isometry and two triangles with the same side length are related by a pair of reflections; this amounts to a proof of SSS as a nontrivial result.

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Alternatively, we can start with Euclidean geometry without a notion of distance built in. Now "same length" is defined in terms of reflections ($\overline{AB}$ and $\overline{CD}$ are defined to have the same length if there is a pair of reflections whose composition sends $A$ to $C$ and $B$ to $D$). SSS can then be stated as follows:

Suppose $\{A,B,C\},\{D,E,F\}$ are triples of distinct points such that $\overline{AB}$ and $\overline{DE}$ have the same length, $\overline{BC}$ and $\overline{EF}$ have the same length, and $\overline{CA}$ and $\overline{FD}$ have the same length. Then there is a pair of reflections whose composition sends $A$ to $D$, $B$ to $E$, and $C$ to $F$.

This is noncircular since it's formulated entirely in terms of the primitive notions of Euclidean geometry without distance - note that reflections can be defined by thinking about perpendicular bisectors - and it is nontrivial since a priori the three pairs of reflections involved in the hypothesis could be quite different. Finally, this formulation of SSS can also be proved from the appropriate axioms.