![]() One can view the Euclidean plane as the complex plane, that is, as a 2-dimensional space over the reals. The similarities group S is itself a subgroup of the affine group, so every similarity is an affine transformation. The direct similitudes form a normal subgroup of S and the Euclidean group E( n) of isometries also forms a normal subgroup. The similarities of Euclidean space form a group under the operation of composition called the similarities group S. Similarities preserve angles but do not necessarily preserve orientation, direct similitudes preserve orientation and opposite similitudes change it. Similarities preserve planes, lines, perpendicularity, parallelism, midpoints, inequalities between distances and line segments. Where A ∈ O n(ℝ) is an n × n orthogonal matrix and t ∈ ℝ n is a translation vector. Symbolically, we write the similarity and dissimilarity of two triangles △ ABC and △ A ′B ′C ′ as follows: A B C ∼ A ′ B ′ C ′ The "SAS" is a mnemonic: each one of the two S's refers to a "side" the A refers to an "angle" between the two sides. ![]() This is known as the SAS similarity criterion. ![]()
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