equal in measure to , and is equal in measure to , then this implies that is equal in measure to and the triangles are similar. All the corresponding sides are proportional: . This is equivalent to saying that one triangle (or its mirror image) is an enlargement of the other. Any two pairs of sides are proportional, and the angles included between these sides are congruent: and is equal in measure to . This is known as the SAS similarity criterion. 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. Symbolically, we write the similarity and dissimilarity of two triangles and as follows: There are several elementary results concerning similar triangles in Euclidean geometry: Any two equilateral triangles are similar. Two triangles, both similar to a third triangle, are similar to each other (transitivity of similarity of triangles). Corresponding altitudes of similar triangles have the same ratio as the corresponding sides. Two right triangles are similar if the hypotenuse and one other side have lengths in the same ratio. There are several equivalent conditions in this case, such as the right triangles having an acute angle of the same measure, or having the lengths of the legs (sides) being in the same proportion. Given a triangle and a line segment one can, with a ruler and compass, find a point such that . The statement that point satisfying this condition exists is Wallis's postulate and is logically equivalent to Euclid's parallel postulate. In hyperbolic geometry (where Wallis's postulate is false) similar triangles are congruent. In the axiomatic treatment of Euclidean geometry given by George David Birkhoff (see Birkhoff's axioms) the SAS similarity criterion given above was used to replace both Euclid's parallel postulate and the SAS axiom which enabled the dramatic shortening of Hilbert's axioms. Similar triangles provide the basis for many synthetic (without the use of coordinates) proofs in Euclidean geometry. Among the elementary results that can be proved this way are: the angle bisector theorem, the geometric mean theorem, Ceva's theorem, Menelaus's theorem and the Pythagorean theorem. Similar triangles also provide the foundations for right triangle trigonometry. Other similar polygons The concept of similarity extends to polygons with more than three sides. Given any two similar polygons, corresponding sides taken in the same sequence (even if clockwise for one polygon and