Congruence

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K-12: Materials at high school level.



We would like to be able to say if two shapes are "the same" or not. And the same can mean slightly different things. For instance, we can say that two shapes are the same if they have exactly the same size and shape. Below you see some shapes that we would consider the same. The two modern windmills on the left are exactly the smae shape and size. In the picture in the center we see as many as 9 different windmills and they look like they may very well all be the same shape and size. On the right you see some pre-fabricated houses. The three shown here look like all three would have exactly the same shape and size. In mathematics we would say the shapes are congruent.

557px-Guantanamo Bay windmills.jpg 800px-Overview windmills Kinderdijk.jpg New Karavilas.jpg
Two modern windmills Dutch windmills Houses


The notion of similar indicates that they will have the same shape, but they will be different size. Some examples are shown. If we ignore the decoration on the larger elephant's trunk, the top elephant is just a scaled down (smaller) version of the larger one. The other picture shows a pair of Somali Wild Asses. The young animal is just a smaller version of the larger one. In mathematics we would NOT call the animals below congruent. We would call them similar, and similar shapes are discussed in another section.

ParkBlocksElephantPortland.jpg Somali Wild Ass pair.jpg
Statue of two elephants in a park in Portland, OR A Somali Wild Ass and it's foal


We can introduce two notions of "sameness" for triangles:

Congruent triangles
Two triangles are congruent if they have the same side lengths and angle measures. They are the same size and shape.
Similar triangles
Two triangles are similar if they have the same angle measures. They are the same shape, but may be different sizes. We will talk more about similar shapes in another section.