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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 . In geometry , two figures or objects are congruent if they have the same shape and size , or if one has the same shape and size as the mirror image of the other.
The first four axioms establish some elementary properties of the two primitive relations. For instance, Reflexivity and Transitivity of Congruence establish that congruence is an equivalence relation over line segments. The Identity of Congruence and of Betweenness govern the trivial case when those relations are applied to nondistinct points.
An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.
A reflexive and symmetric relation is a dependency relation (if finite), and a tolerance relation if infinite. A preorder is reflexive and transitive. A congruence relation is an equivalence relation whose domain is also the underlying set for an algebraic structure, and which respects the
Reflexive relation, a relation where elements of a set are self-related; Reflexive user interface, an interface that permits its own command verbs and sometimes underlying code to be edited; Reflexive operator algebra, an operator algebra that has enough invariant subspaces to characterize it; Reflexive space, a subset of Banach spaces
In universal algebra and lattice theory, a tolerance relation on an algebraic structure is a reflexive symmetric relation that is compatible with all operations of the structure. Thus a tolerance is like a congruence , except that the assumption of transitivity is dropped. [ 1 ]
The pons asinorum in Oliver Byrne's edition of the Elements [1]. In geometry, the theorem that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum (/ ˈ p ɒ n z ˌ æ s ɪ ˈ n ɔːr ə m / PONZ ass-ih-NOR-əm), Latin for "bridge of asses", or more descriptively as the isosceles triangle theorem.
Similarly, the reflexive transitive symmetric closure of , denoted , is a congruence, meaning it is an equivalence relation (by definition) and it is also compatible with string concatenation. The relation ↔ R ∗ {\displaystyle {\overset {*}{\underset {R}{\leftrightarrow }}}} is called the Thue congruence generated by R {\displaystyle R} .