Build That Triangle
Teacher Notes
This pair of related tasks is designed to develop the concept of triangle congruence based on a valid subset of corresponding parts of congruent triangles—leading to the discovery of the ASA (and AAS),
SSS, and SAS triangle congruence postulates. They will fit well right after the introduction of the idea that congruent triangles have six corresponding parts and, perhaps, after students have had some experience with the notational bookkeeping of writing congruence statements so that the corresponding parts “line up” correctly in those statements.
Objectives:
In Game A, the focus will be on the concept of being able to use a minimum number of corresponding parts to be able to guarantee congruence. While students should be gaining some ideas about whether specific combinations always work or only sometimes, do not discuss these ideas until after Game B.
The conceptual objective of Game A will be that students will understand that:

4 congruent corresponding parts will always be enough to guarantee that two triangles are congruent

2 congruent corresponding parts will never be enough

3 congruent corresponding parts will sometimes be enough
Furthermore, they will be able to justify why they believe these things.
In Game B, the focus turns to discovering exactly which sets of 3 congruent corresponding parts will
always guarantee that two triangles are congruent and which do not.
The conceptual objective of Game B will be that students will understand that:

SSS combination always guarantees congruence, but AAA does not.

Two angles and a side always guarantees congruence (AAS and ASA) no matter whether the side is between the angles (included) or not.

Two sides and an angle sometimes works, but only if the angle is between the two sides (SAS). If the angle is not between the angle (SSA), congruence is not guaranteed.
Launching the Tasks:
It’s a good idea to be prepared to demonstrate the exploratory process using some think-aloud modeling.
For Game A, choose any of the envelopes and choose one part, then illustrate how it is easy to build two non-congruent triangles that have that one corresponding part congruent. Then choose a second part and illustrate how it is still possible to build two non-congruent triangles that have both corresponding parts congruent. Remind them about how to check if two triangles are congruent by superimposition
(rotating, reflecting, and translating so that one is on top of the other).

Stop after demonstrating 2 parts. It doesn’t give anything away, but it should be sufficient for students to access the task fully. Emphasize that you aren’t finished with a combination until after you’ve: a) found one triangle that works AND b) tried your best to find a second, different (non-congruent) triangle using the same combination. Students stopping their explorations after completing a) will be the major issue as they start the task.
If the students participated in Game A previously, then they will have a good idea of how the exploration process should work and the launch can be very brief. Have them glance at Part 2 of Game B to see the objectives that they will be aiming for while they explore.
Preparing the Materials:
While you can use paper for the corresponding parts envelopes, it’s best if you can use overhead transparency acetates instead. Not only will the transparencies withstand heavy student handling for a longer period of time, students will find it easier to explore if they can layer the parts on top of each other and see them all clearly.
The templates for the envelope sets are at the back of this task. Notice that there are small circled numbers on each piece that will help identify which set a piece belongs with should they get separated by students during the exploration. Be sure that these marks stay with the part as you
separate them.
Each envelope for Game A should have all six parts from the same triangle, cut into separate pieces.
There are six different triangle templates included. Each group will need quick access (though it doesn’t necessarily have to be simultaneous access) to three different envelopes during the explore phase of
Game A.
Each envelope for Game B should have only three parts—a different combination of parts from a single triangle. There are 20 different combinations from the triangle included on the Game B template.
Again, groups should have access to three different sets during the Explore phase.
Students should also have access to compasses, protractors, rulers and patty paper if they wish to use them. The instructions for both tasks say “build” instead of “construct” or “draw” in order to let the students create an accurate triangle any way they wish.

The Setup: Each group will get an envelope containing six pieces of information about a triangle—three side lengths and three angle measures—on separate notes. Take out one note at a time from your envelope until you have just enough information to build only one triangle with exactly those measurements. You can use any construction or drawing tools or techniques you want: compass, ruler, protractor, paper folding, and so on. Obviously, all six pieces can be used to form a unique triangle. The challenge of the game is to find out if there are combinations of fewer pieces that will determine a unique triangle as well.
For each trial: a.
Find a valid triangle using the parts you have already pulled. b.
Find a different triangle—not congruent to the first one—that uses all of the parts you have already pulled. If you believe that a second triangle is not possible with your current combination, write down the combination you have and write down why you think there’s only one triangle possible in this case.
Once you believe you have created a unique triangle:
1.
Check your solution by pulling out the remaining notes and matching them up on your triangle to make sure you made the correct triangle.
2.
Repeat the process at least 2 more times with different envelopes.
3.
Be sure that you and your group have recorded your findings (“name” of triangle, how many notes it took, which particular notes it took, any other observations or thoughts you had while working on this).

After you have worked with 3 different triangles:
1.
Make a conjecture: What is the largest number of notes you would ever have to take out of the envelope before you build a unique triangle? Give justifications for your conjecture.
2.
Make another conjecture: What is the smallest number of notes you could take out and still build one triangle that uses all the information? Will any collection of that many notes always be enough? Give justifications for your conjecture.

After the whole class shares their results and their conjectures:
1.
Summarize what you now believe to be true conjectures. Is there a “magic number,” where fewer pieces are never enough and more pieces are always enough? Is the “magic number” of pieces always “magic”, always creating a unique triangle?
Exit Slip A
One student playing this game, pulled out two notes, 𝑚∠𝐴 = 60° and ∠𝐵 = 90° . a.
Build two different triangles that fit this information. b.
Are more than two different triangles possible? c.
Name one additional piece of information the student could pull out next so that only one triangle is possible?

The Setup: This each envelope contains exactly 3 of 6 measurements of a triangle. Each group tries to make a unique triangle using only that set of clues. Call your set of clues
“good” if it lets you make exactly one triangle. Call the set “bad” if it doesn’t fit any triangle or if it fits more than one. Be sure to repeat the game for three different sets.
1.
Record your work carefully. For each set, be sure to list the parts that were included, using “S” for a side and “A” for an angle.
2.
Decide whether the set of clues are “good” or “bad”. If it’s good, show the unique triangle that the set builds. If it’s “bad,” show why.

Based on the results of your group’s work:
1.
Write a conjecture about which combinations are always good? (We will call these conjectures the triangle congruence postulates)
2.
Write a conjecture about which combinations are always bad?
3.
Write a conjecture about which combinations may be good or bad?

Triangle congruence postulates say that if two triangles share certain pieces of information, for example, SSS, then they are congruent. They also say that it is possible to build only one triangle from these three pieces of information. Well, isn’t there a contradiction? Are there
two triangles or only one?
When people say “Given three sides, only one triangle can be built,” they really mean that any
two triangles built from these sides (or sides congruent to them) must be congruent.
1.
Are triangles ABC and ADC congruent? If they are congruent, which congruence postulate helped you decide?
2.
Are the two triangles pictured below definitely congruent or not necessarily congruent?
If they are congruent, which congruence postulate helped you decide?

3.
In two ways, show that a diagonal divides a square into two congruent triangles. Use two different congruence postulates to show that the triangles are, indeed congruent.
4.
Find a counterexample to this statement: “AAA (all three pairs of corresponding angles have equal measure) is sufficient information to prove congruence in two triangles.”
Explain why it disproves the conjecture. Include a picture in your explanation.

5.
The picture below is a “proof without words” that two congruent pairs of corresponding sides and one congruent pair of non-included angles (SSA) is not enough information to guarantee that two triangles to be congruent. Add the words (and labels if you want) that explain the proof.

Game A Triangle Sets (all six pieces in each envelope, each piece separated from the others)
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Game B Sets (three per envelope, each piece separated from each other)
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