## If Triangle Proofs are the aspirin, what is the headache?

Prove that two triangles are congruent. Sometimes seen as the “first real proofs” of a Geometry course (but they should probably not be the first proofs done, and the types usually done aren’t much of proofs– but let that be for now).

How do you get students to feel the need to prove? How do we give them a reasonable headache alleviated by the learning target’s aspirin?

The headache-aspirin idea can be illustrated like this: add the first 100 integers. 1+2+3+…+100. The headache is that it seems like a lot of tedious work, but using Gauss’ arithmetic series folding idea, we fold the long sum onto itself and add pairs inward: 1+100, 2+99, … getting 50 sums each equal to 101, thus finding the sum of integers from 1 to 100 is 5050. This legend is oft repeated as Gauss’ clever way to get around a teacher’s tedious punishment: notice how even in legend we appreciate the origin of a clever idea as a way to reduce tedium.

Back to triangle proofs. We want students to see SSS, SAS, ASA, HL (SS-rightA) as useful tools to show that two triangles are congruent. Dan Meyer made an astute comment: If proofs are the aspirin, the doubt is the headache. The congruency shortcuts are very abstract, but furthermore, a student may not have reason to believe or disbelieve whatever you are going to say on the subject of congruecy. Two threads emerge here:

(1) *why* do we want to show triangles congruent? How can we make a student *doubt* that triangles may be congruent?

Triangle congruence is our tool of two dimensional congruence. So we need to have students care about congruence in general. If your students are philosophical (and many are) then studying the idea of sameness/congruency has some grounding there. But all students are going to want something concrete as well.

(2) why do we use these little 3 part shortcuts?

The accepted shortcuts are an efficient result. Nothing wrong with showing triangles are congruent by matching all six parts: SASASA — nothing wrong except efficiency. Arriving at the result of the triangle congruency shortcuts can be done by students themselves.

So here’s my scratchwork for a lesson idea: work from the opposite case, can we make a student doubt that different triangles exist? **Building Different Triangles** We infuse doubt by assigning students to find non-congruent triangles. **When they run up against their sandbox’s boundary– the conditions that cause some triangles to be automatically congruent– they can all of a sudden doubt that triangles can always be made differently.**

Students in groups are given the task to build different triangles than their groupmates, given a set of matching triangle parts. The given parts are either physical manipulatives or digital (advantages with each one)

0. Do we agree that if three sides and three angles are given to us, we can make only one triangle? Why or why not?

0.5 Think ahead, what if we are given only 5 parts, can we make two different triangles? What about given other numbers of parts, like 2? Can we make different triangles then? How many different triangles?

1. Given A=30, AB=3, BC=2. Make as many different triangles as possible. Related concept: SAS, Law of Cosines eventually. Geogebra interactive.

2. Given AB=3, BC=2. Make as many different triangles as possible. Related concept: triangle inequality. Geogebra interactive.

3. Given AB=5, BC = 4, CA = 2. Make as many different triangles as possible. Related concept: converse of Pythagorean theorem. Geogebra Interactive.

4. Given A=40, B=30, AB=5. Make as many different triangles as possible.

5. Given A=40, B=30, BC=5. Make as many different triangles as possible.

6. Given A=53, B=57, BC=5.35, AC=5.1. Make as many different triangles as possible.

Be careful with the chosen values. Sometimes (especially the first one) you want the students to arrive at different triangles. Save the “we can’t!”/”they’re all the same” moment (SSS (3) / AASS (6) ) for later in the progression.

We also don’t need to restrict ourselves to tech-based manipulatives. Straws and string work great for the SSS case.

What kind of conjectures would students make when exposed to this kind of task? Many will apply the triangle angle sum, or have it solidified in their mind: meaning they can conjecture *we at most need to be given two angles*. Hey, that’s a proposition we can prove! That’s one **students** can prove, or at least justify. And its not arbitrary or needless: its something that is a significant step towards reducing tedium of triangle differentiation. If we know two angles, we know the third.

And how about conjecturing and proving statements like: to show two triangles are congruent, at least three parts must be known. (note, hypotenuse-leg gives three parts: two sides and the right angle) Or, at least one side must be known. These are within the grasp of proof by the students because they will have had the experience of seeing the counter examples and of building intuition about what *forces *the known congruency shortcuts to be what they are.

Thoughts? extensions?

It seems to me that there is a more general question of, when are two objects necessarily the same, and when could they be different?

The classic triangle congruence proofs boil down to sides and angles. Why? Are there other ways to characterize a triangle? Well, of course; but why do we choose sides and angles as our go-tos? I will leave that question open-ended.

Separately, here is a possible extension (in the “branching off” sense): Let’s modify our thinking about sides for a moment. A side length is the distance between adjacent vertices. Let us consider instead the length from a vertex to an interior point. As an explicit example, suppose you have a triangle with one interior point marked. If the distances from the three vertices to the interior point are 3, 4, and 5, does this correspond to a unique triangle?

No: And I think fiddling with straws or pictures or Geogebra can provide convincing evidence that there are multiple triangles with such a property.

But wait! We have modified our thinking about sides, yet done nothing with angles. Okay: How about if all the interior angles of the triangle are equal? Yes, in fairness, this has an effect on the sides: The triangle must be equilateral. Still, we have arrived at a problem of a rather different nature.

Re-stating: Suppose we have an equilateral triangle, and there is an interior point with distances 3, 4, and 5 to the three vertices. Does this characterize a unique triangle?

Lest this all seem like dilly-dallying, the answer is: yes, it characterizes a unique triangle. Why? How does one prove this with certitude?

[One can fiddle with the problem further, asking about a, b, and c rather than 3, 4, and 5; or perhaps by modifying our thinking around angles…]

I don’t know if such an “extension” is of interest; I leave it here on the off-chance that it could be.

That’s how I’ve taught it, but with construction using compass and protractor. “How many different triangles can be drawn from this information?” Include sides in the same ratio and you can investigate similarity too.

MQ, yeah including the angles can take you down the road to similarity. Great to either hint that way early– or come back around to this activity as you begin that unit.

I’m interested in your interior point approach. I think that would be a nice unusual way to institute some doubt in students minds– in other words, a proposition that is not completely obvious while still being within the reach of the novice student. I’m wondering about the choice of the numbers. Do you use arbitrary (distinct) numbers first? Do students make a jump to conjecture about having two/three of the values equal?

Andrew, I like compass and protractor/s.e. stuff, but I’ve found using the tools can bog students down. (On the other hand, the geogebra interactives I posted above might go too quickly) What do you do to help strike a balance for a productive pace?

“I’m interested in your interior point approach. I think that would be a nice unusual way to institute some doubt in students minds– in other words, a proposition that is not completely obvious while still being within the reach of the novice student. I’m wondering about the choice of the numbers. Do you use arbitrary (distinct) numbers first? Do students make a jump to conjecture about having two/three of the values equal?”

The choice of numbers here (viz., 3, 4, 5) is to allow a certain sort of reflection that incorporates the Pythagorean Theorem. There are a few ways to broach it, but here is one realizable path:

Begin by questioning whether you can draw “different” triangles once you know 3 side-lengths. This connects well with the traditional body of work on SSS congruence. (You could also ask the same question with angles, and let it serve as a reminder with respect to what AAA shows.) I might then try to elicit other examples of what you need to establish two triangles are congruent. Likely students will be a bit stuck at first, and eventually try some of the side and angle stuff that they would’ve already gone through. But somebody might have another idea (or be scaffolded outside of trad’l examples).

One nice example is area and isosceles: This doesn’t give you quite enough information to determine a unique triangle. (A point worth investigating!) But area and equilateral does, and this suggests that knowing a triangle is equilateral really tells you a lot (even without knowing its side-lengths). You might segue into what you can figure out about equilateral triangles. I’m a bit inclined to let the ideas develop organically, but if you want to lead them towards the aforementioned interior point problem:

Does an equilateral triangle and its side uniquely determine the triangle? (Yes.) What about an equilateral triangle and the distance to its center (e.g., centroid)? Yes. From there, you might muse about the vertex to centroid distance. In an equilateral triangle, it is 1/3 the height. Meaning, if you add up the distance from each vertex (there are 3 of them!) to the centroid, then you get the height. This is a pretty cool property — but a property of what? The centroid? Try a different point in the equilateral triangle. (Geogebra could be helpful here.) What happens if you drag around the interior point and sum the three vertex-point distances? (Sorry if this ruins a surprise but) It is constant, i.e., always the height; this result is known as Viviani’s Theorem.

https://en.wikipedia.org/wiki/Viviani%27s_theorem

Now you are comfortably in the realm of asking questions about an interior point and its distance to the vertices of (at least) an equilateral triangle. And so you can wend your way back to the earlier problem suggested, if you wish (and if there is interest).

As a final, related remark: I have found that interior points can lead to some interesting mathematical discussions. An outline of another task that I recently used is provided here:

http://blog.mrmeyer.com/2015/if-graphing-linear-inequalities-are-aspirin-then-how-do-you-create-the-headache/#comment-2408543