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Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid: It has two adjacent angles that are supplementary, that is, they add up to 180 degrees. The angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal.
Any non-self-crossing quadrilateral with exactly one axis of symmetry must be either an isosceles trapezoid or a kite. [5] However, if crossings are allowed, the set of symmetric quadrilaterals must be expanded to include also the crossed isosceles trapezoids, crossed quadrilaterals in which the crossed sides are of equal length and the other sides are parallel, and the antiparallelograms ...
A kite with three 108° angles and one 36° angle forms the convex hull of the lute of Pythagoras, a fractal made of nested pentagrams. [22] The four sides of this kite lie on four of the sides of a regular pentagon, with a golden triangle glued onto the fifth side. [16] Part of an aperiodic tiling with prototiles made from eight kites
Trapezium, plural trapezia, may refer to: Trapezium, in British and other forms of English, a trapezoid, a quadrilateral that has exactly one pair of parallel sides; Trapezium, in North American English, an irregular quadrilateral with no sides parallel; Trapezium (bone), a bone in the hand; Trapezium Cluster, a group of stars in the Orion Nebula
The formula for the area of a trapezoid can be simplified using Pitot's theorem to get a formula for the area of a tangential trapezoid. If the bases have lengths a, b, and any one of the other two sides has length c, then the area K is given by the formula [2] (This formula can be used only in cases where the bases are parallel.)
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. One pair of opposite sides is parallel and equal in length. Adjacent angles are supplementary. Each diagonal divides the quadrilateral into two congruent ...
Finally, the angles CME and FMA are the same. Hence, AFM is an isosceles triangle, and thus the sides AF and FM are equal. The proof that FD = FM goes similarly: the angles FDM, BCM, BME and DMF are all equal, so DFM is an isosceles triangle, so FD = FM. It follows that AF = FD, as the theorem claims.
A cyclic polygon with an even number of sides has all angles equal if and only if the alternate sides are equal (that is, sides 1, 3, 5, … are equal, and sides 2, 4, 6, … are equal). [11] A cyclic pentagon with rational sides and area is known as a Robbins pentagon. In all known cases, its diagonals also have rational lengths, though ...