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Reference.org
Similarity (geometry)
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<p>In <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a>, two objects are similar if they have the same <a href="/facts/Shape/nww0x85d">shape</a>, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly <a href="/facts/Scaling_(geometry)/pFm6fvpu">scaling</a> (enlarging or reducing), possibly with additional <a href="/facts/Translation_(geometry)/KlpWmnfX">translation</a>, <a href="/facts/Rotation_(mathematics)/6gtkkcy0">rotation</a> and <a href="/facts/Reflection_(mathematics)/5JHWnrql">reflection</a>. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is <a href="/facts/Congruence_(geometry)/2tKJ6AhB">congruent</a> to the result of a particular uniform scaling of the other. </p> <table><tbody><tr><td></td><td></td><td></td><td></td></tr></tbody></table> <p>For example, all <a href="/facts/Circle/9fy5ggKn">circles</a> are similar to each other, all <a href="/facts/Square/JHJeMvmL">squares</a> are similar to each other, and all <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangles</a> are similar to each other. On the other hand, <a href="/facts/Ellipse/1wUDnGxY">ellipses</a> are not all similar to each other, <a href="/facts/Rectangle/mdLfIGKA">rectangles</a> are not all similar to each other, and <a href="/facts/Isosceles_triangle/rvLe3yqK">isosceles triangles</a> are not all similar to each other. 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. </p> <p>If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. </p><p>Two <a href="/facts/Congruence_(geometry)/2tKJ6AhB">congruent</a> shapes are similar, with a scale factor of 1. 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. </p>