Our students come to us knowing that the base angles of an isosceles triangle are congruent. But they don’t know why.

**CCSS-M.G-CO.C.10: **Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; **base angles of isosceles triangles are congruent**; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

We leave this proof as an exercise on the unit assessment, which is why there is so much setup in the exercise. (Didn’t you always love math textbooks that left proofs as exercises at the end of the section for you to do instead of actually working through the proofs during the section?) I do wonder what would happen if there were no hint. And if this were an exercise in class, of course I wouldn’t give a written hint. But since this is on the test, I am admittedly limiting the amount of productive struggle that I expect from my students.

How would you expect your geometry students to prove the base angles of an isosceles triangle are congruent? What misconceptions might your students have?

This year we got several of the traditional SAS (and SSS) proofs:

And we got a few of the rigid motion – reflection proofs:

I think we still need some work on these proofs … like explicitly stating that A lies on the perpendicular bisector of segment BC because it is the same distance from B as it is from C.

We got a long paragraph proof with the misconception that the two smaller triangles formed by the altitude will be 45˚-45˚-90˚, but then ending with an argument for reflecting the triangle about its altitude to show why the base angles are congruent.

We got another argument for constructing the altitude/angle bisector/perpendicular bisector/median from the vertex and using HL to show that the decomposed triangles are congruent.

And an argument I haven’t seen before using inequalities in triangles.

What we didn’t get were blank responses. Our students are learning to **make sense of problems and persevere in solving them**. Our students are learning to **look for and make use of structure**. Our students are learning to **construct viable arguments and critique the reasoning of others**. As the journey continues, our students are becoming the mathematically proficient students that we want them to become.

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