Why isn't ASS a member of the "Fab Five"?
I have taught Geometry for many years, and a big part of the curriculum has been proving congruent triangles. If you teach this, you know about what I refer to as the "Fab Five":
- Angle-Side-Angle (ASA)
- Side-Angle-Side (SAS)
- Side-Side-Side (SSS)
- Angle-Angle-Side (AAS)
- Hypotenuse-Leg of a Right Triangle (HL Rt. △)
I introduce this list with accompanying diagrams illustrating each method. Here is a sample:
After generating the list of the "Fab Five", I emphasize to the students that each method requires three parts from each triangle. This, in turn, always results with students asking: "Hey Mr. K. What about AAA and ASS?", the latter of which usually produces a number of giggles.
So I ask them: "If you know that three angles in one triangle are congruent to the three corresponding angles in another triangle, why are the triangles not necessarily congruent?" After a little guidance, students will recall our work from earlier in the year with dilations and scale factors and the fact that a dilation preserves angle measures. This becomes our first introduction into similarity. So, AAA is not a member of the "Fab Five".
This brings us to Angle-Side-Side, aka "ASS". Now I understand the teenage mind pretty well, even if it's been quite a while since I had one. The mere mention of the term and writing it on the board is usually an opportunity to have a few laughs - always an important part of my class culture. Some teachers explain that "It's a bad word, so you can't use it as a proof reason." That's cute, but I'm not sure if teachers really explain why it doesn't work.
So I present the students with this:
In the past, students used their compasses, protractors, and straightedges to draw triangles with these measurements. Then I would display the triangles on the board and ask the students for their observations. Some cases produced congruent triangles, some did not. Of course, this depended on their construction & measurement abilities.
This year, I had the students construct the triangles using the Desmos Geometry app. I had them first make a triangle and show the side and angle measures. Then, the students manipulated the triangle to match the given parameters:
This brought up issues of accuracy (how close should it be?) and placement of measurements (does m∠ABC have to equal 60° or can m∠CAB = 60°?).
After they made their triangles, I gave the students patty paper and asked them to trace their triangles. I then collected their papers and laid/stacked them under the document camera to see which parameters produced congruent triangles.
The results and conversations were really powerful. Many "ah-ha" moments were made. And most importantly, the students realized why ASS is not a member of the "Fab Five"!
Finally, here is a LINK to a presentation which includes screenshots and links to each of the Desmos Geometry constructions.
Thanks for reading!