Chapter 4

Congruent Triangles

Much of what we can learn about geometry derives from a study of congruent triangles. A pair of triangles are congruent if they are the same size and the same shape, that is, if all the sides and angles of one triangle are equal to all the sides and angles of the other triangle.

In this chapter, you will learn more about what congruent triangles are, and how we can be sure they are congruent. By the end of the chapter, you will have some Postulates that you may use to verify that a given pair of triangles are congruent. They are called Postulates rather than Theorems because, although we may be convinced that they are reasonable and true, we have not proved that they are true. They are assumptions that we make based on our experiments with triangles. We use these postulates as the basis of most of the proofs about the properties of geometric figures. In the next chapter, we will get into the meat of the matter: why would we want to prove a pair of triangles congruent, and what further information can we gain from this?

A very interesting activity involving congruent triangles and other geometric shapes is called a Tangram. Tangrams were invented over a century ago, in China, and have fascinated people ever since! A Tangram is a puzzle, somewhat like a jigsaw puzzle, that is made of congruent triangles and a few other geometric shapes.

Tangram puzzles have been made from paper, wood, ceramic tiles, and recently, on the computer. In the Geometer's Sketchpad activities you will have a chance to try a Tangram puzzle.

When you begin a Tangram puzzle, the shapes are arranged like this:

How many congruent triangles can you find in this figure? What kinds of symmetry do these triangles have? When you flip or rotate the figures while solving a Tangram puzzle, you should keep the symmetries in mind, as it will save you unnecessary steps! There would be no purpose in reflecting the red triangle across a vertical line, for example, as the triangle would look exactly the same!

The challenge of a Tangram puzzle is to move the 7 figures given above (which are called tiles): slide them, flip them over, rotate them, and combine them to fit a particular shape. The shapes can be abstract, or look somewhat like people, animals, even letters of the alphabet. Here is an example of a Tangram: the challenge, and the solution:

(This clever Tangram challenge was designed by the people at the website below)

You can, of course, design your own challenge, and ask others to solve them!

There are some very interesting web sites related to Tangrams. To try some colorful and interesting tangram puzzles on the computer, click on the link below and download the demo software:

http://wuarchive.wustl.edu/systems/mac/info-mac/game/tangrams-demo-102.hqx

Another fascinating website is based on a Chinese Tangram puzzle invented by Mr. Tong, Ye'geng in July 1893, Yi Zhi Tu©. The patterns of the puzzle were drawn by his five sons including Mr. Tong, Da'nian. More than a hundred years later, in June 1996, his great-grandson, Harry Yu Tong, designed Tongram®, which is a JavaTM applet that allows people all over the world to play Yi Zhi Tu© on Internet. To try this tangram puzzle, click on the link below:

http://math.rutgers.edu/~ttu/tongram/index.html

Yet another interesting website is:

http://www.creativeimaginations.net

When you get there click on Picture This-Tangrams. I hope you enjoy your visit!

Project

After solving the tangram puzzles in the Geometer's Sketchpad activities, design your own tangram challenge, using GSP. Then create a second file with the solution to your challenge.

Go to Chapter 5 Triangle Properties

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