![]() ![]() B’’C’’=EF because a rotation preserves congruence. Rotating ∆A’B’C’ about E using angle C’EF will leave E=B’=B’’. Then what rotation will ensure that ∠B’ maps onto ∠E? We know that ∆A’B’C’ is congruent to ∆ABC because a translation preserves congruence. We can use a translation and a rotation, but we need to map ∆ABC to ∆DEF with B to E using vector BE. It takes a lot of Math Practice 3 for us to make it through explanations for why SSS, SAS, and ASA provide sufficient information for proving triangles congruent. I cannot remember where I recently read (a Tweet? a blog post?) that students need to be convinced a statement is true before they will expend effort proving it. We know that using dynamic geometry software doesn’t prove our results for us.īut using dynamic geometry software does help convince us that we are proving the right thing. Once you’ve mapped C to F using vector CF, the student suggests rotating the new triangle 180˚ about C. But have we used the given SAS? We know that ∠B≅∠E, not that ∠C≅∠F. We’ve mapped ∆ABC to ∆DEF with C to F using vector CF, and rotating ∆A’B’C’ about F using angle C’A’D will map one triangle on top of the other. How can we use rigid motions to show that SAS always works? (I at least knew that the proofs weren’t left as an “exercise” for students at the end of the section on congruent triangles.) We’ve always proved SAA and HL, but for some reason I thought the others were in the back of the book in a section of more challenging proofs of theorems. This standard made me realize that the textbooks I had used for a long time allowed the ASA, SAS, and SSS Triangle Congruence Theorems into our deductive system as postulates. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. ![]()
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