Chapter - 16 Parallelograms | Mathematics (Maths) Class 8 PDF Download

 Page 1


Q u e s t i o n : 1
ABCD is a parallelogram in which ?A = 110°. Find the measure of each of the angles ?B, ?C and ?D.
S o l u t i o n :
It is given that ABCD is a parallelogram in which ?A is equal to 110°. Sum of the adjacent angles of a parallelogram is 180°. ? ?A + ?B = 180° ? 110°+ ?B = 180° ? ?B = (180°-110
Also, ?B + ?C = 180° ? 70°+ ?C = 180° ? ?C = (180°-70°) ? ?C = 110° ? ?C = 110°Further, ?C + ?D = 180° ? 110°+ ?D = 180° ? ?D = (180°-110°) ? ?D = 70° ? ?D = 70°
Q u e s t i o n : 2
Two adjacent angles of a parallelogram are equal. What is the measure of each of these angles?
S o l u t i o n :
Let the required angle be x°. As the adjacent angles are equal, we have:    x +x = 180            (since the sum of adjacent angles of a parallelogram is 180°) ? 2x = 180 ? x =
180
2
? x = 90
Q u e s t i o n : 3
Two adjacent angles of a parallelogram are in the ratio 4 : 5. Find the measure of each of its angles.
S o l u t i o n :
 
Let ABCD be the parallelogram. Then, ?A and ?B are its adjacent angles. Let ?A = (4x)° ?B = (5x)°  ? ?A + ?B = 180°      [since sum of the adjacent angles of a parallelogram is 180°
                                
Q u e s t i o n : 4
Two adjacent angles of a parallelogram are (3x - 4)° and (3x + 16)°. Find the value of x and hence find the measure of each of its angles.
S o l u t i o n :
Let ABCD be a parallelogram. Let ?A = (3x -4)° ?B = (3x +16)°  ? ?A + ?B = 180°       [since the sum of adjacent angles of a parallelogram is 180°] ? 3x -4 +3x +16 = 180 ? 3x -4 +3x +
Q u e s t i o n : 5
The sum of two opposite angles of a parallelogram is 130°. Find the measure of each of its angles.
S o l u t i o n :
Let ABCD be a parallelogram and let the sum of its opposite angles be 130°. ?A + ?C = 130°The opposite angles are equal in a parallelogram.    ?  ?A = ?C = x° ? x +x = 130 ? 2x =
Q u e s t i o n : 6
Two sides of a parallelogram are in the ratio 5 : 3. If its perimeter is 64 cm, find the lengths of its sides.
S o l u t i o n :
Let the lengths of two sides of the parallelogram be 5x cm and 3x cm, respectively. Then, its perimeter = 2(5x +3x) cm
                             = 16x cm
  ? 16x = 64 ? x =
64
16
? x = 4 ? One side ? (5 ×4) cm = 20 cmOther side ? (3 ×4) cm = 12 cm
Q u e s t i o n : 7
The perimeter of a parallelogram is 140 cm. If one of the sides is longer than the other by 10 cm, find the length of each of its sides.
S o l u t i o n :
Let the lengths of two sides of the parallelogram be x cm and  (x +10) cm, respectively. Then, its perimeter = 2[x +(x +10)] cm
                           
                         = 2[x +x +10] cm = 2[2x +10] cm = 4x +20 cm
     4x +20 = 140 ? 4x = 140 -20 ? 4x = 120 ? x =
120
4
? x = 30Length of one side = 30 cm Length of the other side ? (30 +10) cm = 40 cm
Q u e s t i o n : 8
In the adjacent figure, ABCD is a rectangle. If BM and DN are perpendiculars from B and D on AC, prove that ?BMC ? ?DNA. Is it true that BM = DN?
S o l u t i o n :
Refer to the figure given in the book.
Page 2


Q u e s t i o n : 1
ABCD is a parallelogram in which ?A = 110°. Find the measure of each of the angles ?B, ?C and ?D.
S o l u t i o n :
It is given that ABCD is a parallelogram in which ?A is equal to 110°. Sum of the adjacent angles of a parallelogram is 180°. ? ?A + ?B = 180° ? 110°+ ?B = 180° ? ?B = (180°-110
Also, ?B + ?C = 180° ? 70°+ ?C = 180° ? ?C = (180°-70°) ? ?C = 110° ? ?C = 110°Further, ?C + ?D = 180° ? 110°+ ?D = 180° ? ?D = (180°-110°) ? ?D = 70° ? ?D = 70°
Q u e s t i o n : 2
Two adjacent angles of a parallelogram are equal. What is the measure of each of these angles?
S o l u t i o n :
Let the required angle be x°. As the adjacent angles are equal, we have:    x +x = 180            (since the sum of adjacent angles of a parallelogram is 180°) ? 2x = 180 ? x =
180
2
? x = 90
Q u e s t i o n : 3
Two adjacent angles of a parallelogram are in the ratio 4 : 5. Find the measure of each of its angles.
S o l u t i o n :
 
Let ABCD be the parallelogram. Then, ?A and ?B are its adjacent angles. Let ?A = (4x)° ?B = (5x)°  ? ?A + ?B = 180°      [since sum of the adjacent angles of a parallelogram is 180°
                                
Q u e s t i o n : 4
Two adjacent angles of a parallelogram are (3x - 4)° and (3x + 16)°. Find the value of x and hence find the measure of each of its angles.
S o l u t i o n :
Let ABCD be a parallelogram. Let ?A = (3x -4)° ?B = (3x +16)°  ? ?A + ?B = 180°       [since the sum of adjacent angles of a parallelogram is 180°] ? 3x -4 +3x +16 = 180 ? 3x -4 +3x +
Q u e s t i o n : 5
The sum of two opposite angles of a parallelogram is 130°. Find the measure of each of its angles.
S o l u t i o n :
Let ABCD be a parallelogram and let the sum of its opposite angles be 130°. ?A + ?C = 130°The opposite angles are equal in a parallelogram.    ?  ?A = ?C = x° ? x +x = 130 ? 2x =
Q u e s t i o n : 6
Two sides of a parallelogram are in the ratio 5 : 3. If its perimeter is 64 cm, find the lengths of its sides.
S o l u t i o n :
Let the lengths of two sides of the parallelogram be 5x cm and 3x cm, respectively. Then, its perimeter = 2(5x +3x) cm
                             = 16x cm
  ? 16x = 64 ? x =
64
16
? x = 4 ? One side ? (5 ×4) cm = 20 cmOther side ? (3 ×4) cm = 12 cm
Q u e s t i o n : 7
The perimeter of a parallelogram is 140 cm. If one of the sides is longer than the other by 10 cm, find the length of each of its sides.
S o l u t i o n :
Let the lengths of two sides of the parallelogram be x cm and  (x +10) cm, respectively. Then, its perimeter = 2[x +(x +10)] cm
                           
                         = 2[x +x +10] cm = 2[2x +10] cm = 4x +20 cm
     4x +20 = 140 ? 4x = 140 -20 ? 4x = 120 ? x =
120
4
? x = 30Length of one side = 30 cm Length of the other side ? (30 +10) cm = 40 cm
Q u e s t i o n : 8
In the adjacent figure, ABCD is a rectangle. If BM and DN are perpendiculars from B and D on AC, prove that ?BMC ? ?DNA. Is it true that BM = DN?
S o l u t i o n :
Refer to the figure given in the book.
In ? BMC and ? DNA: ?DNA = ?BMC = 90° ?BCM = ?DAN     (alternate angles)BC = DA     (opposite sides)By AAS congruency criteria: ? BMC ? ? DNAYes, it is true that BM is equal
Q u e s t i o n : 9
In the adjacent figure, ABCD is a parallelogram and line segments AE and CF bisect the angles A and C respectively. Show that AE||CF.
S o l u t i o n :
Refer to the figure of the book.
?A = ?C                                        (opposite angles of a parallelogram are equal) ?
1
2
?A =
1
2
?C = > ?EAD = ?FCB                  AE and CF bisect the angles A and C, respectively In ?
Q u e s t i o n : 1 0
The lengths of the diagonals of a rhombus are 16 cm and 12 cm respectively. Find the length of each of its sides.
S o l u t i o n :
Let ABCD be a rhombus. Let AC and BD be the diagonals of the rhombus intersecting at a point O. Let AC = 16 cm BD = 12 cmWe know that the diagonals of a rhombus bisect each other
Q u e s t i o n : 1 1
In the given figure ABCD is a square. Find the measure of ?CAD.
S o l u t i o n :
Refer to the figure given in the book.
In ? ADC: DA = DC                 all sides of a square are equal ? ?ACD = ?CADLet ?ACD = ?CAD = x°     Angle opposite to the equal sides are equal x +x +90 = 180                     [since
Q u e s t i o n : 1 2
The sides of a rectangle are in the ratio 5 : 4 and its perimeter is 90 cm. Find its length and breadth.
S o l u t i o n :
Let the length of two sides of the rectangle be 5x cm and 4x cm, respectively. Then, its perimeter = 2(5x +4x) cm
                            = 18x cm
 ? 18x = 90 ? x =
90
18
? x = 5Length of one side ? (5 ×5) cm = 25 cmLength of the other side ? (4 ×5) cm = 20 cm ? Length of the rectangle = 25 cm Breadth = 20 cm
Q u e s t i o n : 1 3
Name each of the following parallelograms.
i
The diagonals are equal and the adjacent sides are unequal.
ii
The diagonals are equal and the adjacent sides are equal.
iii
The diagonals are unequal and the adjacent sides are equal.
iv
All the sides are equal and one angle is 60°.
v
All the sides are equal and one angle is 90°.
vi
All the angles are equal and the adjacent sides are unequal.
S o l u t i o n :
(i) The diagonals are equal and the adjacent sides are unequal.      Hence, the given parallelogram is a rectangle. (ii) The diagonals are equal and the adjacent sides are equal.      Hence
Q u e s t i o n : 1 4
Which of the following statements are true and which are false?
i
The diagonals of a parallelogram are equal.
ii
The diagonals of a rectangle are perpendicular to each other.
iii
The diagonals of a rhombus are equal.
iv
Every rhombus is a kite.
v
Every rectangle is a square.
vi
( )
( ) [ ]
Page 3


Q u e s t i o n : 1
ABCD is a parallelogram in which ?A = 110°. Find the measure of each of the angles ?B, ?C and ?D.
S o l u t i o n :
It is given that ABCD is a parallelogram in which ?A is equal to 110°. Sum of the adjacent angles of a parallelogram is 180°. ? ?A + ?B = 180° ? 110°+ ?B = 180° ? ?B = (180°-110
Also, ?B + ?C = 180° ? 70°+ ?C = 180° ? ?C = (180°-70°) ? ?C = 110° ? ?C = 110°Further, ?C + ?D = 180° ? 110°+ ?D = 180° ? ?D = (180°-110°) ? ?D = 70° ? ?D = 70°
Q u e s t i o n : 2
Two adjacent angles of a parallelogram are equal. What is the measure of each of these angles?
S o l u t i o n :
Let the required angle be x°. As the adjacent angles are equal, we have:    x +x = 180            (since the sum of adjacent angles of a parallelogram is 180°) ? 2x = 180 ? x =
180
2
? x = 90
Q u e s t i o n : 3
Two adjacent angles of a parallelogram are in the ratio 4 : 5. Find the measure of each of its angles.
S o l u t i o n :
 
Let ABCD be the parallelogram. Then, ?A and ?B are its adjacent angles. Let ?A = (4x)° ?B = (5x)°  ? ?A + ?B = 180°      [since sum of the adjacent angles of a parallelogram is 180°
                                
Q u e s t i o n : 4
Two adjacent angles of a parallelogram are (3x - 4)° and (3x + 16)°. Find the value of x and hence find the measure of each of its angles.
S o l u t i o n :
Let ABCD be a parallelogram. Let ?A = (3x -4)° ?B = (3x +16)°  ? ?A + ?B = 180°       [since the sum of adjacent angles of a parallelogram is 180°] ? 3x -4 +3x +16 = 180 ? 3x -4 +3x +
Q u e s t i o n : 5
The sum of two opposite angles of a parallelogram is 130°. Find the measure of each of its angles.
S o l u t i o n :
Let ABCD be a parallelogram and let the sum of its opposite angles be 130°. ?A + ?C = 130°The opposite angles are equal in a parallelogram.    ?  ?A = ?C = x° ? x +x = 130 ? 2x =
Q u e s t i o n : 6
Two sides of a parallelogram are in the ratio 5 : 3. If its perimeter is 64 cm, find the lengths of its sides.
S o l u t i o n :
Let the lengths of two sides of the parallelogram be 5x cm and 3x cm, respectively. Then, its perimeter = 2(5x +3x) cm
                             = 16x cm
  ? 16x = 64 ? x =
64
16
? x = 4 ? One side ? (5 ×4) cm = 20 cmOther side ? (3 ×4) cm = 12 cm
Q u e s t i o n : 7
The perimeter of a parallelogram is 140 cm. If one of the sides is longer than the other by 10 cm, find the length of each of its sides.
S o l u t i o n :
Let the lengths of two sides of the parallelogram be x cm and  (x +10) cm, respectively. Then, its perimeter = 2[x +(x +10)] cm
                           
                         = 2[x +x +10] cm = 2[2x +10] cm = 4x +20 cm
     4x +20 = 140 ? 4x = 140 -20 ? 4x = 120 ? x =
120
4
? x = 30Length of one side = 30 cm Length of the other side ? (30 +10) cm = 40 cm
Q u e s t i o n : 8
In the adjacent figure, ABCD is a rectangle. If BM and DN are perpendiculars from B and D on AC, prove that ?BMC ? ?DNA. Is it true that BM = DN?
S o l u t i o n :
Refer to the figure given in the book.
In ? BMC and ? DNA: ?DNA = ?BMC = 90° ?BCM = ?DAN     (alternate angles)BC = DA     (opposite sides)By AAS congruency criteria: ? BMC ? ? DNAYes, it is true that BM is equal
Q u e s t i o n : 9
In the adjacent figure, ABCD is a parallelogram and line segments AE and CF bisect the angles A and C respectively. Show that AE||CF.
S o l u t i o n :
Refer to the figure of the book.
?A = ?C                                        (opposite angles of a parallelogram are equal) ?
1
2
?A =
1
2
?C = > ?EAD = ?FCB                  AE and CF bisect the angles A and C, respectively In ?
Q u e s t i o n : 1 0
The lengths of the diagonals of a rhombus are 16 cm and 12 cm respectively. Find the length of each of its sides.
S o l u t i o n :
Let ABCD be a rhombus. Let AC and BD be the diagonals of the rhombus intersecting at a point O. Let AC = 16 cm BD = 12 cmWe know that the diagonals of a rhombus bisect each other
Q u e s t i o n : 1 1
In the given figure ABCD is a square. Find the measure of ?CAD.
S o l u t i o n :
Refer to the figure given in the book.
In ? ADC: DA = DC                 all sides of a square are equal ? ?ACD = ?CADLet ?ACD = ?CAD = x°     Angle opposite to the equal sides are equal x +x +90 = 180                     [since
Q u e s t i o n : 1 2
The sides of a rectangle are in the ratio 5 : 4 and its perimeter is 90 cm. Find its length and breadth.
S o l u t i o n :
Let the length of two sides of the rectangle be 5x cm and 4x cm, respectively. Then, its perimeter = 2(5x +4x) cm
                            = 18x cm
 ? 18x = 90 ? x =
90
18
? x = 5Length of one side ? (5 ×5) cm = 25 cmLength of the other side ? (4 ×5) cm = 20 cm ? Length of the rectangle = 25 cm Breadth = 20 cm
Q u e s t i o n : 1 3
Name each of the following parallelograms.
i
The diagonals are equal and the adjacent sides are unequal.
ii
The diagonals are equal and the adjacent sides are equal.
iii
The diagonals are unequal and the adjacent sides are equal.
iv
All the sides are equal and one angle is 60°.
v
All the sides are equal and one angle is 90°.
vi
All the angles are equal and the adjacent sides are unequal.
S o l u t i o n :
(i) The diagonals are equal and the adjacent sides are unequal.      Hence, the given parallelogram is a rectangle. (ii) The diagonals are equal and the adjacent sides are equal.      Hence
Q u e s t i o n : 1 4
Which of the following statements are true and which are false?
i
The diagonals of a parallelogram are equal.
ii
The diagonals of a rectangle are perpendicular to each other.
iii
The diagonals of a rhombus are equal.
iv
Every rhombus is a kite.
v
Every rectangle is a square.
vi
( )
( ) [ ]
Every square is a a parallelogram.
vii
Every square is a rhombus.
viii
Every rectangle is a parallelogram.
ix
Every parallelogram is a rectangle.
x
Every rhombus is a parallelogram.
S o l u t i o n :
(i) The given statement is false. The diagonals of a parallelogram bisect each other, but they are not equal in length. (ii) The given statement is false. The diagonals of a rectangle are equal
Q u e s t i o n : 1 5
Tick
? the correct answer
The two diagonals are not necessarily equal in a
a
rectangle
b
square
c
rhombus
d
isosceles trapezium
S o l u t i o n :
(c) rhombus In a rhombus, the two diagonals are not necessarily equal.
Q u e s t i o n : 1 6
Tick
? the correct answer
The lengths of the diagonals of a rhombus are 16 cm and 12 cm. The length of each side of the rhombus is
a
8 cm
b
9 cm
c
10 cm
d
12 cm
S o l u t i o n :
(c) 10 cm
                 
Let ABCD be a rhombus. Let AC and BD be the diagonals of the rhombus intersecting at a point O. AC = 16 cm BD = 12 cmWe know that the diagonals of a rhombus bisect each other at right
Q u e s t i o n : 1 7
Tick
? the correct answer
Two adjacent angles of a parallelogram are (2x + 25)° and (3x - 5)°. The value of x is
a
28
b
32
c
36
d
42
S o l u t i o n :
b 32We know that the sum of adjacent angles of a parallelogram is180°. ? 2x +25 +3x -5 = 180 ? 5x +20 = 180 ? 5x = 180 -20 ? 5x = 160 ? x =
160
5
? x = 32Therefore, the value of x
Q u e s t i o n : 1 8
Tick
? the correct answer:
The diagonals do not necessarily intersect at right angles in a
a
parallelogram
b
rectangle
c
rhombus
d
kite
S o l u t i o n :
(a) parallelogramIn a parallelogram, the diagonals do not necessarily intersect at right angles.
Q u e s t i o n : 1 9
Tick
? the correct answer:
The length and breadth of a rectangle are in the ratio 4 : 3. If the diagonal measures 25 cm then the perimeter of the rectangle is
( )
Page 4


Q u e s t i o n : 1
ABCD is a parallelogram in which ?A = 110°. Find the measure of each of the angles ?B, ?C and ?D.
S o l u t i o n :
It is given that ABCD is a parallelogram in which ?A is equal to 110°. Sum of the adjacent angles of a parallelogram is 180°. ? ?A + ?B = 180° ? 110°+ ?B = 180° ? ?B = (180°-110
Also, ?B + ?C = 180° ? 70°+ ?C = 180° ? ?C = (180°-70°) ? ?C = 110° ? ?C = 110°Further, ?C + ?D = 180° ? 110°+ ?D = 180° ? ?D = (180°-110°) ? ?D = 70° ? ?D = 70°
Q u e s t i o n : 2
Two adjacent angles of a parallelogram are equal. What is the measure of each of these angles?
S o l u t i o n :
Let the required angle be x°. As the adjacent angles are equal, we have:    x +x = 180            (since the sum of adjacent angles of a parallelogram is 180°) ? 2x = 180 ? x =
180
2
? x = 90
Q u e s t i o n : 3
Two adjacent angles of a parallelogram are in the ratio 4 : 5. Find the measure of each of its angles.
S o l u t i o n :
 
Let ABCD be the parallelogram. Then, ?A and ?B are its adjacent angles. Let ?A = (4x)° ?B = (5x)°  ? ?A + ?B = 180°      [since sum of the adjacent angles of a parallelogram is 180°
                                
Q u e s t i o n : 4
Two adjacent angles of a parallelogram are (3x - 4)° and (3x + 16)°. Find the value of x and hence find the measure of each of its angles.
S o l u t i o n :
Let ABCD be a parallelogram. Let ?A = (3x -4)° ?B = (3x +16)°  ? ?A + ?B = 180°       [since the sum of adjacent angles of a parallelogram is 180°] ? 3x -4 +3x +16 = 180 ? 3x -4 +3x +
Q u e s t i o n : 5
The sum of two opposite angles of a parallelogram is 130°. Find the measure of each of its angles.
S o l u t i o n :
Let ABCD be a parallelogram and let the sum of its opposite angles be 130°. ?A + ?C = 130°The opposite angles are equal in a parallelogram.    ?  ?A = ?C = x° ? x +x = 130 ? 2x =
Q u e s t i o n : 6
Two sides of a parallelogram are in the ratio 5 : 3. If its perimeter is 64 cm, find the lengths of its sides.
S o l u t i o n :
Let the lengths of two sides of the parallelogram be 5x cm and 3x cm, respectively. Then, its perimeter = 2(5x +3x) cm
                             = 16x cm
  ? 16x = 64 ? x =
64
16
? x = 4 ? One side ? (5 ×4) cm = 20 cmOther side ? (3 ×4) cm = 12 cm
Q u e s t i o n : 7
The perimeter of a parallelogram is 140 cm. If one of the sides is longer than the other by 10 cm, find the length of each of its sides.
S o l u t i o n :
Let the lengths of two sides of the parallelogram be x cm and  (x +10) cm, respectively. Then, its perimeter = 2[x +(x +10)] cm
                           
                         = 2[x +x +10] cm = 2[2x +10] cm = 4x +20 cm
     4x +20 = 140 ? 4x = 140 -20 ? 4x = 120 ? x =
120
4
? x = 30Length of one side = 30 cm Length of the other side ? (30 +10) cm = 40 cm
Q u e s t i o n : 8
In the adjacent figure, ABCD is a rectangle. If BM and DN are perpendiculars from B and D on AC, prove that ?BMC ? ?DNA. Is it true that BM = DN?
S o l u t i o n :
Refer to the figure given in the book.
In ? BMC and ? DNA: ?DNA = ?BMC = 90° ?BCM = ?DAN     (alternate angles)BC = DA     (opposite sides)By AAS congruency criteria: ? BMC ? ? DNAYes, it is true that BM is equal
Q u e s t i o n : 9
In the adjacent figure, ABCD is a parallelogram and line segments AE and CF bisect the angles A and C respectively. Show that AE||CF.
S o l u t i o n :
Refer to the figure of the book.
?A = ?C                                        (opposite angles of a parallelogram are equal) ?
1
2
?A =
1
2
?C = > ?EAD = ?FCB                  AE and CF bisect the angles A and C, respectively In ?
Q u e s t i o n : 1 0
The lengths of the diagonals of a rhombus are 16 cm and 12 cm respectively. Find the length of each of its sides.
S o l u t i o n :
Let ABCD be a rhombus. Let AC and BD be the diagonals of the rhombus intersecting at a point O. Let AC = 16 cm BD = 12 cmWe know that the diagonals of a rhombus bisect each other
Q u e s t i o n : 1 1
In the given figure ABCD is a square. Find the measure of ?CAD.
S o l u t i o n :
Refer to the figure given in the book.
In ? ADC: DA = DC                 all sides of a square are equal ? ?ACD = ?CADLet ?ACD = ?CAD = x°     Angle opposite to the equal sides are equal x +x +90 = 180                     [since
Q u e s t i o n : 1 2
The sides of a rectangle are in the ratio 5 : 4 and its perimeter is 90 cm. Find its length and breadth.
S o l u t i o n :
Let the length of two sides of the rectangle be 5x cm and 4x cm, respectively. Then, its perimeter = 2(5x +4x) cm
                            = 18x cm
 ? 18x = 90 ? x =
90
18
? x = 5Length of one side ? (5 ×5) cm = 25 cmLength of the other side ? (4 ×5) cm = 20 cm ? Length of the rectangle = 25 cm Breadth = 20 cm
Q u e s t i o n : 1 3
Name each of the following parallelograms.
i
The diagonals are equal and the adjacent sides are unequal.
ii
The diagonals are equal and the adjacent sides are equal.
iii
The diagonals are unequal and the adjacent sides are equal.
iv
All the sides are equal and one angle is 60°.
v
All the sides are equal and one angle is 90°.
vi
All the angles are equal and the adjacent sides are unequal.
S o l u t i o n :
(i) The diagonals are equal and the adjacent sides are unequal.      Hence, the given parallelogram is a rectangle. (ii) The diagonals are equal and the adjacent sides are equal.      Hence
Q u e s t i o n : 1 4
Which of the following statements are true and which are false?
i
The diagonals of a parallelogram are equal.
ii
The diagonals of a rectangle are perpendicular to each other.
iii
The diagonals of a rhombus are equal.
iv
Every rhombus is a kite.
v
Every rectangle is a square.
vi
( )
( ) [ ]
Every square is a a parallelogram.
vii
Every square is a rhombus.
viii
Every rectangle is a parallelogram.
ix
Every parallelogram is a rectangle.
x
Every rhombus is a parallelogram.
S o l u t i o n :
(i) The given statement is false. The diagonals of a parallelogram bisect each other, but they are not equal in length. (ii) The given statement is false. The diagonals of a rectangle are equal
Q u e s t i o n : 1 5
Tick
? the correct answer
The two diagonals are not necessarily equal in a
a
rectangle
b
square
c
rhombus
d
isosceles trapezium
S o l u t i o n :
(c) rhombus In a rhombus, the two diagonals are not necessarily equal.
Q u e s t i o n : 1 6
Tick
? the correct answer
The lengths of the diagonals of a rhombus are 16 cm and 12 cm. The length of each side of the rhombus is
a
8 cm
b
9 cm
c
10 cm
d
12 cm
S o l u t i o n :
(c) 10 cm
                 
Let ABCD be a rhombus. Let AC and BD be the diagonals of the rhombus intersecting at a point O. AC = 16 cm BD = 12 cmWe know that the diagonals of a rhombus bisect each other at right
Q u e s t i o n : 1 7
Tick
? the correct answer
Two adjacent angles of a parallelogram are (2x + 25)° and (3x - 5)°. The value of x is
a
28
b
32
c
36
d
42
S o l u t i o n :
b 32We know that the sum of adjacent angles of a parallelogram is180°. ? 2x +25 +3x -5 = 180 ? 5x +20 = 180 ? 5x = 180 -20 ? 5x = 160 ? x =
160
5
? x = 32Therefore, the value of x
Q u e s t i o n : 1 8
Tick
? the correct answer:
The diagonals do not necessarily intersect at right angles in a
a
parallelogram
b
rectangle
c
rhombus
d
kite
S o l u t i o n :
(a) parallelogramIn a parallelogram, the diagonals do not necessarily intersect at right angles.
Q u e s t i o n : 1 9
Tick
? the correct answer:
The length and breadth of a rectangle are in the ratio 4 : 3. If the diagonal measures 25 cm then the perimeter of the rectangle is
( )
a
56 cm
b
60 cm
c
70 cm
d
80 cm
S o l u t i o n :
c
70 cm
Let ABCD be a rectangle and let the diagonal AC be 25 cm, length AB be 4x cm and breadth BC be 3x cm. Each angle of a rectangle is a right angle. ? ?ABC = 90°From the right ? ABC
x
2
= 
625
25
= 25 ? x = 5 ? Length = 4 ×5 = 20 cmBreadth = 3 ×5 = 15 cm 
? Perimeter of the rectangle = 2(20+15) cm
                                         = 70 cm
Q u e s t i o n : 2 0
Tick
? the correct answer:
The bisectors of any two adjacent angles of a parallelogram intersect at
a
30°
b
45°
c
60°
d
90°
S o l u t i o n :
(d) 90°The bisectors of any two adjacent angles of a parallelogram intersect at 90°.
Q u e s t i o n : 2 1
Tick
? the correct answer:
If an angle of a parallelogram is two-thirds of its adjacent angle, the smallest angle of the parallelogram is
a
54°
b
72°
c
81°
d
108°
S o l u t i o n :
b 72°Let x° be the angle of the parallelogram. Sum of the adjacent angles of a parallelogram is 180°. ? x +
2
3
×x = 180 ? x +
2x
3
= 180 ? x +
2x
3
= 180 ?
5x
3
= 180 ? x = 180 ×
3
5
?
Q u e s t i o n : 2 2
Tick
? the correct answer:
The diagonals do not necessarily bisect the interior angles at the vertices in a
a
rectangle
b
square
c
rhombus
d
all of these
S o l u t i o n :
(a) rectangle In a rectangle, the diagonals do not necessarily bisect the interior angles at the vertices.
Q u e s t i o n : 2 3
Tick
? the correct answer:
In a square ABCD, AB = (2x + 3) cm and BC = (3x - 5) cm. Then, the value of x is
a
4
b
5
c
6
d
8
S o l u t i o n :
(d) 8All the sides of a square are equal. ? AB = BC ? 2x +3 = 3x -5 ? 3 +5 = 3x -2x ? 8 = xTherefore, the value of x is 8. 
Q u e s t i o n : 2 4
Tick
? the correct answer:
If one angle of a parallelogram is 24° less than twice the smallest angle then the largest angle of the parallelogram is
a
68°
b
( ) ( ) ( ) ( )
Page 5


Q u e s t i o n : 1
ABCD is a parallelogram in which ?A = 110°. Find the measure of each of the angles ?B, ?C and ?D.
S o l u t i o n :
It is given that ABCD is a parallelogram in which ?A is equal to 110°. Sum of the adjacent angles of a parallelogram is 180°. ? ?A + ?B = 180° ? 110°+ ?B = 180° ? ?B = (180°-110
Also, ?B + ?C = 180° ? 70°+ ?C = 180° ? ?C = (180°-70°) ? ?C = 110° ? ?C = 110°Further, ?C + ?D = 180° ? 110°+ ?D = 180° ? ?D = (180°-110°) ? ?D = 70° ? ?D = 70°
Q u e s t i o n : 2
Two adjacent angles of a parallelogram are equal. What is the measure of each of these angles?
S o l u t i o n :
Let the required angle be x°. As the adjacent angles are equal, we have:    x +x = 180            (since the sum of adjacent angles of a parallelogram is 180°) ? 2x = 180 ? x =
180
2
? x = 90
Q u e s t i o n : 3
Two adjacent angles of a parallelogram are in the ratio 4 : 5. Find the measure of each of its angles.
S o l u t i o n :
 
Let ABCD be the parallelogram. Then, ?A and ?B are its adjacent angles. Let ?A = (4x)° ?B = (5x)°  ? ?A + ?B = 180°      [since sum of the adjacent angles of a parallelogram is 180°
                                
Q u e s t i o n : 4
Two adjacent angles of a parallelogram are (3x - 4)° and (3x + 16)°. Find the value of x and hence find the measure of each of its angles.
S o l u t i o n :
Let ABCD be a parallelogram. Let ?A = (3x -4)° ?B = (3x +16)°  ? ?A + ?B = 180°       [since the sum of adjacent angles of a parallelogram is 180°] ? 3x -4 +3x +16 = 180 ? 3x -4 +3x +
Q u e s t i o n : 5
The sum of two opposite angles of a parallelogram is 130°. Find the measure of each of its angles.
S o l u t i o n :
Let ABCD be a parallelogram and let the sum of its opposite angles be 130°. ?A + ?C = 130°The opposite angles are equal in a parallelogram.    ?  ?A = ?C = x° ? x +x = 130 ? 2x =
Q u e s t i o n : 6
Two sides of a parallelogram are in the ratio 5 : 3. If its perimeter is 64 cm, find the lengths of its sides.
S o l u t i o n :
Let the lengths of two sides of the parallelogram be 5x cm and 3x cm, respectively. Then, its perimeter = 2(5x +3x) cm
                             = 16x cm
  ? 16x = 64 ? x =
64
16
? x = 4 ? One side ? (5 ×4) cm = 20 cmOther side ? (3 ×4) cm = 12 cm
Q u e s t i o n : 7
The perimeter of a parallelogram is 140 cm. If one of the sides is longer than the other by 10 cm, find the length of each of its sides.
S o l u t i o n :
Let the lengths of two sides of the parallelogram be x cm and  (x +10) cm, respectively. Then, its perimeter = 2[x +(x +10)] cm
                           
                         = 2[x +x +10] cm = 2[2x +10] cm = 4x +20 cm
     4x +20 = 140 ? 4x = 140 -20 ? 4x = 120 ? x =
120
4
? x = 30Length of one side = 30 cm Length of the other side ? (30 +10) cm = 40 cm
Q u e s t i o n : 8
In the adjacent figure, ABCD is a rectangle. If BM and DN are perpendiculars from B and D on AC, prove that ?BMC ? ?DNA. Is it true that BM = DN?
S o l u t i o n :
Refer to the figure given in the book.
In ? BMC and ? DNA: ?DNA = ?BMC = 90° ?BCM = ?DAN     (alternate angles)BC = DA     (opposite sides)By AAS congruency criteria: ? BMC ? ? DNAYes, it is true that BM is equal
Q u e s t i o n : 9
In the adjacent figure, ABCD is a parallelogram and line segments AE and CF bisect the angles A and C respectively. Show that AE||CF.
S o l u t i o n :
Refer to the figure of the book.
?A = ?C                                        (opposite angles of a parallelogram are equal) ?
1
2
?A =
1
2
?C = > ?EAD = ?FCB                  AE and CF bisect the angles A and C, respectively In ?
Q u e s t i o n : 1 0
The lengths of the diagonals of a rhombus are 16 cm and 12 cm respectively. Find the length of each of its sides.
S o l u t i o n :
Let ABCD be a rhombus. Let AC and BD be the diagonals of the rhombus intersecting at a point O. Let AC = 16 cm BD = 12 cmWe know that the diagonals of a rhombus bisect each other
Q u e s t i o n : 1 1
In the given figure ABCD is a square. Find the measure of ?CAD.
S o l u t i o n :
Refer to the figure given in the book.
In ? ADC: DA = DC                 all sides of a square are equal ? ?ACD = ?CADLet ?ACD = ?CAD = x°     Angle opposite to the equal sides are equal x +x +90 = 180                     [since
Q u e s t i o n : 1 2
The sides of a rectangle are in the ratio 5 : 4 and its perimeter is 90 cm. Find its length and breadth.
S o l u t i o n :
Let the length of two sides of the rectangle be 5x cm and 4x cm, respectively. Then, its perimeter = 2(5x +4x) cm
                            = 18x cm
 ? 18x = 90 ? x =
90
18
? x = 5Length of one side ? (5 ×5) cm = 25 cmLength of the other side ? (4 ×5) cm = 20 cm ? Length of the rectangle = 25 cm Breadth = 20 cm
Q u e s t i o n : 1 3
Name each of the following parallelograms.
i
The diagonals are equal and the adjacent sides are unequal.
ii
The diagonals are equal and the adjacent sides are equal.
iii
The diagonals are unequal and the adjacent sides are equal.
iv
All the sides are equal and one angle is 60°.
v
All the sides are equal and one angle is 90°.
vi
All the angles are equal and the adjacent sides are unequal.
S o l u t i o n :
(i) The diagonals are equal and the adjacent sides are unequal.      Hence, the given parallelogram is a rectangle. (ii) The diagonals are equal and the adjacent sides are equal.      Hence
Q u e s t i o n : 1 4
Which of the following statements are true and which are false?
i
The diagonals of a parallelogram are equal.
ii
The diagonals of a rectangle are perpendicular to each other.
iii
The diagonals of a rhombus are equal.
iv
Every rhombus is a kite.
v
Every rectangle is a square.
vi
( )
( ) [ ]
Every square is a a parallelogram.
vii
Every square is a rhombus.
viii
Every rectangle is a parallelogram.
ix
Every parallelogram is a rectangle.
x
Every rhombus is a parallelogram.
S o l u t i o n :
(i) The given statement is false. The diagonals of a parallelogram bisect each other, but they are not equal in length. (ii) The given statement is false. The diagonals of a rectangle are equal
Q u e s t i o n : 1 5
Tick
? the correct answer
The two diagonals are not necessarily equal in a
a
rectangle
b
square
c
rhombus
d
isosceles trapezium
S o l u t i o n :
(c) rhombus In a rhombus, the two diagonals are not necessarily equal.
Q u e s t i o n : 1 6
Tick
? the correct answer
The lengths of the diagonals of a rhombus are 16 cm and 12 cm. The length of each side of the rhombus is
a
8 cm
b
9 cm
c
10 cm
d
12 cm
S o l u t i o n :
(c) 10 cm
                 
Let ABCD be a rhombus. Let AC and BD be the diagonals of the rhombus intersecting at a point O. AC = 16 cm BD = 12 cmWe know that the diagonals of a rhombus bisect each other at right
Q u e s t i o n : 1 7
Tick
? the correct answer
Two adjacent angles of a parallelogram are (2x + 25)° and (3x - 5)°. The value of x is
a
28
b
32
c
36
d
42
S o l u t i o n :
b 32We know that the sum of adjacent angles of a parallelogram is180°. ? 2x +25 +3x -5 = 180 ? 5x +20 = 180 ? 5x = 180 -20 ? 5x = 160 ? x =
160
5
? x = 32Therefore, the value of x
Q u e s t i o n : 1 8
Tick
? the correct answer:
The diagonals do not necessarily intersect at right angles in a
a
parallelogram
b
rectangle
c
rhombus
d
kite
S o l u t i o n :
(a) parallelogramIn a parallelogram, the diagonals do not necessarily intersect at right angles.
Q u e s t i o n : 1 9
Tick
? the correct answer:
The length and breadth of a rectangle are in the ratio 4 : 3. If the diagonal measures 25 cm then the perimeter of the rectangle is
( )
a
56 cm
b
60 cm
c
70 cm
d
80 cm
S o l u t i o n :
c
70 cm
Let ABCD be a rectangle and let the diagonal AC be 25 cm, length AB be 4x cm and breadth BC be 3x cm. Each angle of a rectangle is a right angle. ? ?ABC = 90°From the right ? ABC
x
2
= 
625
25
= 25 ? x = 5 ? Length = 4 ×5 = 20 cmBreadth = 3 ×5 = 15 cm 
? Perimeter of the rectangle = 2(20+15) cm
                                         = 70 cm
Q u e s t i o n : 2 0
Tick
? the correct answer:
The bisectors of any two adjacent angles of a parallelogram intersect at
a
30°
b
45°
c
60°
d
90°
S o l u t i o n :
(d) 90°The bisectors of any two adjacent angles of a parallelogram intersect at 90°.
Q u e s t i o n : 2 1
Tick
? the correct answer:
If an angle of a parallelogram is two-thirds of its adjacent angle, the smallest angle of the parallelogram is
a
54°
b
72°
c
81°
d
108°
S o l u t i o n :
b 72°Let x° be the angle of the parallelogram. Sum of the adjacent angles of a parallelogram is 180°. ? x +
2
3
×x = 180 ? x +
2x
3
= 180 ? x +
2x
3
= 180 ?
5x
3
= 180 ? x = 180 ×
3
5
?
Q u e s t i o n : 2 2
Tick
? the correct answer:
The diagonals do not necessarily bisect the interior angles at the vertices in a
a
rectangle
b
square
c
rhombus
d
all of these
S o l u t i o n :
(a) rectangle In a rectangle, the diagonals do not necessarily bisect the interior angles at the vertices.
Q u e s t i o n : 2 3
Tick
? the correct answer:
In a square ABCD, AB = (2x + 3) cm and BC = (3x - 5) cm. Then, the value of x is
a
4
b
5
c
6
d
8
S o l u t i o n :
(d) 8All the sides of a square are equal. ? AB = BC ? 2x +3 = 3x -5 ? 3 +5 = 3x -2x ? 8 = xTherefore, the value of x is 8. 
Q u e s t i o n : 2 4
Tick
? the correct answer:
If one angle of a parallelogram is 24° less than twice the smallest angle then the largest angle of the parallelogram is
a
68°
b
( ) ( ) ( ) ( )
102°
c
112°
d
176°
S o l u t i o n :
c 112°Let x° be the smallest angle of the parallelogram. The sum of adjacent angles of a parallelogram is 180°. ? x +2x -24 = 180 ? 3x -24 = 180 ? 3x = 180 +24 ? 3x = 204 ? x =
204
( )