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What proof of the converse of the Pythagorean Theorem will Grade 8 students be able to understand? The common one requires students to understand the SSS congruence criteria, and I don’t believe they have learned that yet.
Well, one way to do it would be to indeed prove the SSS criterion. The triangle congruence criteria are quite fun to do with transformations, and remember that in Grade 8 we are not expecting very formal proofs. But another way to go would be something like the following proof, which essentially pulls in the necessary piece of the SSS proof.
Given a triangle whose three side lengths $a$, $b$, and $c$ satisfy $a^2+b^2= c^2$, construct a right triangle with legs of length $a$ and $b$. Then, by Pythagoras’ theorem, its hypotenuse has length $c$. Now put the two triangles together along their sides of length $c$, flipping one of the triangles if necessary to get a kite shaped figure (because of the corresponding sides of lengths $a$ and $b$). Drawing the other diagonal you can see the kite as two isosceles triangles matched along their bases. The base angles of the isosceles triangles are equal, so the opposite angles of the kite that you just joined are also equal. But one of those is the right angle of your right triangle. Therefore the original triangle also has a right angle, and you have proved the converse.
(Probably should have tried to draw a figure for this.)
You need to know that the base angles of an isosceles triangle are equal. That has a very nice proof using reflection about the angle bisector of the vertex.
Bill,
I expect that you mean ‘very informal proofs’. But your description of how to go about it is right on. Of course, the diagram you describe is just the one used to prove SSS – join along a pair of sides of equal measure, construct a diagonal, and then invoke the Isosceles Triangle Theorem. As you say, the best way to prove the ITT is to construct the angle bisector and then ask what happens when we fold over it. Students get that right away.
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