G-CO 8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
I have to admit that I wasn’t ready for proving the congruence theorems on our first day of Congruent Triangles. My previous experience with teaching methods for proving triangles congruent is that students need to understand what they are doing – and how SAS is different from ASA – before they are ready to develop an argument that some methods work and some do not. So we explored possible methods for proving triangles congruent and let ASA, SAS, and SSS in to our deductive system as postulates – pending going back and using rigid motions to prove them as theorems. We used a Geometry Nspired activity to explore whether the given information seemed enough to prove the triangles congruent – Congruent Triangles. Our goal as we moved from page to page was to create a different triangle from everyone else in the room with the given conditions. We started with S.
It didn’t take long for two students to have different triangles. We had a counterexample and were ready to try a different method, SSS.
The students understood that they were trying to make a triangle different from others, and while we had all sorts of transformations, all triangles were congruent, which we showed by measuring the angles in our “finished” triangle. We agreed to let in SSS as a postulate for proving triangles congruent.
For the next few methods that we tested, each student had two triangles on their page, with the same goal of making two different triangles. No one was successful with SAS.
We tried ASA next, and agreed to let it in as a method for proving triangles congruent. And then we proved that AAS worked as a result of ASA and the Third Angles Theorem.
Several students successfully made two different triangles when we had the given conditions of SSA.
So we concluded that SSA doesn’t prove triangles congruent when the given angle is acute. Next we tried making the given angle obtuse, which sometimes results in no triangle (a need to know about the Triangle Inequality Theorem) and sometimes results in exactly one triangle.
Next we made the given angle right.
And went on to prove that HL is a method for proving triangles congruent (using the Pythagorean Theorem).
I have tried activities before where students draw triangles with a protractor and straightedge using the given conditions. We ended up spending the entire period not getting anywhere, mainly because it was difficult for students to see from a list of three conditions how they might construct two different triangles. I have been using this Geometry Nspired activity successfully for 3-4 years now. They really get that our goal is to find a counterexample to show that the given conditions do not necessarily make congruent triangles, and they work hard to find one. I found a MAP lesson on Analyzing Congruency Proofs, but I wasn’t convinced that it was going to be more effective than what I had been using. I also found a disturbing statement in the lesson which didn’t help convince me to try it:
If the triangles are ‘AAS’ they will always be similar, but not necessarily congruent.
And so the journey continues ….