The traditional approach to congruence is fraught with difficulties, leaving triangle congruence properties based on assumptions instead of proofs. A transformation approach overcomes those difficulties, opening a path for students to confront and solve the challenge of proving SSS as a theorem.

Technology Tips:

Web Sketchpad (WSP) is our dynamic geometry platform. Each WSP activity provides only a small number of tools, simplifying the user interface and encouraging productive exploration.

When you tap to activate a tool, the entire construction appears, ready for you to place its glowing objects where you want them, or to match them to existing objects in the sketch.

On each page of this activity you’ll try to construct multiple reflect functions to produce the exact same result as other transformations. The functions you'll try to construct this way include a rotate function (page 2), a translate function (page 3), a glide reflection (page 4), and a dilate function (page 5). (Hint: all of these except one are possible.)

Using the traditional approach to congruence (defining congruence in terms of equal measures), the distinction between two segments having equal lengths and two segments being congruent was meaningless. Students found the distinction silly: "Of course two segments of equal length are congruent, because equal length and congruent mean the same thing!"

With a transformation approach, the distinction becomes meaningful. To claim that two segments are congruent, you have to do more than just measure them: you have to show that one can be transformed so its image is superposed on the other.

Our ultimate goal will be to prove the SSS Theorem, but as with so many difficult problems, it's very useful to do a simpler problem first. If we can prove the Segment Congruence Theorem, you'll find that proving the SSS Theorem is a piece of cake.

Remember that congruent figures are figures that can be superposed using isometries,transformations that preserve distance and that any other isometryrotation, translation, or glide reflection can be produced by a composition of reflections.

Because this is a proof task, you must be able to explain why each step of your construction works.

On page 1 you will develop your strategy for superposing an image of `overline(AB)` on `overline(CD)`. The given fact of the proof `(AB = CD)` has already been applied, so you just need to figure out how to construct the transformations.

How many perpendicular bisectors, and how many reflections, do you need to superpose your segments?

On page 2 the given fact has not yet been applied. You need to figure out when, and why, you need to use the given fact to make your construction work correctly.

As you do your construction, when is it most convincing, or most logical, to apply the given fact?

On page 3, as you do each step of the construction, describe that step and explain why it works the way it does.

Organize your page 3 descriptions and explanations in a clear logical way (paragraph, two-column format, or other format specified by your teacher).

By describing what you did, and explaining why it works, you have proved the Segment Congruence Theorem.

Remember that congruent figures are figures that can be superposed using isometries, and that any isometry can be produced by a composition of reflections.

On page 1 you will develop your strategy for superposing an image of `triABC` on `triDEF`. The givens of the proof `(AB = DE`, `BC = EF`, and `CA = FD)` have already been applied, so you just need to figure out how to construct the transformations.

How many perpendicular bisectors, and how many reflections, do you need to superpose your triangles?

On page 2 the givens have not yet been applied. You need to figure out when, and why, you need to use the three given facts to make your construction work correctly.

As you do your construction, when is it most convincing, or most logical, to apply each given fact?

On page 3, as you do each step of the construction, describe that step and explain why it works the way it does.

Organize your page 3 descriptions and explanations in a clear logical way (paragraph, two-column format, or other format specified by your teacher).

By describing what you did, and explaining why it works, you have proved the Side-Side-Side Triangle Congruence Theorem.