Lesson #2 - Augsburg College

Geometry: A Shapely Approach to Math
The following activity is aimed at helping the students discover these answers through trial and error.
First, have the group cut out the pre-made triangles and Information Slips. To begin the activity, have students
pair off. Then have one student in each pair pick a triangle while the other takes an Information Slip.
The student with the Information Slip should then ask his or her partner for pieces of information.
* These can be any combination of angles and sides, but be specific. For example: “Give me
angle b, angle c, side A, and side C,” rather than “Give me two angles and two sidese.” You may
want to emphasize the importance of keeping track of the angle and side names. (We named them
for a reason!)
The student with the triangle should then take the appropriate measurements and tell his or her partner,
who can then write them down on the Information Slip. Now the task is for the student with the Information
Slip to try to create (draw) a congruent triangle with the specified information. Remember: sometimes this
will be possible and sometimes it will not. It all depends on how much and which pieces of information a
student has about a triangle. It may be worthwhile to have both students in each pair take a triangle and an
Information Slip. This way, both can be trying to construct each other’s triangle at the same time, and the
group will have twice as many trials.
The following discussion questions may be appropriate to ask during the activity, or they may be more
appropriate for after the activity. Use your judgment. The goal of asking these questions is to guide discovery
and help students summarize their findings. Here are the questions:
• Is it necessary to have all six pieces of information (i.e.: all three angle and all three side measure-
ments) in order to construct a congruent triangle?
* Why do you think this is?
• Was anybody able to do it with five pieces of information? Four? Three? Two?
* Why do you think this is?
• Of the people who tried using three pieces of information, were you always able to successfully
construct a congruent triangle? (Hopefully someone answers no!)
* What happens if the three pieces of information are all angle measures? If nobody has
tried this, have the group try it and see what happens. (It is impossible to construct a congru-
ent triangle knowing only the angle measures. These triangles will have the same shape but
not necessarily be the same size. These are called similar triangles. There are pic-
tures of examples of these at the end of the lesson.)
* What happens if the three pieces of information are all side lengths? Again, if nobody has
tried this, do so. (This, perhaps surprisingly, will produce a congruent triangle! Given three
lengths, there is only one way for them to fit together and make a triangle. Cool!)
• What combinations of three pieces of information do not work?
* For a specific combination that doesn’t work, have one student who tried this combination
explain in his or her own words why this is.
• What combinations of pieces of information do work? (Below are some definitions that help us talk
about these combinations. Refer to the pictures at the end of the lesson for examples of each.)
* S-A-S stands for “side-angle-side” and means that if you tell me the lengths of two sides
and the measure of the angle between them, I can construct a triangle congruent to yours.
* A-S-A stands for “angle-side-angle” and means that the measures of two angles and the
length of the side in between them is also sufficient information.
* S-S-S, or “side-side-side” also works. That is, if you tell me the lengths of all three sides
(but don’t tell me any of the angle measures!), I can construct a congruent triangle.
• Why was it important to name the sides and the angles?
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