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PRERNA PUBLIC SCHOOL MATHS ASSIGNMENT ON PARALELLOGRAM AND QUADRILATERAL BY TUSHAR M. JOSHI

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A B D C A B D C QUADRILATERAL PAPELLELOGRAM

Characterizations:

Characterizations A simple (non self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements is true: 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 are parallel and equal in length. Adjacent angles are supplementary. Each diagonal divides the quadrilateral into two congruent triangles. The sum of the squares of the sides equals the sum of the squares of the diagonals. (This is the parallelogram law.) It has rotational symmetry of order 2.

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Parallelograms In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean Parallel Postulate and neither condition can be proven without appealing to the Euclidean Parallel Postulate or one of its equivalent formulations.

Properties:

Properties Diagonals of a parallelogram bisect each other, Opposite sides of a parallelogram are parallel (by definition) and so will never intersect. The area of a parallelogram is twice the area of a triangle created by one of its diagonals. The area of a parallelogram is also equal to the magnitude of the vector cross product of two adjacent sides. Any line through the midpoint of a parallelogram bisects the area.

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Any non-degenerate affine transformation takes a parallelogram to another parallelogram. A parallelogram has rotational symmetry of order 2 (through 180°). If it also has two lines of relational symmetry then it must be a rhombus or an oblong. The perimeter of a parallelogram is 2( a + b) where a and b are the lengths of adjacent sides.

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The sum of the distances from any interior point of a parallelogram to the sides is independent of the location of the point. (This is an extension of Viviane's theorem). The converse also holds: If the sum of the distances from a point in the interior of a quadrilateral to the sides is independent of the location of the point, then the quadrilateral is a parallelogram.

Types of parallelogram:

Types of parallelogram Rhomboid – A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not right angles. Rectangle – A parallelogram with four angles of equal size Rhombus – A parallelogram with four sides of equal length. Square – A parallelogram with four sides of equal length and four angles of equal size (right angles).

Area formulas:

Area formulas The area K of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles. The area of the rectangle is Arect = (B + A) x H and the area of a single orange triangle is Atri = 1\2 A x H. Therefore, the area of the parallelogram is K = Arect – 2 x Atri = [( B + A ) x H] – ( A x H) = B x H

Quadrilaterals:

Quadrilaterals 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 origin of the word "quadrilateral" is the two Latin words quadri, a variant of four, and latus, meaning "side."

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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 (and planar) quadrilateral ABCD add up to 360 degrees of arc, that is: L A + L B + L C + L D = 360 o

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This is a special case of the n-gon interior angle sum formula ( n − 2) × 180°. In a crossed quadrilateral, the interior angles on either side of the crossing add up to 720°. All convex quadrilaterals tile the plane by repeated rotation around the midpoints of their edges.

Convex Quadrilaterals – Parallelograms:

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.

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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 diagonal 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."

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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 (four equal sides and four equal angles).

Convex quadrilaterals – other:

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. Right kite: a kite with two opposite right angles.

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Trapezoid (North American English) or Trapezium (British English): at least one pair of opposite sides are parallel. Trapezium (NAm.): no sides are parallel. (In British English this would be called an irregular quadrilateral, and was once called a trapezoid). Isosceles trapezoid (NAm.) or isosceles trapezium (Brit.): one pair of opposite sides are parallel and the other two sides are of equal length. This implies that the base angles are equal in measure, 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.

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Tangential trapezoid : a trapezoid where the four sides are tangents to an inscribed circle. Tangential quadrilateral: the four sides are tangents to an inscribed circle. A convex quadrilateral is tangential if and only if opposite sides have equal sums. Cyclic quadrilateral: the four vertices lie on a circumscribed circle. A convex quadrilateral is cyclic if and only if opposite angles sum to 180°.

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Bicentric quadrilateral : it is both tangential and cyclic. Orthodiagonal quadrilateral: the diagonals cross at right angles. Equidiagonal quadrilateral: the diagonals are of equal length. Ex-tangential quadrilateral: the four extensions of the sides are tangent to an excircle.

More quadrilaterals:

More quadrilaterals An equilic quadrilateral has two opposite equal sides that, when extended, meet at 60°. A Watt quadrilateral is a quadrilateral with a pair of opposite sides of equal length. A quadric quadrilateral is a convex quadrilateral whose four vertices all lie on the perimeter of a square. A geometric chevron (dart or arrowhead) is a concave quadrilateral with bilateral symmetry like a kite, but one interior angle is reflex.

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A self-intersecting quadrilateral is called variously a cross-quadrilateral, crossed quadrilateral, butterfly quadrilateral or bow-tie quadrilateral. A special case of crossed quadrilaterals are the anti parallelograms, crossed quadrilaterals in which (like a parallelogram) each pair of nonadjacent sides has equal length. The diagonals of a crossed or concave quadrilateral do not intersect inside the shape.

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A non-planar quadrilateral is called a skew quadrilateral. Formulas to compute its dihedral angles from the edge lengths and the angle between two adjacent edges were derived for work on the properties of molecules such as cyclobutane that contain a "puckered" ring of four atoms. See skew polygon for more. Historically the term gauche quadrilateral was also used to mean a skew quadrilateral.

Special line segments:

Special line segments The two diagonals of a convex quadrilateral are the line segments that connect opposite vertices. The two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides. The two diagonals of a convex quadrilateral are the line segments that connect opposite vertices. The two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides. They intersect at the "vertex centroid" of the quadrilateral .

Notations in metric formulas:

Notations in metric formulas In the metric formulas below, the following notations are used. A convex quadrilateral ABCD has the sides a = AB, b = BC, c = CD, and d = DA. The diagonals are p = AC and q = BD, and the angle between them is θ. The semi-perimeter s is defined as: 1\2 = (a + b + c + d).

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