Triangle congruence criteria (G-CO.8)

[tomergal]

I wonder how students are expected to “explain how the criteria for triangle congruence follow from the definition of congruence in terms of rigid motions.”

Let’s take SSS for example. In the “traditional” approach (i.e. triangles are congruent iff they have congruent side lengths and angle measures), we could justify it by showing how when we know all the side lengths of a triangle, there’s only one triangle we can construct (…and therefore the angle measures must be congruent as well).

But how would we do that with the rigid motions definition of congruence? The general plan is clear: assume two triangles have the same side lengths, and come up with a sequence of rigid transformations that maps one onto the other. But when trying to work this line of reasoning for two non-specific triangles, I found I had to reason in a way similar to that described above (i.e. there’s only one way to construct a triangle from three given side lengths).

It might be that I’m aiming too high and all that is necessary is to have concrete examples with two concrete triangles and a sequence of concrete rigid transformations.

I will appreciate any help with this issue.

[Sarah Stevens]

It won’t be easy but you should find your answers in this progression written by Dr. Hung Hsi Wu. It is difficult reading but mathematically beautiful! https://math.berkeley.edu/~wu/CCSS-Geometry_1.pdf

Also, Dr. Zal Usiskin has written Geometry texts using this approach which can be ordered off Amazon: http://www.amazon.com/gp/product/067345956X?psc=1&redirect=true&ref_=oh_aui_detailpage_o02_s00

Finally, I had the pleasure of meeting Dr. Usiskin at the annual NCTM conference and he confirmed that this tiny little book is another great resource. http://www.amazon.com/gp/product/0866514651?psc=1&redirect=true&ref_=oh_aui_detailpage_o05_s00

In regards to your question, it is possible to do rigorous proofs of triangle congruence using rigid transformations. From reading Wu, you will see that a reflection along the perpendicular bisector will guarantee to carry one point to another. Then a reflection across the angle bisector will guarantee to take one side to another. Then it’s simply am matter of proving that the vertex opposite that side must be at the same location. To wrap my head around these proofs, I got a box of patty paper and worked on the transformations until I understand why one triangle was guaranteed to be concurrent with another. It really is quite a powerful tool.

[tomergal]

Thank you Sarah!

This definitely helps. I’m pretty certain I can devise nice, flowing, proofs of SAS, ASA, and SSS based on Wu’s highly rigorous proofs.

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