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Quadrilaterals

Trapezium Trapezoid Isosceles trapezoid Second Book of Nephi Book of Jacob Book of Enos Book of Jarom Book of Omni Words of Mormon Large Plates of Nephi Book of Mosiah Book of Alma Book of Helaman Third Nephi Fourth Nephi Book of Mormon Additions by Moroni Book of Ether Book of Moroni

Set of uniform antiprisms
Type	uniform polyhedron
Faces	2 p-gons, 2p triangles
Edges	4p
Vertices	2p
Vertex configuration	3.3.3.p
Coxeter-Dynkin diagram
Symmetry group	Dpd
Dual polyhedron	trapezohedron
Properties	convex, semi-regular vertex-transitive

In geometry, a quadrilateral is a polygon with four sides and four vertices. Sometimes, the term quadrangle is used, for etymological symmetry with triangle, and sometimes tetragon for consistence with pentagon, a five-sided shape.

Quadrilaterals are either simple (not self-intersecting) or complex (self-intersecting). Simple quadrilaterals are either convex or concave. Convex quadrilaterals are further classified as follows:

• Trapezium (American English): no sides are parallel.
• Trapezium (British English) or trapezoid (Amer.): two opposite sides are parallel.
• Isosceles trapezium (Brit.) or isosceles trapezoid (Amer.): two opposite sides are parallel, the two other sides are of equal length, and the two ends of each parallel side have equal angles. This implies that the diagonals are of equal length.
• Parallelogram: both pairs of opposite sides are parallel. This implies that opposite sides are of equal length, opposite angles are equal, and the diagonals bisect each other.
• Kite: two adjacent sides are of equal length and the other two sides also of equal length. This implies that one set of opposite angles is equal, and that one diagonal perpendicularly bisects the other. (It is common, especially in the discussions on plane tessellations, to refer to a concave kite as a dart.)
• Rhombus: all four sides are of equal length. This implies that opposite sides are parallel, opposite angles are equal, and the diagonals perpendicularly bisect each other.
• Rhomboid: a parallelogram in which adjacent sides are of unequal lengths and angles are oblique (not right angles).
• Rectangle (or Oblong): all four angles are right angles. This implies that opposite sides are parallel and of equal length, and the diagonals bisect each other and are equal in length.
• Square (regular quadrilateral): all four sides are of equal length, and each angle is a right angle. This implies that opposite sides are parallel, and that the diagonals perpendicularly bisect each other and are of equal length.
• Cyclic quadrilateral: the four vertices lie on a circumscribed circle.
• Tangential quadrilateral: the four edges are tangential to an inscribed circle. Another term for a tangential polygon is inscriptible.
• Bicentric quadrilateral: both cyclic and tangential.

One special kind of concave quadrilateral is an arrowhead, which is like a kite, but the top concaves inwards.

[edit] Taxonomy

A taxonomy of quadrilaterals is illustrated by the following graph. Lower forms are special cases of higher forms. Note that "trapezium" here is referring to the British definition (the American equivalent is a trapezoid).