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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 (5-sided), hexagon (6-sided) and so on. The word quadrilateral is made of the words quad (meaning "four") and lateral (meaning "of sides").

Quadrilaterals are simple (not self-intersecting) or complex (self-intersecting), 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.

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 pushed-over square" (including a square).
• Rhomboid: a parallelogram in which adjacent sides are of unequal lengths and angles are oblique (not right angles). Informally: "a pushed-over 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).

• 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.

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 two-dimensional Euclidean space, expressing vector AC as a free vector in Cartesian space equal to (x1,y1) and BD as (x2,y2), this can be rewritten as:

Question:1)Give the best name for the quadrilateral: a quadrilateral whose diagonals are both congruent and perpendicular 2) Fill in the blank with almost, sometimes, or never a) a rectangle that has perpendicular diagonals is _________a square b)A parallelogram that is not equilateral is __________ a rectangle

Answers:1) I don't know, but "square" isn't it. The quadrilateral with vertices (0,3), (2,0), (0,-1), (-2,0) has congruent, perpendicular diagonals--but it isn't a square. That example is a kite, but here's another example that isn't a kite: (-1,0), (0,-1), (3,0), (0,3). This one's an isosceles trapezoid. 2) a. Always: a rectangle is divided by its diagonals into four isosceles triangles, all with congruent legs. If they all have right angles at the vertex, then by SAS they are all congruent, giving us a square. b. Sometimes. A 1x2 rectangle shows that this happens at least sometimes; however, the parallelogram (0,0), (2,0), (3,1), (1,1) shows that this doesn't happen always.

Question:

Answers:A trapezoid comes to mind. Namely, an isosceles one. It's also possible to have a kite with congruent diagonals that isn't a parallelogram. Somebody else here said "a rectangle or a square". First of all, those are both examples of parallelograms. Second of all, a square IS a special case of a rectangle. "Rectangles" include any parallelograms with a right angle.