Triangle Congruence Explorations with Wikki Sticks

This is a great method of exploring the triangle congruence shortcuts.  I tried doing this with uncooked spaghetti a number of years ago, but it was a big pain.  Trying to break the spaghetti into “congruent” segments was very difficult, and the stupid things rolled around on the desk and never stayed where you put them.  One of my daughters brought home a project with Wikki Sticks/Bendaroos, and the rest is history.  Bendaroos are a wax-coated string.  They cut to length easily, and are slightly sticky (think post-it notes), so they stay where you put them.  They are available cheaply from Amazon.com, or probably from other school supply sites as well.

Students are paired up.  Each pair gets a snack bag of supplies.  Each snack bag contains:

• two red sticks about 9 cm long
• two blue sticks about 7 cm long
• two black sticks about 12 cm long
• several (at least four) white (or some other color) sticks that are full length (app. 15 cm)
• two 40 degree angles
• two 80 degree angles
• two 60 degree angles

Divide the materials between the two students so that each student has one of each item (two of the whites).

Students explore the congruence shortcuts by trying to build different triangles given certain criteria.  For example, if you’re exploring SSS, each student should use a black, blue and red stick to build any triangle they can.  They then compare triangles to decide if they are congruent or if they are different, or if it’s even possible to create triangles that are different given that the three sides are the same:

There’s few rules:  the colored sticks must be straight and connect at the ends.  The white sticks are “magic” and can connect anywhere (not necessarily the endpoints).  Students use the white ones to form sides of any length.

Since the resulting triangles are always congruent, SSS is enough to prove triangle congruence.

Another example:  SAS.  Each student must use a blue stick, a red stick, and the 60 degree angle between them (each triangle has SAS in common).  Then see if it is possible to build different triangles:

Since only one triangle results from this initial setup, SAS is enough to prove triangle congruence.

SSA (or ASS) is the tricky one, of course.  For this setup, tell students to use the red stick, the blue stick, and the 40 degree angle.  Tell them the 40 degree angle must be at the end of the red stick, but not between the blue and the red sticks.  Some students should be able to come up with the two different triangles possible with this scenario:

Which means that SSA doesn’t work as a congruence theorem (SSA is not sufficient to prove triangle congruence)

All of the triangle congruence possibilities can be explored with this kit.  Have fun!