Chapter 8 - Quadrilaterals, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 PDF Download

INTRODUCTION
We have studied in detail about the properties of a triangle. We also know that the triangle is a figure obtained
by joining three non collinear points in pair. In this chapter we shall discuss about four non-collinear points such that no three of them are collinear.

1. QUADRILATERALS
We know that the figure obtained on joining three non-collinear points in pairs is a triangle. If we mark four
points and join them in some order, then there are three possibilities for the figure obtained:
(i) If all the points are collinear (in the same line), we obtain a line segment.

(ii) If three out of four points are collinear, we get a triangle.

(iii) If no three points out of four are collinear, we obtain a closed figure with four sides.
                         

Each of the figure obtained by joining four points in order is called a quadrilateral. (quad means four and lateral for sides).

2. CONSTITUENTS OF A QUADRILATERAL
A quadrilateral has four sides, four angles and four vertices.
                        

In quadrilateral ABCD, AB, BC, CD and DA are the four sides; A, B, C and D are the four vertices and A,
B, C and D are the four angles formed at the vertices.
If we join the opposite vertices A to C and B to D, then AC and BD are the two diagonals of the quadrilateral ABCD.

3. QUADRILATERALS IN PRACTICAL LIFE
We find so many objects around us which are of the shape of a quadrilateral the floor, walls, ceiling, windows
of our classroom, the blackboard, each face of the duster, each page of our mathematics book, the top of our
study table, etc. Some of these are given below.

4. SOME RELATED TERMS TO QUADRILATERALS

In a quadrilateral ABCD, we have

(i) VERTICES :- The points A, B, C and D are called the vertices of quadrilateral ABCD.

(ii) SIDES :- The line segments AB, BC, CD and DA are called the sides of quadrilateral ABCD.

(iii) DIAGONALS :- The line segments AC and BD are called the diagonals of quadrilateral ABCD.

(iv) ADJACENT SIDES :- The sides of a quadrilateral are said to be adjacent sides if they have a common end point.
Here, in the above figure, (AB,BC),(BC,CD),(CD,DA)and(DA,AB) are four pairs of adjacent sides or consecutive sides of quadrilateral ABCD.

(v) OPPOSITE SIDES :- Two sides of a quadrilateral are said to be opposite sides if they have no common end point.
Here, in the above figure, (AB, DC) and (BC, AD) are two pairs of opposite sides of quadrilateral ABCD.

(vi) CONSECUTIVE ANGLES :- Two angles of a quadrilateral are said to be consecutive angles if they have a common arm.
Here, in the above figure, (A,B), (B,C), (C, D) and (D, A) are four pairs of consecutive angles.

(vii) OPPOSITE ANGLES :- Two angles of a quadrilateral are said to be opposite angles if they have no common arm.
Here, in the given figure, (A, C) and (B, D) are two pairs of opposite angles of quadrilateral ABCD.
 

5. TYPES OF QUADRILATERALS

(i) PARALLELOGRAM :- A quadrilateral in which Dboth pair of opposite Csides are parallel is called a parallelogram.

In figure, ABCD is a quadrilateral in which AB || DC, BC || AD.
 quadrilateral ABCD is a parallelogram.

(ii) RHOMBUS :- A parallelogram whose all sides are equal is called rhombus.

In figure, ABCD is a parallelogram in which AB = BC = CD = DA, AB | DC and BC || AD .
 parallelogram ABCD is a rhombus.

(iii) RECTANGLE :- A parallelogram whose each angle is equal to 90°, is called a rectangle.
In figure, ABCD is a parallelogram in which
A = B = C = D = 90° ,
AB || DC and BC || AD.
 Parallelogram ABCD is a rectangle.

(iv) SQUARE :- A rectangle in which a pair of adjacent sides are equal is said to be a square.
In figure, ABCD is a rectangle in which A = B = C = D = 90° ,
AB = BC, BC = CD, CD = DA, DA= AB.
i.e., AB = BC = CD = DA .

 rectangle ABCD is a square.

(v) TRAPEZIUM :- A quadrilateral in which exactly one pair of opposite sides is parallel, is called a trapezium.
A In figure, ABCD is a quadrilateral in which ABDC.
 ABCD is trapezium.

(vi) ISOSCELES TRAPEZIUM :- A trapezium whose non-parallel sides are equal is called an isosceles trapezium. 

In figure, ABCD is a trapezium in which AB || DC and BC = AD .
 trapezium ABCD is isosceles trapezium.

(vii) KITE :- A quadrilateral in which two pairs of adjacent sides are equal is called a kite.
In figure, ABCD is a quadrilateral in which AB = AD and BC = CD.

 quadrilateral ABCD is a kite.

6. ANGLE SUM PROPERTY OF A QUADRILATERAL

THEOREM-I : The sum of the four angles of a quadrilateral is 360°.
Given : A quadrilateral ABCD.
To Prove :A + B + C + D = 360°

Construction : Join AC. 2 1
 Proof :

Hence, proved.

Ex.1 Three angles of a quadrilateral measure 56°, 100° and 88°. Find the measure of the fourth angle.

Sol. Let the measure of the fourth angle be x°.
 56° + 100° + 88° + x° = 360°                       [Sum of all the angles of quadrileteral is 360°]
⇒ 244+x=360
⇒ x = 360 – 244 = 116
Hence, the measure of the fourth angle is 116°.

Ex.2 In figure, ABCD is a trapezium in which AB || CD. If D = 45° and C = 75°, find A and B.
 Sol. We have, AB || CD and AD is a transversal.
so, A + D = 180° [Interior angles on the same side of the transversal]

Ex. 3 In the given figure, sides AB and CD of the quadrilateral ABCD are produced. Find the value of x.

Sol.

Ex.4 In the given figure, ABCD is a quadrilateral in which AE and BE are the angle bisectors of A and B. Prove that C + D = 2AEB.

Sol. Given : ABCD is a quadrilateral in which AE and BE are the

Hence proved.

Ex.5 In figure, ABCD is a quadrilateral in which AB = AD and BC = CD . Prove that
 (i) AC bisects A and C

(ii) BE = DE.

Sol. Given : ABCD is a quadrilateral in which AB = AD and BC = CD
To Prove :

(i) AC bisects A and C

(ii) BE = DE.
Proof :

Hence proved

Ex.6 In a quadrilateral ABCD, AO and BO are the bisectors of A and B respectively. Prove that

Sol. Given : In a quadrilateal ABCD, AO and BO are the bisectors of A and B



Hence proved

Ex.7 In fig. bisectors of B and D of quadrilateral ABCD meet CD and AB produced at P and
 Q respectively. Prove that 

Sol. Given : bisectors of B and D of quadrilateral ABCD meet CD and AB produced at P and Q

To Prove : 

Proof :

                     
[Sum of the angles of a quadrilateral equals is 360°]

Hence proved

Ex.8 In quadrilateral ABCD B = 90°, C – D = 60° and A – C – D = 10°. Find A, C and D.

Sol. A + B + C + D = 360° (Sum of the four angles of a quadrilateral is 360°)

Ex.9 In quadrilateral ABCD
A + C = 140°, A : C = 1 : 3 and B : D = 5 : 6. Find the A, B, C and D.

Sol. 

Hence, A = 35°, B = 100°, C = 105° and D = 120°

Ex.10 In figure, ABCD is a parallelogram in which D = 72°. Find A, B and C.

Sol. We have D = 72°
But B = D [Opposite angles of the parallelogram]
B = 72°

Now, AB || CD and AD and BC are two transversals.
So, A + D = 180° [Interior angles on the same side of the transversal AD]