![]() ![]() The similarity theorem may be reformulated as the AAA (angle-angle-angle) similarity theorem: two triangles have their corresponding angles equal if and only if their corresponding sides are proportional. The fundamental theorem of similarity states that a line segment splits two sides of a triangle into proportional segments if and only if the segment is parallel to the triangle’s third side. Segments of lengths a, b, c, and d are said to be proportional if a: b = c: d (read, a is to b as c is to d in older notation a: b:: c: d). Shape is intimately related to the notion of proportion, as ancient Egyptian artisans observed long ago. Similar figures, on the other hand, have the same shape but may differ in size. (For an illustrated exposition of the proof, see Sidebar: The Bridge of Asses.) The Bridge of Asses opens the way to various theorems on the congruence of triangles.Īs indicated above, congruent figures have the same shape and size. Euclid’s proof of this theorem was once called Pons Asinorum (“ Bridge of Asses”), supposedly because mediocre students could not proceed across it to the farther reaches of geometry. The first very useful theorem derived from the axioms is the basic symmetry property of isosceles triangles-i.e., that two sides of a triangle are equal if and only if the angles opposite them are equal. ![]() Following this, there are corresponding angle-side-angle (ASA) and side-side-side (SSS) theorems. The first such theorem is the side-angle-side (SAS) theorem: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent. Two triangles are said to be congruent if one can be exactly superimposed on the other by a rigid motion, and the congruence theorems specify the conditions under which this can occur.
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