Monday, 6 May 2013

Quadrilaterals | Quadrangles | Types and Properties

A quadrilateral is a polygon with four sides (edges) or four vertices or four corners


Sum of Interior Angles of  Quadrilateral

(n-2) Χ 180             where n=the number of sides the shape has. quadrilateral has four sides so
(4-2) Χ 180
= 2 Χ 180
= 360


∠1 + ∠2   ∠3 + ∠4 = 360°


Types of Quadrilaterals


Concave Quadrilaterals

A quadrilateral that contains a reflex angle.

 1. Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel and equal sides

2. Rhombus (Rhomb) {Equilateral Quadrangle}

A rhombus is a parallelogram whose all four sides are equal

3. Rhomboid

A rhomboid is a parallelogram in which adjacent sides are of unequal lengths and oblique angles

4. Rectangle [Equiangular Quadrangle]

A rectangle is a plane figure with four straight sides and four right angles 

5. Square {Regular Quadrilateral}

A square is a rhombus whose all angles are right angles

6. Oblong

A oblong is a rectangle which has unequal adjacent sides.

7. Kite

A kite is a quadrilateral whose two pair of adjacent sides are equal

8. Trapezoid (Trapezium)

A quadrilateral with at least one pair of parallel sides is known as a trapezoid


Cyclic Quadrilaterals

cyclic quadrilateral (inscribed quadrilateral) is a quadrilateral drawn inside a circle so that its corners lie on the circumference of the circle whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. 

Properties of Cyclic Quadrilaterals

(a) the opposite angles of a cyclic quadrilateral sum to 180°
i.e. a+ c = 180°
b + d = 180°
(b) the exterior angle of a cyclic quadrilateral is equal to the interior
opposite angle

i.e. e = c
Summary
Here is a list of all the properties of quadrilaterals that we have mentioned along with the classes of the quadrilaterals that possess those properties:
PropertyQuadrilaterals
OrthodiagonalKite, Dart, Rhombus, Square
CyclicSquare, Rectangle, Isosceles Trapezoid
InscriptibleKite, Dart, Rhombus, Square
Having two parallel sidesRhombus, Square, Rectangle, Parallelogram, Trapezoid
Having two pairs of parallel sidesRhombus, Square, Rectangle, Parallelogram
EquilateralRhombus, Square
EquiangularRectangle, Square

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