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  2. Corresponding sides and corresponding angles - Wikipedia

    en.wikipedia.org/wiki/Corresponding_sides_and...

    The orange and green quadrilaterals are congruent; the blue one is not congruent to them. Congruence between the orange and green ones is established in that side BC corresponds to (in this case of congruence, equals in length) JK, CD corresponds to KL, DA corresponds to LI, and AB corresponds to IJ, while angle ∠C corresponds to (equals) angle ∠K, ∠D corresponds to ∠L, ∠A ...

  3. Congruence (geometry) - Wikipedia

    en.wikipedia.org/wiki/Congruence_(geometry)

    In a Euclidean system, congruence is fundamental; it is the counterpart of equality for numbers. In analytic geometry , congruence may be defined intuitively thus: two mappings of figures onto one Cartesian coordinate system are congruent if and only if, for any two points in the first mapping, the Euclidean distance between them is equal to ...

  4. Equivalence class - Wikipedia

    en.wikipedia.org/wiki/Equivalence_class

    Congruence is an example of an equivalence relation. The leftmost two triangles are congruent, while the third and fourth triangles are not congruent to any other triangle shown here. Thus, the first two triangles are in the same equivalence class, while the third and fourth triangles are each in their own equivalence class.

  5. Hilbert's third problem - Wikipedia

    en.wikipedia.org/wiki/Hilbert's_third_problem

    The third of Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedra of equal volume , is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second?

  6. Hilbert's axioms - Wikipedia

    en.wikipedia.org/wiki/Hilbert's_axioms

    This was, in considerable part, influenced by the example Hilbert set in the Grundlagen. A 2003 effort (Meikle and Fleuriot) to formalize the Grundlagen with a computer, though, found that some of Hilbert's proofs appear to rely on diagrams and geometric intuition, and as such revealed some potential ambiguities and omissions in his definitions.

  7. Similarity (geometry) - Wikipedia

    en.wikipedia.org/wiki/Similarity_(geometry)

    Similar figures. In Euclidean geometry, two objects are similar if they have the same shape, 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 scaling (enlarging or reducing), possibly with additional translation, rotation and reflection.

  8. Triangle inequality - Wikipedia

    en.wikipedia.org/wiki/Triangle_inequality

    The top example shows a case where z is much less than the sum x + y of the other two sides, and the bottom example shows a case where the side z is only slightly less than x + y. In mathematics , the triangle inequality states that for any triangle , the sum of the lengths of any two sides must be greater than or equal to the length of the ...

  9. Synthetic geometry - Wikipedia

    en.wikipedia.org/wiki/Synthetic_geometry

    Synthetic proofs of geometric theorems make use of auxiliary constructs (such as helping lines) and concepts such as equality of sides or angles and similarity and congruence of triangles. Examples of such proofs can be found in the articles Butterfly theorem, Angle bisector theorem, Apollonius' theorem, British flag theorem, Ceva's theorem ...