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Page 1
Question:56
Three angles of a quadrilateral are 80°, 95° and 112°. Its fourth angle is
a 78°
b 73°
c 85°
d 100°
Solution:
b
73° ? Explanation:
Let the measure of the fourth angle be x
o
.
Since the sum of the angles of a quadrilateral is 360
o
, we have:
80
o
+ 95
o
+ 112
o
+ x = 360
o
? 287
o
+ x = 360
o
? ? x = 73
o
Hence, the measure of the fourth angle is 73
o
.
Question:57
The angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. The smallest of these angles is
a 45°
b 60°
c 36°
Page 2
Question:56
Three angles of a quadrilateral are 80°, 95° and 112°. Its fourth angle is
a 78°
b 73°
c 85°
d 100°
Solution:
b
73° ? Explanation:
Let the measure of the fourth angle be x
o
.
Since the sum of the angles of a quadrilateral is 360
o
, we have:
80
o
+ 95
o
+ 112
o
+ x = 360
o
? 287
o
+ x = 360
o
? ? x = 73
o
Hence, the measure of the fourth angle is 73
o
.
Question:57
The angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. The smallest of these angles is
a 45°
b 60°
c 36°
d 48°
Solution:
b 60° ? Explanation:
Let ?A = 3x ?, ?B = 4x, ?C = 5x and ?D = 6x.
Since the sum of the angles of a quadrilateral is 360
o
, we have:
3x + 4x + 5x + 6x = 360
o
? 18x = 360
o
? ? x = 20
o
? ?A = 60
o
?, ?B = 80
o
, ?C = 100
o
and ?D = 120
o
Hence, the smallest angle is 60
°.
Question:58
In the given figure, ABCD is a parallelogram in which ?BAD = 75° and ?CBD = 60°. Then, ?BDC = ?
a 60°
b 75°
c 45°
d 50°
Solution:
c
45°
Explanation:
?B = 180
o
- ?A
? ?B = 180
o
- 75
o
= 105
o
Now, ?B = ? ?ABD + ?CBD
? ?? 105
o
? = ?ABD + 60
o
? ?ABD ? = 105
o
- 60
o
= 45
o
? ?ABD = ??BDC ? = 45
o
Alternateangles
Question:59
ABCD is a rhombus such that ?ACB = 50°. Then, ?ADB = ?
a
40°
b
25°
c
65°
d
130°
Solution:
Page 3
Question:56
Three angles of a quadrilateral are 80°, 95° and 112°. Its fourth angle is
a 78°
b 73°
c 85°
d 100°
Solution:
b
73° ? Explanation:
Let the measure of the fourth angle be x
o
.
Since the sum of the angles of a quadrilateral is 360
o
, we have:
80
o
+ 95
o
+ 112
o
+ x = 360
o
? 287
o
+ x = 360
o
? ? x = 73
o
Hence, the measure of the fourth angle is 73
o
.
Question:57
The angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. The smallest of these angles is
a 45°
b 60°
c 36°
d 48°
Solution:
b 60° ? Explanation:
Let ?A = 3x ?, ?B = 4x, ?C = 5x and ?D = 6x.
Since the sum of the angles of a quadrilateral is 360
o
, we have:
3x + 4x + 5x + 6x = 360
o
? 18x = 360
o
? ? x = 20
o
? ?A = 60
o
?, ?B = 80
o
, ?C = 100
o
and ?D = 120
o
Hence, the smallest angle is 60
°.
Question:58
In the given figure, ABCD is a parallelogram in which ?BAD = 75° and ?CBD = 60°. Then, ?BDC = ?
a 60°
b 75°
c 45°
d 50°
Solution:
c
45°
Explanation:
?B = 180
o
- ?A
? ?B = 180
o
- 75
o
= 105
o
Now, ?B = ? ?ABD + ?CBD
? ?? 105
o
? = ?ABD + 60
o
? ?ABD ? = 105
o
- 60
o
= 45
o
? ?ABD = ??BDC ? = 45
o
Alternateangles
Question:59
ABCD is a rhombus such that ?ACB = 50°. Then, ?ADB = ?
a
40°
b
25°
c
65°
d
130°
Solution:
We know that diagonals of rhombus bisect each other at 90°.
Then, in ?BOC,
90° + 50° + ?OBC = 180°
Anglesumpropertyoftriangle
? ?OBC = 180°
- 140°
?
?OBC = 40°
But ?OBC = ?ADB
Alternateinteriorangles
Thus, ?ADB = 40°
Hence, the corerct option is a
.
Question:60
In which of the following figures are the diagonals equal?
a
Parallelogram
b
Rhombus
c
Trapezium
d
Rectangle
Solution:
d
Rectangle.
The diagonals of a rectangle are equal.
Question:61
If the diagonals of a quadrilateral bisect each other at right angles, then the figure is a
a
trapezium
b
parallelogram
c
rectangle
d
rhombus
Solution:
Page 4
Question:56
Three angles of a quadrilateral are 80°, 95° and 112°. Its fourth angle is
a 78°
b 73°
c 85°
d 100°
Solution:
b
73° ? Explanation:
Let the measure of the fourth angle be x
o
.
Since the sum of the angles of a quadrilateral is 360
o
, we have:
80
o
+ 95
o
+ 112
o
+ x = 360
o
? 287
o
+ x = 360
o
? ? x = 73
o
Hence, the measure of the fourth angle is 73
o
.
Question:57
The angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. The smallest of these angles is
a 45°
b 60°
c 36°
d 48°
Solution:
b 60° ? Explanation:
Let ?A = 3x ?, ?B = 4x, ?C = 5x and ?D = 6x.
Since the sum of the angles of a quadrilateral is 360
o
, we have:
3x + 4x + 5x + 6x = 360
o
? 18x = 360
o
? ? x = 20
o
? ?A = 60
o
?, ?B = 80
o
, ?C = 100
o
and ?D = 120
o
Hence, the smallest angle is 60
°.
Question:58
In the given figure, ABCD is a parallelogram in which ?BAD = 75° and ?CBD = 60°. Then, ?BDC = ?
a 60°
b 75°
c 45°
d 50°
Solution:
c
45°
Explanation:
?B = 180
o
- ?A
? ?B = 180
o
- 75
o
= 105
o
Now, ?B = ? ?ABD + ?CBD
? ?? 105
o
? = ?ABD + 60
o
? ?ABD ? = 105
o
- 60
o
= 45
o
? ?ABD = ??BDC ? = 45
o
Alternateangles
Question:59
ABCD is a rhombus such that ?ACB = 50°. Then, ?ADB = ?
a
40°
b
25°
c
65°
d
130°
Solution:
We know that diagonals of rhombus bisect each other at 90°.
Then, in ?BOC,
90° + 50° + ?OBC = 180°
Anglesumpropertyoftriangle
? ?OBC = 180°
- 140°
?
?OBC = 40°
But ?OBC = ?ADB
Alternateinteriorangles
Thus, ?ADB = 40°
Hence, the corerct option is a
.
Question:60
In which of the following figures are the diagonals equal?
a
Parallelogram
b
Rhombus
c
Trapezium
d
Rectangle
Solution:
d
Rectangle.
The diagonals of a rectangle are equal.
Question:61
If the diagonals of a quadrilateral bisect each other at right angles, then the figure is a
a
trapezium
b
parallelogram
c
rectangle
d
rhombus
Solution:
d
rhombus
The diagonals of a rhombus bisect each other at right angles.
Question:62
The lengths of the diagonals of a rhombus are 16 cm and 12 cm. The length of each side of the rhombus is
a
10 cm
b
12 cm
c
9 cm
d
8 cm
Solution:
a
10 cm
Explanation:
Let ABCD be the rhombus.
? AB = BC = CD = DA
Here, AC and BD are the diagonals of ABCD, where AC = 16 cm and BD = 12 cm.
Let the diagonals intersect each other at O.
We know that the diagonals of a rhombus are perpendicular bisectors of each other.
? ? ? ?AOB is a right angle triangle, in which OA = AC /2 = 16/2 = 8 cm and OB = BD/2 = 12/2 = 6 cm.
Now, AB
2
= OA
2
+ OB
2
Pythagorastheorem
? ?AB
2
=
8
2
+
6
2
? ? AB
2
= ? 64 + 36 = 100
? ? AB = ? 10 cm
Hence, the side of the rhombus is 10 cm.
Question:63
The length of each side of a rhombus is 10 cm and one if its diagonals is of length 16 cm. The length of the other
diagonal is
a
13 cm
b
12 cm
c
2
v
39 cm
Page 5
Question:56
Three angles of a quadrilateral are 80°, 95° and 112°. Its fourth angle is
a 78°
b 73°
c 85°
d 100°
Solution:
b
73° ? Explanation:
Let the measure of the fourth angle be x
o
.
Since the sum of the angles of a quadrilateral is 360
o
, we have:
80
o
+ 95
o
+ 112
o
+ x = 360
o
? 287
o
+ x = 360
o
? ? x = 73
o
Hence, the measure of the fourth angle is 73
o
.
Question:57
The angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. The smallest of these angles is
a 45°
b 60°
c 36°
d 48°
Solution:
b 60° ? Explanation:
Let ?A = 3x ?, ?B = 4x, ?C = 5x and ?D = 6x.
Since the sum of the angles of a quadrilateral is 360
o
, we have:
3x + 4x + 5x + 6x = 360
o
? 18x = 360
o
? ? x = 20
o
? ?A = 60
o
?, ?B = 80
o
, ?C = 100
o
and ?D = 120
o
Hence, the smallest angle is 60
°.
Question:58
In the given figure, ABCD is a parallelogram in which ?BAD = 75° and ?CBD = 60°. Then, ?BDC = ?
a 60°
b 75°
c 45°
d 50°
Solution:
c
45°
Explanation:
?B = 180
o
- ?A
? ?B = 180
o
- 75
o
= 105
o
Now, ?B = ? ?ABD + ?CBD
? ?? 105
o
? = ?ABD + 60
o
? ?ABD ? = 105
o
- 60
o
= 45
o
? ?ABD = ??BDC ? = 45
o
Alternateangles
Question:59
ABCD is a rhombus such that ?ACB = 50°. Then, ?ADB = ?
a
40°
b
25°
c
65°
d
130°
Solution:
We know that diagonals of rhombus bisect each other at 90°.
Then, in ?BOC,
90° + 50° + ?OBC = 180°
Anglesumpropertyoftriangle
? ?OBC = 180°
- 140°
?
?OBC = 40°
But ?OBC = ?ADB
Alternateinteriorangles
Thus, ?ADB = 40°
Hence, the corerct option is a
.
Question:60
In which of the following figures are the diagonals equal?
a
Parallelogram
b
Rhombus
c
Trapezium
d
Rectangle
Solution:
d
Rectangle.
The diagonals of a rectangle are equal.
Question:61
If the diagonals of a quadrilateral bisect each other at right angles, then the figure is a
a
trapezium
b
parallelogram
c
rectangle
d
rhombus
Solution:
d
rhombus
The diagonals of a rhombus bisect each other at right angles.
Question:62
The lengths of the diagonals of a rhombus are 16 cm and 12 cm. The length of each side of the rhombus is
a
10 cm
b
12 cm
c
9 cm
d
8 cm
Solution:
a
10 cm
Explanation:
Let ABCD be the rhombus.
? AB = BC = CD = DA
Here, AC and BD are the diagonals of ABCD, where AC = 16 cm and BD = 12 cm.
Let the diagonals intersect each other at O.
We know that the diagonals of a rhombus are perpendicular bisectors of each other.
? ? ? ?AOB is a right angle triangle, in which OA = AC /2 = 16/2 = 8 cm and OB = BD/2 = 12/2 = 6 cm.
Now, AB
2
= OA
2
+ OB
2
Pythagorastheorem
? ?AB
2
=
8
2
+
6
2
? ? AB
2
= ? 64 + 36 = 100
? ? AB = ? 10 cm
Hence, the side of the rhombus is 10 cm.
Question:63
The length of each side of a rhombus is 10 cm and one if its diagonals is of length 16 cm. The length of the other
diagonal is
a
13 cm
b
12 cm
c
2
v
39 cm
d
6 cm
Solution:
b
12 cm
Explanation:
Let ABCD be the rhombus.
? AB = BC = CD = DA = 10 cm
Let AC and BD be the diagonals of the rhombus.
Let AC be x and BD be 16 cm and O be the intersection point of the diagonals.
We know that the diagonals of a rhombus are perpendicular bisectors of each other.
? ? ?AOB is a right angle triangle in which OA = AC
÷
2 = x
÷
2 and OB = BD
÷
2 = 16
÷
2 = 8 cm.
Now, AB
2
= OA
2
+ OB
2
Pythagorastheorem
? 10
2
=
x
2
2
+ 8
2
?
x
2
2
= 36 = 6
2
? x = 2 ×6 = 12 cm
Question:64
A diagonal of a rectangle is inclined to one side of the rectangle at 35°. The acute angle between the diagonals is
a
55°
b
70°
c
45°
d
50°
Solution:
Given: In rectangle ABCD, ?OAD = 35°.
÷
÷
÷
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