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Page 1
?
Question:53
If one angle of a triangle is equal to the sum of the other two angles, then the triangle is
a
an isosceles triangle
b
an obtuse triangle
c
an equilateral triangle
d
a right triangle
Solution:
Let ?ABC be such that ?A = ?B + ?C.
In ?ABC,
?A + ?B + ?C = 180º Anglesumproperty
? ?A + ?A = 180º ?A = ?B + ?C
? 2 ?A = 180º
? ?A = 90º
Therefore, ?ABC is a right triangle.
Thus, if one angle of a triangle is equal to the sum of the other two angles, then the triangle is a right triangle.
Hence, the correct answer is option d
.
Question:54
An exterior angle of a triangle is 110° and its two interior opposite angles are equal. Each of these equal angles is
a
70°
b
55°
c
35°
d
27
1°
2
Solution:
Let the measure of each of the two equal interior opposite angles of the triangle be x.
In a triangle, the exterior angle is equal to the sum of the two interior opposite angles.
? x + x = 110°
Page 2
?
Question:53
If one angle of a triangle is equal to the sum of the other two angles, then the triangle is
a
an isosceles triangle
b
an obtuse triangle
c
an equilateral triangle
d
a right triangle
Solution:
Let ?ABC be such that ?A = ?B + ?C.
In ?ABC,
?A + ?B + ?C = 180º Anglesumproperty
? ?A + ?A = 180º ?A = ?B + ?C
? 2 ?A = 180º
? ?A = 90º
Therefore, ?ABC is a right triangle.
Thus, if one angle of a triangle is equal to the sum of the other two angles, then the triangle is a right triangle.
Hence, the correct answer is option d
.
Question:54
An exterior angle of a triangle is 110° and its two interior opposite angles are equal. Each of these equal angles is
a
70°
b
55°
c
35°
d
27
1°
2
Solution:
Let the measure of each of the two equal interior opposite angles of the triangle be x.
In a triangle, the exterior angle is equal to the sum of the two interior opposite angles.
? x + x = 110°
? 2x = 110°
? x = 55°
Thus, the measure of each of these equal angles is 55°.
Hence, the correct answer is option b
.
Question:55
The angles of a triangle are in the ration 3 : 5 : 7. The triangle is
a acute-angled
b obtuse-angled
c right-angled
d an isosceles triangle
Solution:
a acute-angled
Let the angles measure (3x)°, (5x)° and (7x)°
.
Then,
3x +5x +7x = 180° ? 15x = 180° ? x = 12°
Therefore, the angles are 3(12)° = 36°, 5(12)° = 60° and 7(12)° = 84°
.
Hence, the triangle is acute-angled.
Question:56
If one of the angles of a triangle is 130° then the angle between the bisectors of the other two angles can be
a
50°
b
65°
c
90°
d
155°
Solution:
Let ?ABC be such that ?A = 130°.
Here, BP is the bisector of ?B and CP is the bisector of ?C.
? ?ABP = ?PBC =
1
2
?B .....1
Also, ?ACP = ?PCB =
1
2
?C .....2
In ?ABC,
?A + ?B + ?C = 180° Anglesumproperty
? 130° + ?B + ?C = 180°
? ?B + ?C = 180° - 130° = 50°
?
1
2
?B +
1
2
?C =
1
2
× 50° = 25°
? ?PBC + ?PCB = 25° .....3
Using(1)and(2)
In ?PBC,
?PBC + ?PCB + ?BPC = 180° Anglesumproperty
? 25° + ?BPC = 180°
Page 3
?
Question:53
If one angle of a triangle is equal to the sum of the other two angles, then the triangle is
a
an isosceles triangle
b
an obtuse triangle
c
an equilateral triangle
d
a right triangle
Solution:
Let ?ABC be such that ?A = ?B + ?C.
In ?ABC,
?A + ?B + ?C = 180º Anglesumproperty
? ?A + ?A = 180º ?A = ?B + ?C
? 2 ?A = 180º
? ?A = 90º
Therefore, ?ABC is a right triangle.
Thus, if one angle of a triangle is equal to the sum of the other two angles, then the triangle is a right triangle.
Hence, the correct answer is option d
.
Question:54
An exterior angle of a triangle is 110° and its two interior opposite angles are equal. Each of these equal angles is
a
70°
b
55°
c
35°
d
27
1°
2
Solution:
Let the measure of each of the two equal interior opposite angles of the triangle be x.
In a triangle, the exterior angle is equal to the sum of the two interior opposite angles.
? x + x = 110°
? 2x = 110°
? x = 55°
Thus, the measure of each of these equal angles is 55°.
Hence, the correct answer is option b
.
Question:55
The angles of a triangle are in the ration 3 : 5 : 7. The triangle is
a acute-angled
b obtuse-angled
c right-angled
d an isosceles triangle
Solution:
a acute-angled
Let the angles measure (3x)°, (5x)° and (7x)°
.
Then,
3x +5x +7x = 180° ? 15x = 180° ? x = 12°
Therefore, the angles are 3(12)° = 36°, 5(12)° = 60° and 7(12)° = 84°
.
Hence, the triangle is acute-angled.
Question:56
If one of the angles of a triangle is 130° then the angle between the bisectors of the other two angles can be
a
50°
b
65°
c
90°
d
155°
Solution:
Let ?ABC be such that ?A = 130°.
Here, BP is the bisector of ?B and CP is the bisector of ?C.
? ?ABP = ?PBC =
1
2
?B .....1
Also, ?ACP = ?PCB =
1
2
?C .....2
In ?ABC,
?A + ?B + ?C = 180° Anglesumproperty
? 130° + ?B + ?C = 180°
? ?B + ?C = 180° - 130° = 50°
?
1
2
?B +
1
2
?C =
1
2
× 50° = 25°
? ?PBC + ?PCB = 25° .....3
Using(1)and(2)
In ?PBC,
?PBC + ?PCB + ?BPC = 180° Anglesumproperty
? 25° + ?BPC = 180°
Using(3)
? ?BPC = 180° - 25° = 155°
Thus, if one of the angles of a triangle is 130° then the angle between the bisectors of the other two angles is 155°.
Hence, the correct answer is option d
.
Question:57
In the given figure, AOB is a straight line. The value of x is
a
12
b
15
c
20
d
25
Solution:
It is given that, AOB is a straight line.
? 60º + (5xº + 3xº) = 180º Linearpair
? 8xº = 180º - 60º = 120º
? xº = 15º
Thus, the value of x is 15.
Hence, the correct answer is option b
.
Question:58
The angles of a triangle are in the ratio 2 : 3 : 4. The largest angle of the triangle is
a
12°
b
100°
c
80°
d
60°
Solution:
Suppose ?ABC be such that ?A : ?B : ?C = 2 : 3 : 4.
Let ?A = 2k, ?B = 3k and ?C = 4k, where k is some constant.
In ?ABC,
?A + ?B + ?C = 180º Anglesumproperty
? 2k + 3k + 4k = 180º
? 9k = 180º
? k = 20º
? Measure of the largest angle = 4k = 4 × 20º = 80º
Hence, the correct answer is option c
.
Question:59
In the given figure, ?OAB = 110° and ?BCD = 130° then ?ABC is equal to
a
40°
b
50°
c
60°
Page 4
?
Question:53
If one angle of a triangle is equal to the sum of the other two angles, then the triangle is
a
an isosceles triangle
b
an obtuse triangle
c
an equilateral triangle
d
a right triangle
Solution:
Let ?ABC be such that ?A = ?B + ?C.
In ?ABC,
?A + ?B + ?C = 180º Anglesumproperty
? ?A + ?A = 180º ?A = ?B + ?C
? 2 ?A = 180º
? ?A = 90º
Therefore, ?ABC is a right triangle.
Thus, if one angle of a triangle is equal to the sum of the other two angles, then the triangle is a right triangle.
Hence, the correct answer is option d
.
Question:54
An exterior angle of a triangle is 110° and its two interior opposite angles are equal. Each of these equal angles is
a
70°
b
55°
c
35°
d
27
1°
2
Solution:
Let the measure of each of the two equal interior opposite angles of the triangle be x.
In a triangle, the exterior angle is equal to the sum of the two interior opposite angles.
? x + x = 110°
? 2x = 110°
? x = 55°
Thus, the measure of each of these equal angles is 55°.
Hence, the correct answer is option b
.
Question:55
The angles of a triangle are in the ration 3 : 5 : 7. The triangle is
a acute-angled
b obtuse-angled
c right-angled
d an isosceles triangle
Solution:
a acute-angled
Let the angles measure (3x)°, (5x)° and (7x)°
.
Then,
3x +5x +7x = 180° ? 15x = 180° ? x = 12°
Therefore, the angles are 3(12)° = 36°, 5(12)° = 60° and 7(12)° = 84°
.
Hence, the triangle is acute-angled.
Question:56
If one of the angles of a triangle is 130° then the angle between the bisectors of the other two angles can be
a
50°
b
65°
c
90°
d
155°
Solution:
Let ?ABC be such that ?A = 130°.
Here, BP is the bisector of ?B and CP is the bisector of ?C.
? ?ABP = ?PBC =
1
2
?B .....1
Also, ?ACP = ?PCB =
1
2
?C .....2
In ?ABC,
?A + ?B + ?C = 180° Anglesumproperty
? 130° + ?B + ?C = 180°
? ?B + ?C = 180° - 130° = 50°
?
1
2
?B +
1
2
?C =
1
2
× 50° = 25°
? ?PBC + ?PCB = 25° .....3
Using(1)and(2)
In ?PBC,
?PBC + ?PCB + ?BPC = 180° Anglesumproperty
? 25° + ?BPC = 180°
Using(3)
? ?BPC = 180° - 25° = 155°
Thus, if one of the angles of a triangle is 130° then the angle between the bisectors of the other two angles is 155°.
Hence, the correct answer is option d
.
Question:57
In the given figure, AOB is a straight line. The value of x is
a
12
b
15
c
20
d
25
Solution:
It is given that, AOB is a straight line.
? 60º + (5xº + 3xº) = 180º Linearpair
? 8xº = 180º - 60º = 120º
? xº = 15º
Thus, the value of x is 15.
Hence, the correct answer is option b
.
Question:58
The angles of a triangle are in the ratio 2 : 3 : 4. The largest angle of the triangle is
a
12°
b
100°
c
80°
d
60°
Solution:
Suppose ?ABC be such that ?A : ?B : ?C = 2 : 3 : 4.
Let ?A = 2k, ?B = 3k and ?C = 4k, where k is some constant.
In ?ABC,
?A + ?B + ?C = 180º Anglesumproperty
? 2k + 3k + 4k = 180º
? 9k = 180º
? k = 20º
? Measure of the largest angle = 4k = 4 × 20º = 80º
Hence, the correct answer is option c
.
Question:59
In the given figure, ?OAB = 110° and ?BCD = 130° then ?ABC is equal to
a
40°
b
50°
c
60°
d
70°
Solution:
In the given figure, OA || CD.
Construction: Extend OA such that it intersects BC at E.
Now, OE || CD and BC is a transversal.
? ?AEC = ?BCD = 130° Pairofcorrespondingangles
Also, ?OAB + ?BAE = 180° Linearpair
? 110° + ?BAE = 180°
? ?BAE = 180° - 110° = 70°
In ?ABE,
?AEC = ?BAE + ?ABE Inatriangle, exteriorangleisequaltothesumoftwooppositeinteriorangles
? 130° = 70° + x°
? x° = 130° - 70° = 60°
Thus, the measure of angle ?ABC is 60°.
Hence, the correct answer is option c
.
Question:60
If two angles are complements of each other, then each angle is
a
an acute angle
b
an obtuse angle
c
a right angle
d
a reflex angle
Solution:
a
an acute angle
If two angles are complements of each other, that is, the sum of their measures is 90°
, then each angle is an acute angle.
Question:61
An angle which measures more than 180° but less than 360°, is called
a
an acute angle
b
an obtuse angle
c
a straight angle
d
a reflex angle
Solution:
An angle which measures more than 180° but less than 360° is called a reflex angle.
Hence, the correct answer is option d
.
Question:62
The measure of an angle is five times its complement. The angle measures
a
25°
b
35°
c
65°
d
75°
Solution:
d
75°
Page 5
?
Question:53
If one angle of a triangle is equal to the sum of the other two angles, then the triangle is
a
an isosceles triangle
b
an obtuse triangle
c
an equilateral triangle
d
a right triangle
Solution:
Let ?ABC be such that ?A = ?B + ?C.
In ?ABC,
?A + ?B + ?C = 180º Anglesumproperty
? ?A + ?A = 180º ?A = ?B + ?C
? 2 ?A = 180º
? ?A = 90º
Therefore, ?ABC is a right triangle.
Thus, if one angle of a triangle is equal to the sum of the other two angles, then the triangle is a right triangle.
Hence, the correct answer is option d
.
Question:54
An exterior angle of a triangle is 110° and its two interior opposite angles are equal. Each of these equal angles is
a
70°
b
55°
c
35°
d
27
1°
2
Solution:
Let the measure of each of the two equal interior opposite angles of the triangle be x.
In a triangle, the exterior angle is equal to the sum of the two interior opposite angles.
? x + x = 110°
? 2x = 110°
? x = 55°
Thus, the measure of each of these equal angles is 55°.
Hence, the correct answer is option b
.
Question:55
The angles of a triangle are in the ration 3 : 5 : 7. The triangle is
a acute-angled
b obtuse-angled
c right-angled
d an isosceles triangle
Solution:
a acute-angled
Let the angles measure (3x)°, (5x)° and (7x)°
.
Then,
3x +5x +7x = 180° ? 15x = 180° ? x = 12°
Therefore, the angles are 3(12)° = 36°, 5(12)° = 60° and 7(12)° = 84°
.
Hence, the triangle is acute-angled.
Question:56
If one of the angles of a triangle is 130° then the angle between the bisectors of the other two angles can be
a
50°
b
65°
c
90°
d
155°
Solution:
Let ?ABC be such that ?A = 130°.
Here, BP is the bisector of ?B and CP is the bisector of ?C.
? ?ABP = ?PBC =
1
2
?B .....1
Also, ?ACP = ?PCB =
1
2
?C .....2
In ?ABC,
?A + ?B + ?C = 180° Anglesumproperty
? 130° + ?B + ?C = 180°
? ?B + ?C = 180° - 130° = 50°
?
1
2
?B +
1
2
?C =
1
2
× 50° = 25°
? ?PBC + ?PCB = 25° .....3
Using(1)and(2)
In ?PBC,
?PBC + ?PCB + ?BPC = 180° Anglesumproperty
? 25° + ?BPC = 180°
Using(3)
? ?BPC = 180° - 25° = 155°
Thus, if one of the angles of a triangle is 130° then the angle between the bisectors of the other two angles is 155°.
Hence, the correct answer is option d
.
Question:57
In the given figure, AOB is a straight line. The value of x is
a
12
b
15
c
20
d
25
Solution:
It is given that, AOB is a straight line.
? 60º + (5xº + 3xº) = 180º Linearpair
? 8xº = 180º - 60º = 120º
? xº = 15º
Thus, the value of x is 15.
Hence, the correct answer is option b
.
Question:58
The angles of a triangle are in the ratio 2 : 3 : 4. The largest angle of the triangle is
a
12°
b
100°
c
80°
d
60°
Solution:
Suppose ?ABC be such that ?A : ?B : ?C = 2 : 3 : 4.
Let ?A = 2k, ?B = 3k and ?C = 4k, where k is some constant.
In ?ABC,
?A + ?B + ?C = 180º Anglesumproperty
? 2k + 3k + 4k = 180º
? 9k = 180º
? k = 20º
? Measure of the largest angle = 4k = 4 × 20º = 80º
Hence, the correct answer is option c
.
Question:59
In the given figure, ?OAB = 110° and ?BCD = 130° then ?ABC is equal to
a
40°
b
50°
c
60°
d
70°
Solution:
In the given figure, OA || CD.
Construction: Extend OA such that it intersects BC at E.
Now, OE || CD and BC is a transversal.
? ?AEC = ?BCD = 130° Pairofcorrespondingangles
Also, ?OAB + ?BAE = 180° Linearpair
? 110° + ?BAE = 180°
? ?BAE = 180° - 110° = 70°
In ?ABE,
?AEC = ?BAE + ?ABE Inatriangle, exteriorangleisequaltothesumoftwooppositeinteriorangles
? 130° = 70° + x°
? x° = 130° - 70° = 60°
Thus, the measure of angle ?ABC is 60°.
Hence, the correct answer is option c
.
Question:60
If two angles are complements of each other, then each angle is
a
an acute angle
b
an obtuse angle
c
a right angle
d
a reflex angle
Solution:
a
an acute angle
If two angles are complements of each other, that is, the sum of their measures is 90°
, then each angle is an acute angle.
Question:61
An angle which measures more than 180° but less than 360°, is called
a
an acute angle
b
an obtuse angle
c
a straight angle
d
a reflex angle
Solution:
An angle which measures more than 180° but less than 360° is called a reflex angle.
Hence, the correct answer is option d
.
Question:62
The measure of an angle is five times its complement. The angle measures
a
25°
b
35°
c
65°
d
75°
Solution:
d
75°
Let the measure of the required angle be x°
.
Then, the measure of its complement will be (90 -x)°
.
? x = 5(90 -x) ? x = 450 -5x ? 6x = 450 ? x = 75
Question:63
Two complementary angles are such that twice the measure of one is equal to three times the measure of the other. The measure of larger angle is
a
72°
b
54°
c
63°
d
36°
Solution:
b
54°
Let the measure of the required angle be x°
.
Then, the measure of its complement will be(90 -x)°
.
? 2x = 3(90 -x) ? 2x = 270 -3x ? 5x = 270 ? x = 54
Question:64
In the given figure, AOB is a straight line. If ?AOC = 4x° and ?BOC = 5x°, then ?AOC = ?
a
40°
b
60°
c
80°
d
100°
Solution:
c
80°
We have :
?AOC + ?BOC = 180° [Since AOB is a straight line] ? 4x +5x = 180° ? 9x = 180° ? x = 20° ? ?AOC = 4 ×20° = 80°
Question:65
In the given figure, AOB is a straight line. If ?AOC = (3x + 10)° and ?BOC (4x - 26)°, then ?BOC = ?
a
96°
b
86°
c
76°
d
106°
Solution:
b
86°
We have :
?AOC + ?BOC = 180° [Since AOB is a straight line ] ? 3x +10 +4x -26 = 180° ? 7x = 196° ? x = 28°
? ?BOC = [4 ×28 -26]
°
Hence, ?BOC = 86°.
Question:66
In the given figure, AOB is a straight line. If ?AOC = (3x - 10)°, ?COD = 50° and ?BOD = (x + 20)°, then ?AOC = ?
a 40°
b 60°
c 80°
d 50°
Solution:
c 80°
We have :
?AOC + ?COD + ?BOD = 180° [Since AOB is a straight line ] ? 3x -10 +50 +x +20 = 180 ? 4x = 120 ? x = 30
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