We had spent some time using the Math Nspired activity Congruent Triangles to explore sufficient criteria for proving triangles congruent. Our goal is to make two triangles using the given criteria that are not congruent. We are obviously not proving that the criteria works when we can’t do it, as we might not have extinguished all possibilities, but if we can find a counterexample, then we are proving that the criteria doesn’t work.
I didn’t realize until I began studying CCSS-M that most of our geometry textbooks admit SSS, SAS, and ASA as postulates. With CCSS-M, we prove the triangle criteria using rigid motions. But I digress. I want to write today specifically about SSA.
So we figured out that SSA doesn’t always work. And using our dynamic geometry software, we also figured out that SSA does work when the triangles are right (otherwise known as HL).
After our exploration, I gave students the following diagram and asked whether the triangles were congruent.
The given information shows SA. Is there anything not marked that we can mark? NP=NP by reflexive. (So it wasn’t that I asked the question and students said NP=NP and then I asked why. It really was that students answered NP=NP by reflexive. They are learning that I don’t just care about the answer. I care about why. And they are beginning to include their justifications as part of their answer.) So now the given information shows SSA. Is that sufficient information to prove the triangles congruent? Most students said no. But I had a few dissenters. They were not convinced that the given triangles weren’t congruent. Their initial argument was that P must lie on the perpendicular bisector of segment MO. But does N also have to lie on that perpendicular bisector? This is the beauty of dynamic geometry software. I don’t have to be the expert. Can you convince me that the triangles are congruent? I’ll give problem solving points to anyone who can. (Note: what problem solving points are will be a future post.)
Some students built the diagram using TI-Nspire.
Other students drew the auxiliary line segment MO to show first that triangle PMO is isosceles and ultimately that triangle MNO is isosceles.
We cut out the SSA example that doesn’t work and showed that if NP is one of the congruent sides, then MP and NO would have to be the other pair of congruent sides.
A few days later, I asked about another pair of triangles.
How would your students answer?
I am thankful for a community of students who feel comfortable dissenting. And so the journey continues …