The new “common core” standards for geometry specifically state that congruence of figures is to be defined by the principle of Superposition. In other words, two figures are congruent if and only if they can be superimposed upon each other through a series of rigid transformations so that they are identical i.e. they coincide at in every point. In the past we said described congruence as having the same measurement i.e. being the same size and shape. For polygons that would mean that all corresponding sides are congruent and all corresponding angles are congruent. We essentially, defined it for every type of shape. For angles, we said they were congruent if they had the same measurement, for segments, if they had the same length. for polygons if their corresponding sides and angles are congruent. Before the year began, I wrote out all of the proofs required for students based on the new standards, but struggled with a complete change over to using the definition based on superposition, although conceptually it doesn't seem a far leap from the definition we used in the past.

I didn't fully embrace superposition in my proofs. I think that part of the problem was that when I have heard superposition arguments in the pass it has been in very informal ways. Someone says that “you can see” that corresponding angles, for example, can be superimposed, or that the base angles of an isosceles triangle can be reflected about an axis of symmetry. I always viewed this as less rigorous and it seemed to me like a hand waving exercise, since there generally is no effort was made to explain the  how they could  be superimposed or how one knows that they can be superimposed in perfect correspondence in the first place. The proofs seem generally more of an assertion that borders on being axiomatic. However, upon serious reflection on the matter it seems that this is probably the fault of the presenters and the principle seems more general and rigorous if approached properly. 

At the beginning of the school year I had not yet decided the best course to take for my geometry students. This year they used superposition to prove the bisection of an angles and segments by paper-folding. But, my proofs generally have gone back to the old ways, making a few minor changes. For instance, the fact that  if two parallel lines are crossed by a transversal that the corresponding angles from the two intersections are congruent, possibly seems like an ideal set up for an argument using superposition, but basically I used Euclid’s 5^th axiom instead and stated it in the contrapositive. In the past, corresponding angles have been assumed by our textbook treatments of Euclidean geometry, and the other is proven. The new standards demand that students prove the statement about corresponding angles. Since Euclid’s 5^th axiom and the statement about corresponding angles are equivalent, that is if one is assumed, then the other can be proven. I chose to assume Euclid’s 5^th and to prove corresponding angles, without ever talking about superposition.  As a result I felt that I cheated a little, feeling that the spirit of the new standards was that I did not assume either, but perhaps start by superimposing corresponding angles in a theoretic framework. I have almost wrapped my head around what I want it to look like enough to attack it more vigorously next year.