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Proving Congruence Theorems

29 Oct

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.

Now that we knew what we ASA, SSS, and SAS meant & how to use them, I felt a little more ready to investigate why they were true using rigid motions.

I constructed two congruent triangles using SSS and asked students whether they could come up with a series of rigid motions that mapped one triangle onto another. I used the Illustrative Mathematics task Why does SSS work? as a resource.

I gave students the triangles on paper first. They worked in pairs – cutting out the pre-image on one person’s paper, placing it on top of the pre-image on the other person’s paper, and then literally moving the triangle using rigid motions until it “landed” on top of the image. Then they moved to the TI-Nspire document, where they performed the transformations, listing the order as they went.

 

You can tell that this student was successful because my original triangle DEF was outlined in red. Also, as I checked them, I moved point C to see if their series of rigid motions maintained congruence.

 

This student has some issue with “attend to precision”. We have not talked about reflecting a triangle across an angle. But with a little work, that language can be corrected.

 

This student performed the rigid motions correctly, but did not generalize the results. I tried to get around to all of the students – and made it to several who were doing something similar. (It is difficult to get around to all 34 in a short period of time – and now that I think back, I should have used the Class Capture of TI-Nspire Navigator to manage that more effectively, although there are times I don’t want to show Class Capture to the whole class because I don’t want what someone else is doing to influence their work.)

What is the significance of 94 degrees? Which angle measures 94 degrees? Why does that work?

 

I have a few observations about this lesson.

1. Students were engaged. They were talking a little to their partners, but for the most part, it was quiet in class this day. You could “hear” them working and thinking.

2. All students were not successful. This was my first time to talk about this. I probably should have stopped them earlier so that we could all discuss their results, but the time got away from me. They worked the whole time, but they didn’t all get to the point where they could generalize their results.

3. I just didn’t feel like I had another whole day to spend on this. And so I’m not sure that we have had closure. I think if I asked them, they could tell me that there are rigid motions that map one triangle onto the other, but I don’t know that they could all tell me an order of rigid motions that would always work.

And so the journey continues ….

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1 Comment

Posted by on October 29, 2012 in Angles & Triangles, Geometry, Rigid Motions

 

One response to “Proving Congruence Theorems

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