RS Aggarwal Solutions: Quadrilaterals | Mathematics (Maths) Class 6 PDF Download

 Page 1


Points to Remember :
1. Quadrilateral. A simple closed figure
bounded by four line segments is called a
quadrilateral. In the adjacent figure, ABCD
is a quadrilateral. A quadrilateral ABCD
has :
(i) Four vertices, namely A, B, C and D.
(ii) Four sides, namely AB, BC, CD and DA.
(iii) Four angles, namely DAB, ABC,
BCD and CDA, to be denoted by A,
B, C and D respectively.
(iv) Two diagonals, namely AC and BD.
(i) Adjacent sides. Two sides of a
quadrilateral which have a common end
point are called its adjacent sides.
Thus AB, BC ; BC, CD ; CD, DA and
DA, AB are four pairs of adjacent sides
of the quadrilateral ABCD.
(ii) Opposite sides. Two sides of a
quadrilateral are called its opposite sides
if they do not have a common end point.
Thus AB, DC and AD, BC are two pairs
of the opposite sides of the quadrilateral
ABCD.
(iii) Adjacent angles. Two angles of a
quadrilateral having one common side are
called its adjacent angles.
Thus A, B ; B, C ; C, D and
D, A are four pairs of adjacent angles
of the quadrilateral ABCD.
(iv) Opposite angles. Two angles of a
quadrilateral which are not adjacent angles
are known as the opposite angles of the
quadrilateral.
Thus A, C, and B, D are two pairs
of opposite angles of the quadrilateral
ABCD.
2. Convex and concave quadrilaterals
(i) Convex quadrilateral. A quadrilateral in
which the measure of each angle is less
than 180º is called a convex quadrilateral.
(ii) Concave quadrilateral. A quadrilateral
in which the measure of at least one of
the angles is more than 180º is called a
concave quadrilateral.
In the above figure, PQRS is a concave
quadrilateral in which S > 180º.
In this chapter, by a quadrilateral, we
would mean a convex quadrilateral.
3. Interior and exterior of a quadrilateral
Consider a quadrilateral ABCD. It divides
the whole plane into three parts.
Page 2


Points to Remember :
1. Quadrilateral. A simple closed figure
bounded by four line segments is called a
quadrilateral. In the adjacent figure, ABCD
is a quadrilateral. A quadrilateral ABCD
has :
(i) Four vertices, namely A, B, C and D.
(ii) Four sides, namely AB, BC, CD and DA.
(iii) Four angles, namely DAB, ABC,
BCD and CDA, to be denoted by A,
B, C and D respectively.
(iv) Two diagonals, namely AC and BD.
(i) Adjacent sides. Two sides of a
quadrilateral which have a common end
point are called its adjacent sides.
Thus AB, BC ; BC, CD ; CD, DA and
DA, AB are four pairs of adjacent sides
of the quadrilateral ABCD.
(ii) Opposite sides. Two sides of a
quadrilateral are called its opposite sides
if they do not have a common end point.
Thus AB, DC and AD, BC are two pairs
of the opposite sides of the quadrilateral
ABCD.
(iii) Adjacent angles. Two angles of a
quadrilateral having one common side are
called its adjacent angles.
Thus A, B ; B, C ; C, D and
D, A are four pairs of adjacent angles
of the quadrilateral ABCD.
(iv) Opposite angles. Two angles of a
quadrilateral which are not adjacent angles
are known as the opposite angles of the
quadrilateral.
Thus A, C, and B, D are two pairs
of opposite angles of the quadrilateral
ABCD.
2. Convex and concave quadrilaterals
(i) Convex quadrilateral. A quadrilateral in
which the measure of each angle is less
than 180º is called a convex quadrilateral.
(ii) Concave quadrilateral. A quadrilateral
in which the measure of at least one of
the angles is more than 180º is called a
concave quadrilateral.
In the above figure, PQRS is a concave
quadrilateral in which S > 180º.
In this chapter, by a quadrilateral, we
would mean a convex quadrilateral.
3. Interior and exterior of a quadrilateral
Consider a quadrilateral ABCD. It divides
the whole plane into three parts.
(i) The part of the plane lying inside the
boundary ABCD is called the interior of
the quadrilateral ABCD. Each point of this
part is called an interior point of the
quadrilateral. In the given figure, the
points P, Q, R are the interior points of
the quadrilateral ABCD.
(ii) The part of the plane lying outside the
boundary ABCD is called the exterior of
the quadrilateral ABCD. Each point of this
part is called an exterior point of the
quadrilateral.
In the given figure, the points L and M
are the exterior points of the quadrilateral
ABCD.
(iii) The boundary ABCD.
Clearly, the point E lies on the quadrilateral
ABCD.
Quadrilateral region. The interior of the
quadrilateral ABCD together with its
boundary is called the quadrilateral region
ABCD.
4. Angle sum property of a quadrilateral.
The sum of the angles of a quadrilateral
is 360º.
5. Various Types of Quadrilaterals :
1. Trapezium. A quadrilateral having one
and only one pair of parallel sides is called
a trapezium.
In the adjacent figure, ABCD is a
trapezium in which AB || DC.
A trapezium is said to be an isosceles
trapezium if its nonparallel sides are equal.
In the adjoining figure, PQRS is an
isosceles trapezium in which PQ || SR
and PS = QR.
Remark. The diagonals of an isosceles
trapezium are always equal.
2. Parallelogram . A quadrilateral in which
both pairs of opposite sides are parallel is
called a parallelogram.
In the given figure, ABCD is a
parallelogram in which AB || DC and AD
|| BC. We denote it by ||gm ABCD.
Properties of a parallelogram :
(i) The opposite sides of a ||gm are equal
and parallel.
(ii) The opposite angles of a ||gm are equal.
(iii) The diagonals of a ||gm bisect each other.
Thus, in a ||gm ABCD, we have
(i) AB = DC, AD = BC and AB || DC, AD ||
BC.
(ii) BAD = BCD and ABC = ADC.
Page 3


Points to Remember :
1. Quadrilateral. A simple closed figure
bounded by four line segments is called a
quadrilateral. In the adjacent figure, ABCD
is a quadrilateral. A quadrilateral ABCD
has :
(i) Four vertices, namely A, B, C and D.
(ii) Four sides, namely AB, BC, CD and DA.
(iii) Four angles, namely DAB, ABC,
BCD and CDA, to be denoted by A,
B, C and D respectively.
(iv) Two diagonals, namely AC and BD.
(i) Adjacent sides. Two sides of a
quadrilateral which have a common end
point are called its adjacent sides.
Thus AB, BC ; BC, CD ; CD, DA and
DA, AB are four pairs of adjacent sides
of the quadrilateral ABCD.
(ii) Opposite sides. Two sides of a
quadrilateral are called its opposite sides
if they do not have a common end point.
Thus AB, DC and AD, BC are two pairs
of the opposite sides of the quadrilateral
ABCD.
(iii) Adjacent angles. Two angles of a
quadrilateral having one common side are
called its adjacent angles.
Thus A, B ; B, C ; C, D and
D, A are four pairs of adjacent angles
of the quadrilateral ABCD.
(iv) Opposite angles. Two angles of a
quadrilateral which are not adjacent angles
are known as the opposite angles of the
quadrilateral.
Thus A, C, and B, D are two pairs
of opposite angles of the quadrilateral
ABCD.
2. Convex and concave quadrilaterals
(i) Convex quadrilateral. A quadrilateral in
which the measure of each angle is less
than 180º is called a convex quadrilateral.
(ii) Concave quadrilateral. A quadrilateral
in which the measure of at least one of
the angles is more than 180º is called a
concave quadrilateral.
In the above figure, PQRS is a concave
quadrilateral in which S > 180º.
In this chapter, by a quadrilateral, we
would mean a convex quadrilateral.
3. Interior and exterior of a quadrilateral
Consider a quadrilateral ABCD. It divides
the whole plane into three parts.
(i) The part of the plane lying inside the
boundary ABCD is called the interior of
the quadrilateral ABCD. Each point of this
part is called an interior point of the
quadrilateral. In the given figure, the
points P, Q, R are the interior points of
the quadrilateral ABCD.
(ii) The part of the plane lying outside the
boundary ABCD is called the exterior of
the quadrilateral ABCD. Each point of this
part is called an exterior point of the
quadrilateral.
In the given figure, the points L and M
are the exterior points of the quadrilateral
ABCD.
(iii) The boundary ABCD.
Clearly, the point E lies on the quadrilateral
ABCD.
Quadrilateral region. The interior of the
quadrilateral ABCD together with its
boundary is called the quadrilateral region
ABCD.
4. Angle sum property of a quadrilateral.
The sum of the angles of a quadrilateral
is 360º.
5. Various Types of Quadrilaterals :
1. Trapezium. A quadrilateral having one
and only one pair of parallel sides is called
a trapezium.
In the adjacent figure, ABCD is a
trapezium in which AB || DC.
A trapezium is said to be an isosceles
trapezium if its nonparallel sides are equal.
In the adjoining figure, PQRS is an
isosceles trapezium in which PQ || SR
and PS = QR.
Remark. The diagonals of an isosceles
trapezium are always equal.
2. Parallelogram . A quadrilateral in which
both pairs of opposite sides are parallel is
called a parallelogram.
In the given figure, ABCD is a
parallelogram in which AB || DC and AD
|| BC. We denote it by ||gm ABCD.
Properties of a parallelogram :
(i) The opposite sides of a ||gm are equal
and parallel.
(ii) The opposite angles of a ||gm are equal.
(iii) The diagonals of a ||gm bisect each other.
Thus, in a ||gm ABCD, we have
(i) AB = DC, AD = BC and AB || DC, AD ||
BC.
(ii) BAD = BCD and ABC = ADC.
(iii) If the diagonals AC and BD intersect at
O, then OA = OC and OB = OD.
3. Rhombus. A parallelogram in which all
the sides are equal is called a rhombus.
In the given figure, ABCD is a rhombus
in which AB || DC, AD || BC and AB =
BC = CD = DA.
Properties of a rhombus :
(i) The opposite sides of a rhombus are
parallel.
(ii) All the sides of a rhombus are equal.
(iii) The opposite angles of a rhombus are
equal.
(iv) The diagonals of a rhombus bisect each
other at right angles.
Thus, in a rhombus ABCD, we have :
(i) AB || DC and AD || BC.
(ii) AB = BC = CD = DA.
(iii) DAB = BCD and ABC = CDA.
(iv) Let the diagonals AC and BD intersect at
O. Then, OA = OC, OB = OD and AOB
= COD = BOC = AOD = 1 right
angle.
4. Rectangle. A parallelogram in which
each angle is a right angle, is called a
rectangle.
In the given figure, ABCD is a rectangle.
In which AB || DC, AD || BC and A =
B = C = D = 90º.
Properties of a rectangle :
(i) Opposite sides of a rectangle are equal
and parallel.
(ii) Each angle of a rectangle is 90º.
(iii) Diagonals of a rectangle are equal.
Thus, in a rectangle ABCD, we have :
(i) AB = DC, AD = BC and AB || DC, AD ||
BC.
(ii) A = B = C = D = 1 right angle.
(iii) Diagonal AC = diagonal BD.
5. Square. A parallelogram in which all the
sides are equal and each angle is a right
angle, is called a square.
In the given figure, ABCD is a square in
which AB = BC = CD = DA and A =
B = C = D = 90º.
Properties of a square :
(i) The sides of a square are all equal.
(ii) Each angle of a square is 90º.
(iii) The diagonals of a square are equal and
bisect each other at right angle.
Thus, in a square ABCD, we have :
(i) AB = BC = CD = DA.
Page 4


Points to Remember :
1. Quadrilateral. A simple closed figure
bounded by four line segments is called a
quadrilateral. In the adjacent figure, ABCD
is a quadrilateral. A quadrilateral ABCD
has :
(i) Four vertices, namely A, B, C and D.
(ii) Four sides, namely AB, BC, CD and DA.
(iii) Four angles, namely DAB, ABC,
BCD and CDA, to be denoted by A,
B, C and D respectively.
(iv) Two diagonals, namely AC and BD.
(i) Adjacent sides. Two sides of a
quadrilateral which have a common end
point are called its adjacent sides.
Thus AB, BC ; BC, CD ; CD, DA and
DA, AB are four pairs of adjacent sides
of the quadrilateral ABCD.
(ii) Opposite sides. Two sides of a
quadrilateral are called its opposite sides
if they do not have a common end point.
Thus AB, DC and AD, BC are two pairs
of the opposite sides of the quadrilateral
ABCD.
(iii) Adjacent angles. Two angles of a
quadrilateral having one common side are
called its adjacent angles.
Thus A, B ; B, C ; C, D and
D, A are four pairs of adjacent angles
of the quadrilateral ABCD.
(iv) Opposite angles. Two angles of a
quadrilateral which are not adjacent angles
are known as the opposite angles of the
quadrilateral.
Thus A, C, and B, D are two pairs
of opposite angles of the quadrilateral
ABCD.
2. Convex and concave quadrilaterals
(i) Convex quadrilateral. A quadrilateral in
which the measure of each angle is less
than 180º is called a convex quadrilateral.
(ii) Concave quadrilateral. A quadrilateral
in which the measure of at least one of
the angles is more than 180º is called a
concave quadrilateral.
In the above figure, PQRS is a concave
quadrilateral in which S > 180º.
In this chapter, by a quadrilateral, we
would mean a convex quadrilateral.
3. Interior and exterior of a quadrilateral
Consider a quadrilateral ABCD. It divides
the whole plane into three parts.
(i) The part of the plane lying inside the
boundary ABCD is called the interior of
the quadrilateral ABCD. Each point of this
part is called an interior point of the
quadrilateral. In the given figure, the
points P, Q, R are the interior points of
the quadrilateral ABCD.
(ii) The part of the plane lying outside the
boundary ABCD is called the exterior of
the quadrilateral ABCD. Each point of this
part is called an exterior point of the
quadrilateral.
In the given figure, the points L and M
are the exterior points of the quadrilateral
ABCD.
(iii) The boundary ABCD.
Clearly, the point E lies on the quadrilateral
ABCD.
Quadrilateral region. The interior of the
quadrilateral ABCD together with its
boundary is called the quadrilateral region
ABCD.
4. Angle sum property of a quadrilateral.
The sum of the angles of a quadrilateral
is 360º.
5. Various Types of Quadrilaterals :
1. Trapezium. A quadrilateral having one
and only one pair of parallel sides is called
a trapezium.
In the adjacent figure, ABCD is a
trapezium in which AB || DC.
A trapezium is said to be an isosceles
trapezium if its nonparallel sides are equal.
In the adjoining figure, PQRS is an
isosceles trapezium in which PQ || SR
and PS = QR.
Remark. The diagonals of an isosceles
trapezium are always equal.
2. Parallelogram . A quadrilateral in which
both pairs of opposite sides are parallel is
called a parallelogram.
In the given figure, ABCD is a
parallelogram in which AB || DC and AD
|| BC. We denote it by ||gm ABCD.
Properties of a parallelogram :
(i) The opposite sides of a ||gm are equal
and parallel.
(ii) The opposite angles of a ||gm are equal.
(iii) The diagonals of a ||gm bisect each other.
Thus, in a ||gm ABCD, we have
(i) AB = DC, AD = BC and AB || DC, AD ||
BC.
(ii) BAD = BCD and ABC = ADC.
(iii) If the diagonals AC and BD intersect at
O, then OA = OC and OB = OD.
3. Rhombus. A parallelogram in which all
the sides are equal is called a rhombus.
In the given figure, ABCD is a rhombus
in which AB || DC, AD || BC and AB =
BC = CD = DA.
Properties of a rhombus :
(i) The opposite sides of a rhombus are
parallel.
(ii) All the sides of a rhombus are equal.
(iii) The opposite angles of a rhombus are
equal.
(iv) The diagonals of a rhombus bisect each
other at right angles.
Thus, in a rhombus ABCD, we have :
(i) AB || DC and AD || BC.
(ii) AB = BC = CD = DA.
(iii) DAB = BCD and ABC = CDA.
(iv) Let the diagonals AC and BD intersect at
O. Then, OA = OC, OB = OD and AOB
= COD = BOC = AOD = 1 right
angle.
4. Rectangle. A parallelogram in which
each angle is a right angle, is called a
rectangle.
In the given figure, ABCD is a rectangle.
In which AB || DC, AD || BC and A =
B = C = D = 90º.
Properties of a rectangle :
(i) Opposite sides of a rectangle are equal
and parallel.
(ii) Each angle of a rectangle is 90º.
(iii) Diagonals of a rectangle are equal.
Thus, in a rectangle ABCD, we have :
(i) AB = DC, AD = BC and AB || DC, AD ||
BC.
(ii) A = B = C = D = 1 right angle.
(iii) Diagonal AC = diagonal BD.
5. Square. A parallelogram in which all the
sides are equal and each angle is a right
angle, is called a square.
In the given figure, ABCD is a square in
which AB = BC = CD = DA and A =
B = C = D = 90º.
Properties of a square :
(i) The sides of a square are all equal.
(ii) Each angle of a square is 90º.
(iii) The diagonals of a square are equal and
bisect each other at right angle.
Thus, in a square ABCD, we have :
(i) AB = BC = CD = DA.
(ii) A = B = C = D = 90º.
(iii) Diagonal AC = diagonal BD.
6. Kite. A quadrilateral which has two pairs
of equal adjacent sides but unequal
opposite sides, is called a kite.
In the given figure ABCD is a kite in
which CB = CD and AB = AD but AD 
BC and AB  CD.
( ) EXERCISE 17 A
Q.1. In adjacent figure, a quadrilateral has been
shown.
Name :
(i) its diagonal,
(ii) two pairs of opposite sides,
(iii) two pairs of opposite angles,
(iv) two pairs of adjacent sides,
(v) two pairs of adjacent angles.
D
A B
C
Sol. In the figure, a quadrilateral
(i) Its diagonals are AC and BD
(ii) Two pairs of opposite sides are AB, CD
and AD, BC
(iii) Two pairs of opposite angles are A,
C and B, D
(iv) Two pairs of adjacents sides are AB,
BC and CD and DA
(v) Two pairs of adjacents angles are A,
B and B, C
Q. 2. Draw a parallelogram ABCD in which AB
= 6·5 cm AD = 4·8 cm and BAD = 70º.
Measure its diagonals.
Sol. Steps of construction :
(i) Draw a line segment AB = 6·5 cm.
(ii) At A, draw a ray AE making an angle of
70º with the help of the protractor and
cut off AD = 4·8 cm.
(iii) With centre B and radius 4·8 cm and with
centre D and radius 6·5 cm, draw two
arcs intersecting each other at C.
(iv) Join BC and DC.
Then ABCD is the required parallelogram.
(v) Join AC and BD which measures 9·3 cm
and 6·6 cm respectively.
Q. 3. Two sides of a parallelogram is 4 : 3. If
its perimeter is 56 cm, find the lengths of
the sides of the parallelogram.
Sol. Perimeter of the parallelogram = 56 cm
Ratio in sides = 4 : 3
Let first side = 4 x
Then second side = 3 x
 Perimeter = 2 × sum of two sides
    56 = 2 × (4 x + 3 x)  7 x × 2 = 56
Page 5


Points to Remember :
1. Quadrilateral. A simple closed figure
bounded by four line segments is called a
quadrilateral. In the adjacent figure, ABCD
is a quadrilateral. A quadrilateral ABCD
has :
(i) Four vertices, namely A, B, C and D.
(ii) Four sides, namely AB, BC, CD and DA.
(iii) Four angles, namely DAB, ABC,
BCD and CDA, to be denoted by A,
B, C and D respectively.
(iv) Two diagonals, namely AC and BD.
(i) Adjacent sides. Two sides of a
quadrilateral which have a common end
point are called its adjacent sides.
Thus AB, BC ; BC, CD ; CD, DA and
DA, AB are four pairs of adjacent sides
of the quadrilateral ABCD.
(ii) Opposite sides. Two sides of a
quadrilateral are called its opposite sides
if they do not have a common end point.
Thus AB, DC and AD, BC are two pairs
of the opposite sides of the quadrilateral
ABCD.
(iii) Adjacent angles. Two angles of a
quadrilateral having one common side are
called its adjacent angles.
Thus A, B ; B, C ; C, D and
D, A are four pairs of adjacent angles
of the quadrilateral ABCD.
(iv) Opposite angles. Two angles of a
quadrilateral which are not adjacent angles
are known as the opposite angles of the
quadrilateral.
Thus A, C, and B, D are two pairs
of opposite angles of the quadrilateral
ABCD.
2. Convex and concave quadrilaterals
(i) Convex quadrilateral. A quadrilateral in
which the measure of each angle is less
than 180º is called a convex quadrilateral.
(ii) Concave quadrilateral. A quadrilateral
in which the measure of at least one of
the angles is more than 180º is called a
concave quadrilateral.
In the above figure, PQRS is a concave
quadrilateral in which S > 180º.
In this chapter, by a quadrilateral, we
would mean a convex quadrilateral.
3. Interior and exterior of a quadrilateral
Consider a quadrilateral ABCD. It divides
the whole plane into three parts.
(i) The part of the plane lying inside the
boundary ABCD is called the interior of
the quadrilateral ABCD. Each point of this
part is called an interior point of the
quadrilateral. In the given figure, the
points P, Q, R are the interior points of
the quadrilateral ABCD.
(ii) The part of the plane lying outside the
boundary ABCD is called the exterior of
the quadrilateral ABCD. Each point of this
part is called an exterior point of the
quadrilateral.
In the given figure, the points L and M
are the exterior points of the quadrilateral
ABCD.
(iii) The boundary ABCD.
Clearly, the point E lies on the quadrilateral
ABCD.
Quadrilateral region. The interior of the
quadrilateral ABCD together with its
boundary is called the quadrilateral region
ABCD.
4. Angle sum property of a quadrilateral.
The sum of the angles of a quadrilateral
is 360º.
5. Various Types of Quadrilaterals :
1. Trapezium. A quadrilateral having one
and only one pair of parallel sides is called
a trapezium.
In the adjacent figure, ABCD is a
trapezium in which AB || DC.
A trapezium is said to be an isosceles
trapezium if its nonparallel sides are equal.
In the adjoining figure, PQRS is an
isosceles trapezium in which PQ || SR
and PS = QR.
Remark. The diagonals of an isosceles
trapezium are always equal.
2. Parallelogram . A quadrilateral in which
both pairs of opposite sides are parallel is
called a parallelogram.
In the given figure, ABCD is a
parallelogram in which AB || DC and AD
|| BC. We denote it by ||gm ABCD.
Properties of a parallelogram :
(i) The opposite sides of a ||gm are equal
and parallel.
(ii) The opposite angles of a ||gm are equal.
(iii) The diagonals of a ||gm bisect each other.
Thus, in a ||gm ABCD, we have
(i) AB = DC, AD = BC and AB || DC, AD ||
BC.
(ii) BAD = BCD and ABC = ADC.
(iii) If the diagonals AC and BD intersect at
O, then OA = OC and OB = OD.
3. Rhombus. A parallelogram in which all
the sides are equal is called a rhombus.
In the given figure, ABCD is a rhombus
in which AB || DC, AD || BC and AB =
BC = CD = DA.
Properties of a rhombus :
(i) The opposite sides of a rhombus are
parallel.
(ii) All the sides of a rhombus are equal.
(iii) The opposite angles of a rhombus are
equal.
(iv) The diagonals of a rhombus bisect each
other at right angles.
Thus, in a rhombus ABCD, we have :
(i) AB || DC and AD || BC.
(ii) AB = BC = CD = DA.
(iii) DAB = BCD and ABC = CDA.
(iv) Let the diagonals AC and BD intersect at
O. Then, OA = OC, OB = OD and AOB
= COD = BOC = AOD = 1 right
angle.
4. Rectangle. A parallelogram in which
each angle is a right angle, is called a
rectangle.
In the given figure, ABCD is a rectangle.
In which AB || DC, AD || BC and A =
B = C = D = 90º.
Properties of a rectangle :
(i) Opposite sides of a rectangle are equal
and parallel.
(ii) Each angle of a rectangle is 90º.
(iii) Diagonals of a rectangle are equal.
Thus, in a rectangle ABCD, we have :
(i) AB = DC, AD = BC and AB || DC, AD ||
BC.
(ii) A = B = C = D = 1 right angle.
(iii) Diagonal AC = diagonal BD.
5. Square. A parallelogram in which all the
sides are equal and each angle is a right
angle, is called a square.
In the given figure, ABCD is a square in
which AB = BC = CD = DA and A =
B = C = D = 90º.
Properties of a square :
(i) The sides of a square are all equal.
(ii) Each angle of a square is 90º.
(iii) The diagonals of a square are equal and
bisect each other at right angle.
Thus, in a square ABCD, we have :
(i) AB = BC = CD = DA.
(ii) A = B = C = D = 90º.
(iii) Diagonal AC = diagonal BD.
6. Kite. A quadrilateral which has two pairs
of equal adjacent sides but unequal
opposite sides, is called a kite.
In the given figure ABCD is a kite in
which CB = CD and AB = AD but AD 
BC and AB  CD.
( ) EXERCISE 17 A
Q.1. In adjacent figure, a quadrilateral has been
shown.
Name :
(i) its diagonal,
(ii) two pairs of opposite sides,
(iii) two pairs of opposite angles,
(iv) two pairs of adjacent sides,
(v) two pairs of adjacent angles.
D
A B
C
Sol. In the figure, a quadrilateral
(i) Its diagonals are AC and BD
(ii) Two pairs of opposite sides are AB, CD
and AD, BC
(iii) Two pairs of opposite angles are A,
C and B, D
(iv) Two pairs of adjacents sides are AB,
BC and CD and DA
(v) Two pairs of adjacents angles are A,
B and B, C
Q. 2. Draw a parallelogram ABCD in which AB
= 6·5 cm AD = 4·8 cm and BAD = 70º.
Measure its diagonals.
Sol. Steps of construction :
(i) Draw a line segment AB = 6·5 cm.
(ii) At A, draw a ray AE making an angle of
70º with the help of the protractor and
cut off AD = 4·8 cm.
(iii) With centre B and radius 4·8 cm and with
centre D and radius 6·5 cm, draw two
arcs intersecting each other at C.
(iv) Join BC and DC.
Then ABCD is the required parallelogram.
(v) Join AC and BD which measures 9·3 cm
and 6·6 cm respectively.
Q. 3. Two sides of a parallelogram is 4 : 3. If
its perimeter is 56 cm, find the lengths of
the sides of the parallelogram.
Sol. Perimeter of the parallelogram = 56 cm
Ratio in sides = 4 : 3
Let first side = 4 x
Then second side = 3 x
 Perimeter = 2 × sum of two sides
    56 = 2 × (4 x + 3 x)  7 x × 2 = 56
 14 x = 56   
x
56
14
4
 First side = 4 x = 4 × 4 = 16 cm
and second side = 3 x = 3 × 4 = 12 cm.   Ans.
Q. 4. Name each of the following
parallelograms :
(a) The diagonals are equal and the adjacent
sides are unequal.
(b) The diagonals are equal and the adjacent
sides are equal.
(c) The diagonals are unequal and the adjacent
sides are equal.
Sol. (a) 
.
.
.
 A parallelograms whose diagonals
are equal and adjacent sides are unequal,
is a rectangle.
(b)
.
.
.
 A parallelogram whose diagonal are
equal and also side are equal, is a square.
(c)
.
.
.
 A parallelogram whose diagonal are
unequal but adjacent sides are equal is a
rhombus.
Q. 5. What is a trapezium ? When do you call a
trapezium an isosceles trapezium ?
Draw an isosceles trapezium. Measure
its sides and angles.
Sol. A quadrilateral whose one pair of opposite
sides are equal but other pair non parallel,
is called a trapezium.
When the non-parallel sides of a
trapezium are equal, then it is called an
isosceles trapezium.
ABCD is an isosceles trapezium in which
AD = BC
Then DAB = CBA
On measuring, AD = BC = 3 cm
and DAB = CBA = 60º
D C
B A
Q. 6. Which of the following statements are
true and which are false ?
(a) The diagonals of a parallelogram are equal.
(b) The diagonals of a rectangle are
perpendicular to each other.
(c) The diagonals of a rhombus are equal.
Sol. (a) False 
.
.
.
 Diagonals of a parallelogram
are not equal.
(b) False 
.
.
.
 Diagonals of a rectangle do not
bisect each other at right angles.
(c) False 
.
.
.
 Diagonals of a rhombus are not
equal.
Q.7. Given reasons for the following :
(a) A square can be thought of as a special
rectangle.
(b) A square can be thought of as a special
rhombus.
(c) A rectangle can be thought of as a special
parallelogram.
(d) Square is also a parallelogram.
Sol. (a) Because if each side of a rectangle
are equal it is called a square.
(b) Square is a special rhombus if its each
angle is equal i.e., of 90º.
(c) If in a parallelogram, if each angle is of
90º, it is called a rectangle.
(d) A square is a parallelogram whose each
side and each angle are equal.
Q.8. A figure is said to be regular if its sides
are equal in length and angles are equal
in measure. What do you mean by a
regular quadrilateral ?
Sol. A regular quadrilateral is a quadrilateral
if its each side and angles are equal
square is a regular quadrilateral.
( ) EXERCISE 17 B
Objective questions
Mark ( ) against the correct answer
in each of the following :
Q. 1. The sum of all the angles of a quadrilateral
is
(a) 180º  (b) 270º  (c) 360º  (d) 400º
Sol. (c) 
.
.
.
 Sum of angles of a quadrilateral is
360º.
Q.2. The three angles of a quadrilateral are
80º, 70º and 120º. The fourth angle is
(a) 110º  (b) 100º  (c) 90º  (d) 80º
Sol. Sum of 4 angles of a quadrilateral = 360º