Similarity (geometry)

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Two geometrical objects are called similar if they both have the same shape. More precisely, one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. One can be obtained from the other by uniformly "stretching" the same amount on all directions, possibly with additional rotation and reflection, i.e., both have the same shape, or one has the same shape as the mirror image of the other. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, nor are hyperbolas all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar.

This article assumes that a scaling, enlargement or stretch can have a scale factor of 1, so that all congruent shapes are also similar, but some school text books specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different to qualify as similar.

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Similar triangles

To understand the concept of similarity of triangles, one must think of two different concepts. On the one hand there is the concept of shape and on the other hand there is the concept of scale.

If you were to draw a map, you would probably try to preserve the shape of what you are mapping, while you would make your picture at a unit rate that is in proportion to the original size or value.

In particular, similar triangles are triangles that have the same shape and are up to scale of one another. For a triangle, the shape is determined by its angles, so the statement that two triangles have the same shape simply means that there is a correspondence between angles that preserve their measures.

Formally speaking, we say that two triangles \triangle ABC and \triangle DEF are similar if either of the following conditions holds:

1. Corresponding sides have lengths in the same ratio:

2.  \angle BAC is equal in measure to \angle EDF , and \angle ABC is equal in measure to \angle DEF. This also implies that \angle ACB is equal in measure to \angle DFE.

When two triangles \triangle ABC and \triangle DEF are similar, we write

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