The SSS Theorem: Donâ€™t Assume It, Prove It!

1 Introduction

Scott Steketee, 21st Century Partnership for STEM Education, stek@21pstem.org

Donovan Hayes, Hill-Freedman World Academy, Philadelphia, dhayes@philasd.org

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2 Benefits of a Transformation Approach

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3 The Problem with Congruence

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.

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4 A Line with Five Identities

There are five ways you can think about this line:

- Itâ€™s the locus of all the fixed points of the function that takes `x` to `r(x)`.
- Itâ€™s the mirror line that defines the reflect function that takes `x` to `r(x)`.
- Itâ€™s the locus of points that are the same distance from `x` as they are from `r(x)`.
- Itâ€™s the perpendicular bisector of the segment between `x` and `r(x)`.
- Itâ€™s the line constructed using the intersections of the two circles defined by points `x` and `r(x)`.

Use the tools to construct and/or trace all of these in turn, each on its own page. The tools include Hide/Show buttons you can use to hide or show each way of constructing or tracing this line.

In your own words, describe something that's interesting or important about each of these fiveâ€”five lines that are all the same, but also different, each in its own way!

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**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.

5 Summary and Discussion

Setting the Stage for SSS

To set the stage for our constructive proof of the SSS Theorem, we’ve:

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Big Mathematical Ideas

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6 Mystery Transformations

This lesson is located here: https://geometricfunctions.org/fc/unit2/mystery-transformations/

7 Mirror, Mirror

This lesson is located here: https://geometricfunctions.org/fc/unit2/mirror-mirror/

8 Prove the Segment Congruence Theorem

To prepare for our conclusion, we'll first solve a simpler problem: Proving the Segment Congruence Theorem.

This lesson is located here: https://geometricfunctions.org/fc/unit3/prove-sss/#ProvetheSegment

9 Prove the SSS Theorem

We now find ourselves having done the hard part: figuring out a strategy to use in superposing one geometric shape on another. By superposing several points, the rest of the shape must follow!

This lesson is located here: https://geometricfunctions.org/fc/unit3/prove-sss/#ProvetheSSS

10 Summary and Discussion

- Adopted a transformation approach to solve important problems with congruence
- Connected transformations to function concepts (independent/dependent, rate of change, notation)
- Used our bodies as variables following a function rule
- Used patty paper to experiment with superposition as the criterion for congruence
- Used WSP to establish the important role of perpendicular bisectors and the Perpendicular Bisector Theorem
- Used WSP to experiment with superposing two congruent triangles
- Used WSP, with history icons, to prove the Segment Congruence Theorem
- Left the SSS Theorem as an exercise for participants

11 Resources

WSP Tool Library: geometricfunctions.org/fc/tools/library

WSP Sketch Viewer: geometricfunctions.org/fc/tools/viewer

Sine of the Times Blog: http://sineofthetimes.org

Scott: stek@geometricfunctions.org

Donovan: dhayes@philasd.org

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:**

Record every major revision, in reverse chronological order

14 Feb 2018: Simplified this template.

24 Jan 2018: Created this lesson template.

24 Jan 2018: Created this lesson template.