![]() Note that the "AAA" is a mnemonic: each one of the three A's refers to an "angle". This is known as the AAA similarity theorem. It can be shown that two triangles having congruent angles ( equiangular triangles) are similar, that is, the corresponding sides can be proved to be proportional. Two triangles, △ ABC and △ A'B'C' are similar if and only if corresponding angles have the same measure: this implies that they are similar if and only if the lengths of corresponding sides are proportional. However, some school textbooks specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar. Two congruent shapes are similar, with a scale factor of 1. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. This is because two ellipses can have different width to height ratios, two rectangles can have different length to breadth ratios, and two isosceles triangles can have different base angles.įigures shown in the same color are similar On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other.įor example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection. In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. For other uses, see Similarity (disambiguation) and Similarity transformation (disambiguation).
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