I wrote on this site that the last thing a student wants to see is algebra in a geometry class. But my student didn't have much trouble with the algebra for these proofs -- as long as it's simple algebra and not, say, the quadratic formula or graphing! He appears to be getting a little more comfortable with the comfortable with the idea of proof.
And so we finally reach Section 7-2 of the U of Chicago text, the Triangle Congruence Theorems. I will be able to demonstrate how SSS, SAS, ASA, and AAS follow from the definition of congruence in terms of isometries.
The text begins with SSS. We are given two triangles, ABC and DEF, with
If we're lucky, C will already be mapped to F. Then this same isometry will end up mapping the entire triangle ABC to DEF -- which would already make the two triangles congruent. So let's say that ABC is not mapped to DEF. We already mapped
The text now looks at the quadrilateral DFEC' and notes that it satisfies the definition of one of our special quadrilaterals -- the kite. By the Kite Symmetry Theorem from Chapter 5, the kite contains a symmetry diagonal,
The text moves on to SAS and ASA. These proofs are similar. In each case, we begin with an isometry mapping one of the congruent sides to the other -- without loss of generality, we call these congruent sides
In all three proofs, the trick is to map ABC to a triangle that is, if not DEF itself, then a triangle that reflects to DEF. In no case is the isometry mapping ABC to DEC' made explicit -- only that some isometry must exist. Only the isometry from DEC' to DEF is ever made explicit -- this isometry is always a reflection.
I have much more to say about the proofs of SSS, SAS, and ASA -- but once again, I don't have much time to write it now.
Meanwhile, I'm still having problems figuring things out on the new computer. I have my printer scanner from the old computer. I can send documents to the printer to print, but I just can't get the computer to receive documents from the scanner. This is such an important lesson that I simply must have these images posted to this blog. Hopefully at some point I can finally have time to do things, such as figure out how to get documents to scan properly.