Class-IX Unit-8 Quadilaterals

Download Solution (Quadilaterals)

what is a quadrilateral

definition of quadrilateral:

A quadrilateral is the union of four line-segments determined by four distinct coplanar points of which no three are collinear and the line-segments intersect only at end points.
For ABCD to be quadrilateral, following condition are required
a) The four points A, B, C and D must be distinct and co-planar. 
b) No three of points A, B, C and D are co-linear. 
c) Line segments i.e. AB, BC, CD, DA intersect at their end points only.

1)AB, BC, CD and DA are the four sides.

2)Points A, B, C and D are the four vertices.
3)?ABC, ?BCD, ?CDA and ?DAC are the four angles.
4) AB and CD are the opposite sides.
5) Angle A and C are the opposite angles.
6) AB and BC are the adjacent sides.
7) Angle A and B are the adjacent angles
Angle Property of Quadrilateral
1) Sum of all the interior angles is 360⁰
2) Sum of all the exterior angles is 360⁰

Types of quadrilateral

A quadrilateral is a four-sided polygon with four angles. There are many kinds of quadrilaterals. The five most common types are the parallelogram, the rectangle, the square, the trapezoid, and the rhombus. 

Parallelogram

A quadrilateral which has both pairs of opposite sides parallel is called a parallelogram.
Its properties are:
(a) The opposite sides of a parallelogram are equal. 
(b) The opposite angles of a parallelogram are equal. 
(c) The diagonals of a parallelogram bisect each other. 
(d) The diagonal of a parallelogram divide into two congruent triangles

A quadrilateral is said to a parallelogram if
Opposite sides are equal OR Opposite angles are equal OR Diagonal bisects each other OR A pair of opposite are parallel and equal

Trapezium

A quadrilateral which has one pair of opposite sides parallel is called a trapezium.

Rhombus

Rhombus is a parallelogram in which any pair of adjacent sides is equal.

Properties of a rhombus:
(a)All sides of a rhombus are equal
(b)The opposite angles of a rhombus are equal
(c)The diagonals of a rhombus bisect each other at right angles.

Rectangle

A parallelogram which has one of its angles a right angle is called a rectangle.

Properties of a rectangle are:
(a)The opposite sides of a rectangle are equal
(b) Each angle of a rectangle is a right-angle.
(c) The diagonals of a rectangle are equal.
(d) The diagonals of a rectangle bisect each other.

Square

A quadrilateral, all of whose sides are equal and all of whose angles are right angles.

Properties of square are:
(a)All the sides of a square are equal.
(b) Each of the angles measures 90�.
(c) The diagonals of a square bisect each other at right angles.
(d) The diagonals of a square are equal.

All the quadrilaterals can be shown in Venn diagram like this

We can divide the entire set of quadilateral in three major parts

1) general quadrilaterals
2)parallelograms
3) trapezoids.
Some Other observation from this
a) A square is always a parallelogram Similary a rectangle is always a parallelogram
b) A square is always a rectangle,rhombus
c) A rhombus can be square.
d) A rectangle has four right angles.
e) A rectangle is not always a rhombus
f) A Trapezium is not a parallelogram

Mid-point Theorem for Triangles

Theorem-I

The line segment joining the mid points of the two sides of the triangle is parallel to the third side

Theorem-II

A line drawn through mid point of one side of a triangle and parallel to another side bisect the third side of the triangle

How to solve the angle Problem in Quadilateral

1) We will be given problem where three angles of quadilateral known or two angles are known with other two angles having some relationship
2) Case1 : Three angles are known,How to find the fourth angle

As per angle property of quadilateral
Sum of all the interior angles is 360⁰

So
A+B+C+D=360
If A,B,C are known we can find D easily

3) case 2:two angles are known with other two angles having some relationship

A and B are known and 2C-D= 50

Now again we know that
A+B+C+D=360
Then C+D=

We can find the values of C and D from below two equations
2C-D= 50
C+D=

Example 1:The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral
Solution:
Angles are 3x,5x,9x,13x
So
3x+5x+9x+13x=360
30x=360
x=12

So angles are
36, 60,108, 156

Additional Topics on Quadilateral

1) Quadilateral meams Quad( Four ) ,Lateral means sides
2) A quadrilateral is also sometimes called Quadrangle ("four angles"), so it sounds like "triangle"
or Tetragon ("four and polygon"), so it sounds like "pentagon", "hexagon", etc.
3) Quadilaterals can be classified in terms of parallel pair of lines also
No Pair of Parallel lines: General quadilaterals
One pair of parallel lines: Trapezium
Two pair of parallel lines : square,parallelogram,rectangle,rhombus

All sides congurent: Square,rhombus

One pair of sides are congurent:Isococeles Trapezium
Two pair of sides are congurent: Rectangle,parallelogram,Kite