The SSS Theorem

In this lesson you’ll prove an important geometric fact: The Side-Side-Side Theorem (SSS).

In Activity 1 you will practice by constructing and proving a simpler theorem.

In Activity 2, you will construct the superposition of SSS triangles, and you will explain how to use the steps of your construction to prove the SSS Theorem.

In Activity 3, you will consider some other cases different from the triangles in Activity 2, and figure out how to make your proof from Activity 2 apply to these cases as well.

1 Prove the Segment Congruence Theorem

**Construct and prove the Segment Congruence Theorem:** Given `AB = CD`, the two segments `overline(AB)` and `overline(CD)` are congruent.

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

**You can rephrase the task:** Given `AB = CD`, use reflections to superpose an image of `overline(AB)` on `overline(CD)`.

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.

2 Prove the SSS Theorem

**Construct and prove the SSS Theorem:** Given `triABC` and `triDEF` with `AB = DE`, `BC = EF`, and `CA = FD`, the two triangles are congruent.

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

**You can rephrase the task:** Given `triABC` and `triDEF` with `AB = DE`, `BC = EF`, and `CA = FD`, use reflections to superpose an image of `triABC` on `triDEF`.

**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.

3 The SSS Theorem: Other Cases

**Other Cases for the SSS Theorem:** In the previous activity, you used three examples to construct and prove the SSS Theorem. In all three of the examples `triABC` and `triDEF` were opposite-handed: one clockwise and one counter-clockwise. But what if the two triangles are similarly handed? How can you construct/prove the SSS theorem for similarly-handed triangles?

In this activity you'll do just that, but instead of creating an entirely new proof you'll find a easier way, by using the proof you've already created.

Why is it impossible to put two similarly-handed triangles into superposition using the same reflections you used for opposite-handed triangles?

What's the simplest way you can transform `triABC` to make its image opposite-handed compared to `triDEF`?

**On page 1** transform `triABC` so that the image is opposite-handed to `triDEF`. Then use the construction from the previous activity to superpose an image of `triABC` onto `triDEF`.

**Page 2** contains a different example.

As you do this construction, describe it and explain why it works. You can simplify your explanation by referring to the proof you created for opposite-handed triangles.

**Page 3** contains another special case for you to consider.

What can you do with this case to make it easy to prove?

5 The Fine Print

**Requirements:**

These activities require web access using a browser that supports HTML5 and JavaScript. (That means almost any current browser.) No purchase is required, and there’s no advertising anywhere.

**Release Information**

Release: 2015Q4Update2, Semantic Version: 4.5.1-alpha, Build Number: 1026.6-wsp-widgets, Build Stamp: stek-macbook-pro/20180605163618

Web Sketchpad Copyright © 2017 KCP Technologies, a McGraw-Hill Education Company.

“You may freely use Web Sketchpad in your own Web pages, provided you do so for non-commercial use only” [per kcpt.github.io].

“You may freely use Web Sketchpad in your own Web pages, provided you do so for non-commercial use only” [per kcpt.github.io].

License: Creative Commons CC-BY-NC-SA 4.0

**Update History:**

17 Apr 2018: Now includes construction history, more animations, closer connection to proof, etc.

24 Oct 2017: Created this page.

24 Oct 2017: Created this page.