Monday, 6 May 2013

Types and Properties of Quadrilaterals (Quadrangles)

A quadrilateral is a polygon with four sides (edges) or four vertices or four corners it can also be called as a TETRAGON

 

There are different types and varieties of quadrilaterals having different properties such as different angles and sides. You can learn them here in a very easy and informative way.

After reading this tutorial you can answer these questions like

What are the properties of quadrilateral?

What is a diagonal and what are adjacent sides?

How many types of quadrilaterals are there in Geometry?

What is a cyclic quadrilateral and what are its properties?

And at last but not the least all types and properties of parallelogram, rhombus, square, rectangle, kite, oblong, trapezoid, etc.




Properties of Quadrilaterals:

-Four sides.
-Four vertices (corners).
-Interior angles sum to 360°.
-Exterior angles sum to 360°.



Important Terms Related to Quadrilateral:


Diagonals:-


Diagonals are line segments that join two opposite vertices of a quadrilateral (corners).

A quadrilateral can have maximum two diagonal










Where AD is the line segment joining two opposite vertices A and D



Adjacent Sides:-

Two sides are adjacent, if they share a common vertex.











In the quadrilateral shown above the 4 pair of adjacent sides are:


  1.  AD and DC
  2.  DC and BC
  3.  BC and AB
  4.  AB and AD 



Adjacent Angles:-

Two angles are adjacent, if they share a common side.













In the quadrilateral shown above the 4 pair of adjacent angles are:

  1. ∠1 and ∠3
  2. ∠3 and ∠4
  3. ∠2 and ∠4
  4. ∠1 and ∠2




Types of Quadrilaterals

There are special types of quadrilateral:
Types of Quadrilateral
Some types are also included in the definition of other types! For example a square, rhombus and rectangle are also parallelograms.


Concave Quadrilaterals

A quadrilateral that contains a reflex angle.

1. Parallelogram

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

Parallelogram

Properties of Parallelogram :-

  • Opposite sides are congruent(i.e., Equal in length).
  • Opposite angels are congruent(i.e., Equal in length).
  • Consecutive angles are supplementary (i.e., Sum of consecutive angle = 180°).
  • If one angle is right angle, then all angles are right angle(i.e., It is a square).
  • The diagonals of a parallelogram bisect each other to form two pairs of congruent triangles.

 

2. Rhombus (Rhomb) {Equilateral Quadrangle

A rhombus is a parallelogram all of whose all four sides are congruentRhombus


Properties of Rhombus :-

  • All four sides are congruent(i.e., Equal in length).
  • Opposite sides are parallel(i.e., It's a parallelogram).
  • Opposite angels are congruent(i.e., Equal in length).
  • Consecutive angles are supplementary (i.e., Sum of consecutive angle = 180°).
  • The diagonals of a rhombus bisect pairs of opposite angles.
  • If one angle is right angle, then all angles are right angle(i.e., It is a square).
  • Four congruent triangles are formed by diagonals.

 

3. Rhomboid

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


 

Properties of Rhomboid :-

  • All four sides are congruent(i.e., Equal in length).
  • Opposite angels are congruent(i.e., Equal in length).
  • Opposite angels are parallel(i.e., It's a parallelogram).
  • Consecutive angles are supplementary (i.e., Sum of consecutive angle = 180°).
  • If one angle is right angle, then all angles are right angle(i.e., It is a square).
  • The diagonals bisect each other to form two pairs of congruent triangles.

 

4. Rectangle [Equiangular Quadrangle]

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

Properties of Rhombus :-
  • Opposite sides that are congruent and parallel.
  • Adjacent sides are of unequal length 
  • Consecutive angles are supplementary, and
  • Diagonals bisect each other.

 

5. Square {Regular Quadrilateral}

A square is a rhombus whose all angles are right angles

Properties of Square :-

  • The diagonals bisect each other and at right angle.
  • The diagonals bisect its angles.
  • Opposite sides parallel and congruent.
  • All four angles square are congruent.
  • All four sides square are congruent.
  • The diagonals are congruent.

 

6. Oblong

A oblong is a rectangle which has unequal adjacent sides.

Properties of oblong :-

  • Two sets of parallel lines meeting at right angles. 

 

7. Kite

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

Properties of Kites:-

  • Two sets of parallel lines meeting at right angles. 
  • Two disjoint pairs of consecutive sides are congruent by definition   Note: Disjoint means that the two pairs are totally separate.
  • The diagonals are perpendicular.
  • One diagonal is the perpendicular bisector of the other diagonal
  • The main diagonal bisects a pair of opposite angles.
  • The opposite angles at the endpoints of the cross diagonal are congruent.

 

8. Trapezoid (Trapezium)

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

Properties of Kites:-

  • The bases are parallel by definition.
  • Each lower base angle is supplementary to the upper base angle on the same side.



Cyclic Quadrilaterals

A 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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