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From Wikipedia
In Euclidean plane geometry, a quadrilateral is a polygon with four sides (or 'edges') and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5sided), hexagon (6sided) and so on. The word quadrilateral is made of the words quad (meaning "four") and lateral (meaning "of sides").
Quadrilaterals are simple (not selfintersecting) or complex (selfintersecting), also called crossed. Simple quadrilaterals are either convex or concave.
The interior angles of a simple quadrilateral add up to 360 degrees of arc. In a crossed quadrilateral, the interior angles on either side of the crossing add up to 720Â°.
All convex quadrilaterals tile the plane by repeated rotation around the midpoints of their edges.
Convex quadrilaterals  parallelograms
A parallelogram is a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of equal length; that opposite angles are equal; or that the diagonals bisect each other. Parallelograms also include the square, rectangle, rhombus and rhomboid.
 Rhombus or rhomb: all four sides are of equal length. Equivalent conditions are that opposite sides are parallel and opposite angles are equal, or that the diagonals perpendicularly bisect each other. An informal description is "a pushedover square" (including a square).
 Rhomboid: a parallelogram in which adjacent sides are of unequal lengths and angles are oblique (not right angles). Informally: "a pushedover rectangle with no right angles."
 Rectangle: all four angles are right angles. An equivalent condition is that the diagonals bisect each other and are equal in length. Informally: "a box or oblong" (including a square).
 Square (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are right angles. An equivalent condition is that opposite sides are parallel (a square is a parallelogram), that the diagonals perpendicularly bisect each other, and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle.
 Oblong: a term sometimes used to denote a rectangle which has unequal adjacent sides (i.e. a rectangle that is not a square).
A shape that is both a rhombus (four equal sides) and a rectangle (four equal angles) is a square (four equal sides and four equal angles).
Convex quadrilaterals  other
 Kite: two pairs of adjacent sides are of equal length. This implies that one diagonal divides the kite into congruent triangles, and so the angles between the two pairs of equal sides are equal in measure. It also implies that the diagonals are perpendicular. (It is common, especially in the discussions on plane tessellations, to refer to the concave quadrilateral with these properties as a dart or arrowhead, with term kite being restricted to the convex shape.)
 Orthodiagonal quadrilateral: the diagonals cross at right angles.
 Trapezium (British English) or trapezoid (NAm.): one pair of opposite sides are parallel.
 Isosceles trapezium (Brit.) or isosceles trapezoid (NAm.): one pair of opposite sides are parallel and the base angles are equal in measure. This implies that the other two sides are of equal length, and that the diagonals are of equal length. An alternative definition is: "a quadrilateral with an axis of symmetry bisecting one pair of opposite sides".
 Trapezium (NAm.): no sides are parallel. (In British English this would be called an irregular quadrilateral, and was once called a trapezoid.)
 Cyclic quadrilateral: the four vertices lie on a circumscribed circle. A quadrilateral is cyclic if and only if opposite angles sum to 180Â°.
 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.
Area of a convex quadrilateral
There are various general formulas for the area of a convex quadrilateral.
The area of a quadrilateral ABCD can be calculated using vectors. Let vectors AC and BD form the diagonals from A to C and from B to D. The area of the quadrilateral is then
 \frac{1}{2} {AC}\times{BD},
which is the magnitude of the cross product of vectors AC and BD. In twodimensional Euclidean space, expressing vector AC as a free vector in Cartesian space equal to (x_{1},y_{1}) and BD as (x_{2},y_{2}), this can be rewritten as:
From Yahoo Answers
Answers:Yes, all parallelograms are quadrilaterals. Yes, all rectangles are parallelograms. Yes, all squares are trapezoids. Yes, all trapezoids are quadrilaterals. (A trapezoid, is by definition, any quadrilateral with at least one pair of parallel sides). http://planetmath.org/encyclopedia/Trapezoid.html *From The Words of Mathematics by Steven Schwartzman (1994, Mathematical Association of America): trapezoid (noun); trapezoidal (adjective); trapezium, plural trapezia (noun): The Greek word trapeza "table" was composed of tetra "four" and the IndoEuropean root ped "foot." A Greek table must have had four feet (= legs). The suffix oid (q.v.) means "looking like," so that a trapezoid is a figure that looks like a table (at least in somebody's imagination). Some Americans define a trapezoid as a quadrilateral with at least one pair of parallel sides. Under that definition, a parallelogram is a special kind of trapezoid. For other Americans, however, a trapezoid is a quadrilateral with one and only one pair of parallel sides, in which case a parallelogram is not a trapezoid. The situation is further confused by the fact that in Europe a trapezoid is defined as a quadrilateral with no sides equal. Even more confusing is the existence of the similar word trapezium, which in American usage means "a quadrilateral with no sides equal," but which in European usage is a synonym of what Americans call a trapezoid. Apparently to cut down on the confusion, trapezium is not used in American textbooks. The trapeze used in a circus is also related, since a trapeze has or must once have had four "sides": two ropes, the bar at the bottom, and a support bar at the top. http://mathforum.org/dr.math/faq/formulas/faq.quad.html One last argument for the trapezoid being a quadrilateral with AT LEAST one pair of parallel sides: http://mathforum.org/library/drmath/view/54901.html
Answers:Yes. It is any four sided shape. There are specific quadrilaterals that have there own names, e.g. square, retangle, rhombus, parallelagram, trapezium etc. These all come under the family of quadrilaterals.
Answers:CONGRUENT = the same BISECT = split in half, exactly 1/2 ADJACENT = the two sides that are next to each other (in this case they are touching, like the two lines that connect to form a corner)
Answers:1. False. A rectangle is a parallelogram, so the figure might be a rectangle. 2. True. 3. True. 4. True. 5. True. 6. False. A rectangle is a rhombus. 7. False. A rectangle is a rhombus, so the figure might be a rectangle.
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