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The two triangles on the left are congruent. The third is similar to them. The last triangle is neither congruent nor similar to any of the others. Congruence permits alteration of some properties, such as location and orientation, but leaves others unchanged, like distances and angles. The unchanged properties are called invariants.
Particular definitions of congruence can be made for groups, rings, vector spaces, modules, semigroups, lattices, and so forth. The common theme is that a congruence is an equivalence relation on an algebraic object that is compatible with the algebraic structure, in the sense that the operations are well-defined on the equivalence classes.
In mathematics, the triple bar is sometimes used as a symbol of identity or an equivalence relation (although not the only one; other common choices include ~ and ≈). [7] [8] Particularly, in geometry, it may be used either to show that two figures are congruent or that they are identical. [9]
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for ...
In mathematics, a congruence is an equivalence relation on the integers. The following sections list important or interesting prime-related congruences. The following sections list important or interesting prime-related congruences.
Congruence (general relativity), in general relativity, a congruence in a four-dimensional Lorentzian manifold that is interpreted physically as a model of spacetime or a bundle of world lines; Zeller's congruence, an algorithm to calculate the day of the week for any date; Scissors congruence, related to Hilbert's third problem
Note that the "AAA" is a mnemonic: each one of the three A's refers to an "angle". Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent. [3] There are several criteria each of which is necessary and sufficient for two triangles to be similar:
if p ≡ 3 (mod 8), then p is not a congruent number, but 2 p is a congruent number. if p ≡ 5 (mod 8), then p is a congruent number. if p ≡ 7 (mod 8), then p and 2 p are congruent numbers. It is also known that in each of the congruence classes 5, 6, 7 (mod 8), for any given k there are infinitely many square-free congruent numbers with k ...