Lesson 7-4: Overlapping Triangles (Day 74)

Today I subbed in a high school English class. As it turns out, this teacher has only three classes -- one junior class and two senior classes. So of course there's no "Day in the Life" today.

The juniors have a quick half-page writing assignment on the Romantic Era. Then they continue watching The Crucible. That's right -- it seems as if every time I cover a junior English class, the students are watching The Crucible. Also, I wonder what The Crucible has to do with the Romantic Era, since the play is neither set nor written in that era.

Meanwhile, the seniors are answering questions about Siddhartha. They've just finished reading the chapter where the title character meets Kamala. Indeed, the students are supposed to take the quiz today, but because of the regular teacher's absence, instead the quiz will be given tomorrow.

Fourth period English 12 is the best class of the day, since so many more students complete two chapters' worth of questions here than in fifth period. Third period English 11 is in between.

Lesson 7-4 of the U of Chicago text is called "Overlapping Triangles." In this lesson students will write more sophisticated proofs.

This is what I wrote last year about today's lesson:

Lesson 7-4 of the U of Chicago text covers more proofs. These proofs are trickier, since they involve overlapping triangles.

Because the triangles overlap, it may appear that, in Question 4, we need to show that Triangle SUA is isosceles. But as it turns out, we actually don't need to show this to complete the proof.

The bonus question is somewhat interesting here. It asks whether there is a valid congruence theorem for quadrilaterals, SSASS. Last year I tried to solve it, but got confused, so I want to take the time to set the record straight.

Let's convert this into a multi-day activity -- especially since all the proofs are on the first page, with review on the second page, so we can just replace that second page with this new activity. I added in the Exploration Question from Lesson 7-3, on SSSS, just to put SSASS into perspective -- and it also reflects how I should have set up the projects back at the old charter school, with a simpler question on the first day before the main question on the second day.

As it turns out, SSASS is not a valid congruence theorem for quadrilaterals. A counterexample for SSASS is closely related to a counterexample to SSA for triangles -- we start with two triangles that satisfy SSA yet aren't congruent -- one of these will be acute, the other obtuse. Then we reflect each triangle over the congruent side that is adjacent to the congruent angle. Each triangle becomes a kite -- as the original triangles aren't congruent, the kites can't be congruent either, yet they satisfy SSASS (with the A twice as large as the A of the original triangles).

I tried to prove SSASS by dividing each quadrilaterals into two triangles, then using SAS on the first pair and SSS on the second. The problem with this is that that division doesn't produce two triangles unless the quadrilateral is known to be convex. With our two kites, notice that the acute triangle becomes a convex kite, while the obtuse triangle becomes a nonconvex (or concave) kite -- which is also known as a dart. If both quadrilaterals are already known to be convex, then my proof of SSASS is valid.

One congruence theorem that actually is valid for quadrilaterals is SASAS. We can prove it the same way that we proved SAS for triangles. We put one of the sides -- in this case the congruent side that's between the other two congruent sides -- on the reflecting line. Then we can prove that the two far vertices are on the correct ray, the correct distance from the two vertices on the reflecting line -- this works whether the quadrilateral is convex or concave. We can also prove SASAS by dividing the quadrilateral into triangles. There are separate cases for convex and concave quadrilaterals, but all of them work out.

Other congruence theorems for quadrilaterals are ASASA and AASAS. Another congruence theorem, AAASS, is also valid, but it's similar to AAS in that there's a trivial proof based on the angle-sum that reduces it to ASASA (just as AAS reduces to ASA), only in Euclidean geometry. A neutral proof of AAASS exists, but it's more complicated.

Oh, and if students finish the activity early, it's possible to ask them to solve the Exploration question for Lesson 7-5: Explore this conjecture. If, in quadrilaterals ABCD and EFGH, angles A, C, E, and G are right angles, AB = EF, and BC = FG, then the quadrilaterals are congruent. It turns out that this conjecture is false -- again, a counterexample is a pair of kites, one a square, the other not a square. Then again, if you're tired of giving false conjectures, you can give them one of the valid ones instead like SASAS or ASASA.