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  2. Angular defect - Wikipedia

    en.wikipedia.org/wiki/Angular_defect

    The defect of any of the vertices of a regular dodecahedron (in which three regular pentagons meet at each vertex) is 36°, or π/5 radians, or 1/10 of a circle. Each of the angles measures 108°; three of these meet at each vertex, so the defect is 360° − (108° + 108° + 108°) = 36°.

  3. Antiparallelogram - Wikipedia

    en.wikipedia.org/wiki/Antiparallelogram

    Because of this symmetry, it has two pairs of equal angles and two pairs of equal sides. [2] The four midpoints of its sides lie on a line perpendicular to the axis of symmetry; that is, for this kind of quadrilateral, the Varignon parallelogram is a degenerate quadrilateral of area zero, consisting of four collinear points.

  4. Parallelogram - Wikipedia

    en.wikipedia.org/wiki/Parallelogram

    A simple (non-self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements is true: [2] [3] Two pairs of opposite sides are parallel (by definition). Two pairs of opposite sides are equal in length. Two pairs of opposite angles are equal in measure. The diagonals bisect each other.

  5. List of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/List_of_trigonometric...

    In particular, in these two identities an asymmetry appears that is not seen in the case of sums of finitely many angles: in each product, there are only finitely many sine factors but there are cofinitely many cosine factors. Terms with infinitely many sine factors would necessarily be equal to zero.

  6. Pythagorean triple - Wikipedia

    en.wikipedia.org/wiki/Pythagorean_triple

    If a is replaced with the fraction m/n in the sequence, the result is equal to the 'standard' triple generator (2mn, m 2 − n 2, m 2 + n 2) after rescaling. It follows that every triple has a corresponding rational a value which can be used to generate a similar triangle (one with the same three angles and with sides in the same proportions as ...

  7. Proofs of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/Proofs_of_trigonometric...

    The proofs given in this article use these definitions, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles , see Trigonometric functions . Other definitions, and therefore other proofs are based on the Taylor series of sine and cosine , or on the differential equation f ″ + f = 0 ...

  8. Angles between flats - Wikipedia

    en.wikipedia.org/wiki/Angles_between_flats

    For any pair of flats in a Euclidean space of arbitrary dimension one can define a set of mutual angles which are invariant under isometric transformation of the Euclidean space. If the flats do not intersect, their shortest distance is one more invariant. [1] These angles are called canonical [2] or principal. [3]

  9. List of triangle inequalities - Wikipedia

    en.wikipedia.org/wiki/List_of_triangle_inequalities

    The parameters most commonly appearing in triangle inequalities are: the side lengths a, b, and c;; the semiperimeter s = (a + b + c) / 2 (half the perimeter p);; the angle measures A, B, and C of the angles of the vertices opposite the respective sides a, b, and c (with the vertices denoted with the same symbols as their angle measures);