Page 1
Question:1
Define the following terms:
i
Angle
ii
Interior of an angle
iii
Obtuse angle
iv
Reflex angle
v
Complementary angles
vi
Supplementary angles
Solution:
i
Two rays OA and OB, with a common end-point O, form an angle AOB that is represented as ?AOB
.
ii
The interior of an angle is the set of all points in its plane, which lie on the same side of OA as B and also on the same side of OB as A.
iii
An angle greater than 90°
but less than 180°
is called an obtuse angle.
iv
An angle greater than 180°
but less than 360°
is called a reflex angle.
v
Two angles are said to be complementary if the sum of their measures is 90°
.
vi
Two angles are said to be supplementary if the sum of their measures is 180°
.
Question:2
Find the complement of each of the following angles.
i
55°
ii
16°
iii
iv
2
3
of a right angle
Solution:
Two angles whose sum is 90° are called complementary angles.
i
Complement of 55° = 90° - 55° = 35°
ii
Complement of 16° = (90 -16)°
= 74°
90°
Page 2
Question:1
Define the following terms:
i
Angle
ii
Interior of an angle
iii
Obtuse angle
iv
Reflex angle
v
Complementary angles
vi
Supplementary angles
Solution:
i
Two rays OA and OB, with a common end-point O, form an angle AOB that is represented as ?AOB
.
ii
The interior of an angle is the set of all points in its plane, which lie on the same side of OA as B and also on the same side of OB as A.
iii
An angle greater than 90°
but less than 180°
is called an obtuse angle.
iv
An angle greater than 180°
but less than 360°
is called a reflex angle.
v
Two angles are said to be complementary if the sum of their measures is 90°
.
vi
Two angles are said to be supplementary if the sum of their measures is 180°
.
Question:2
Find the complement of each of the following angles.
i
55°
ii
16°
iii
iv
2
3
of a right angle
Solution:
Two angles whose sum is 90° are called complementary angles.
i
Complement of 55° = 90° - 55° = 35°
ii
Complement of 16° = (90 -16)°
= 74°
90°
iii
Complement of 90° = 90° - 90° = 0°
iv
2
3
of a right angle ? 90 ×
2
3
° = 60°
Complement of
2
3
of a right angle = (90 -60)° = 30°
Question:3
Find the supplement of each of the following angles.
i
42°
ii
90°
iii
124°
iv
3
5
of a right angle
Solution:
Two angles whose sum is 180° are called supplementary angles.
i
Supplement of 42° = 180° - 42° = 138°
ii
Supplement of 90° = 180° - 90° = 90°
iii
Supplement of 124° = 180° - 124° = 56°
iv
3
5
of a right angle ?
3
5
×90 = 54°
Supplement of
3
5
of a right angle = (180 -54)° = 126°
Question:4
Find the measure of an angle which is
i
equal to its complement
ii
equal to its supplement
Solution:
i
Let the measure of the required angle be x°
.
Then, in case of complementary angles:
x +x = 90° ? 2x = 90° ? x = 45°
Hence, measure of the angle that is equal to its complement is 45°
.
ii
Let the measure of the required angle be x°
?.
Then, in case of supplementary angles:
x +x = 180° ? 2x = 180° ? x = 90°
Hence, measure of the angle that is equal to its supplement is 90°
.
Question:5
Find the measure of an angle which is 36° more than its complement.
Solution:
Let the measure of the required angle be x°
.
Then, measure of its complement = (90 -x)°
.
Therefore,
x -(90°-x) = 36° ? 2x = 126° ? x = 63°
Hence, the measure of the required angle is 63°
.
Question:6
Find the measure of an angle which is 30° less than its supplement.
Solution:
Let the measure of the angle be x°.
? Supplement of x° = 180° - x°
It is given that,
( )
( )
Page 3
Question:1
Define the following terms:
i
Angle
ii
Interior of an angle
iii
Obtuse angle
iv
Reflex angle
v
Complementary angles
vi
Supplementary angles
Solution:
i
Two rays OA and OB, with a common end-point O, form an angle AOB that is represented as ?AOB
.
ii
The interior of an angle is the set of all points in its plane, which lie on the same side of OA as B and also on the same side of OB as A.
iii
An angle greater than 90°
but less than 180°
is called an obtuse angle.
iv
An angle greater than 180°
but less than 360°
is called a reflex angle.
v
Two angles are said to be complementary if the sum of their measures is 90°
.
vi
Two angles are said to be supplementary if the sum of their measures is 180°
.
Question:2
Find the complement of each of the following angles.
i
55°
ii
16°
iii
iv
2
3
of a right angle
Solution:
Two angles whose sum is 90° are called complementary angles.
i
Complement of 55° = 90° - 55° = 35°
ii
Complement of 16° = (90 -16)°
= 74°
90°
iii
Complement of 90° = 90° - 90° = 0°
iv
2
3
of a right angle ? 90 ×
2
3
° = 60°
Complement of
2
3
of a right angle = (90 -60)° = 30°
Question:3
Find the supplement of each of the following angles.
i
42°
ii
90°
iii
124°
iv
3
5
of a right angle
Solution:
Two angles whose sum is 180° are called supplementary angles.
i
Supplement of 42° = 180° - 42° = 138°
ii
Supplement of 90° = 180° - 90° = 90°
iii
Supplement of 124° = 180° - 124° = 56°
iv
3
5
of a right angle ?
3
5
×90 = 54°
Supplement of
3
5
of a right angle = (180 -54)° = 126°
Question:4
Find the measure of an angle which is
i
equal to its complement
ii
equal to its supplement
Solution:
i
Let the measure of the required angle be x°
.
Then, in case of complementary angles:
x +x = 90° ? 2x = 90° ? x = 45°
Hence, measure of the angle that is equal to its complement is 45°
.
ii
Let the measure of the required angle be x°
?.
Then, in case of supplementary angles:
x +x = 180° ? 2x = 180° ? x = 90°
Hence, measure of the angle that is equal to its supplement is 90°
.
Question:5
Find the measure of an angle which is 36° more than its complement.
Solution:
Let the measure of the required angle be x°
.
Then, measure of its complement = (90 -x)°
.
Therefore,
x -(90°-x) = 36° ? 2x = 126° ? x = 63°
Hence, the measure of the required angle is 63°
.
Question:6
Find the measure of an angle which is 30° less than its supplement.
Solution:
Let the measure of the angle be x°.
? Supplement of x° = 180° - x°
It is given that,
( )
( )
(180° - x°) - x° = 30°
? 180° - 2x°= 30°
? 2x° = 180° - 30° = 150°
? x° = 75°
Thus, the measure of the angle is 75°.
Question:7
Find the angle which is four times its complement.
Solution:
Let the measure of the required angle be x
.
Then, measure of its complement = (90°-x)
.
Therefore,
x = (90°-x)4 ? x = 360°-4x ? 5x = 360° ? x = 72°
Hence, the measure of the required angle is 72°
.
Question:8
Find the angle which is five times its supplement.
Solution:
Let the measure of the required angle be x
.
Then, measure of its supplement = (180°-x)
.
Therefore,
x = (180°-x)5 ? x = 900°-5x ? 6x = 900° ? x = 150°
Hence, the measure of the required angle is 150°
.
Question:9
Find the angle whose supplement is four times its complement.
Solution:
Let the measure of the required angle be x°
.
Then, measure of its complement = (90 -x)°
.
And, measure of its supplement = (180 -x)°
.
Therefore,
(180 -x) = 4(90 -x) ? 180 -x = 360 -4x ? 3x = 180 ? x = 60
Hence, the measure of the required angle is 60°
.
Question:10
Find the angle whose complement is one-third of its supplement.
Solution:
Let the measure of the required angle be x°
.
Then, the measure of its complement = (90 -x)°
.
And the measure of its supplement = (180 -x)°
.
Therefore,
(90 -x) =
1
3
(180 -x) ? 3(90 -x) = (180 -x) ? 270 -3x = 180 -x ? 2x = 90 ? x = 45
Hence, the measure of the required angle is 45°
.
Question:11
Two complementary angles are in the ratio 4 : 5. Find the angles.
Solution:
Let the two angles be 4x and 5x, respectively.
Then,
4x +5x = 90 ? 9x = 90 ? x = 10°
Hence, the two angles are 4x = 4 ×10° = 40° and 5x = 5 ×10° = 50°
.
Question:12
Find the value of x for which the angles (2x – 5)° and (x – 10)° are the complementary angles.
Solution:
Two angles whose sum is 90° are called complementary angles.
It is given that the angles (2x – 5)° and (x – 10)° are the complementary angles.
? (2x – 5)° + (x – 10)° = 90°
? 3x° – 15° = 90°
Page 4
Question:1
Define the following terms:
i
Angle
ii
Interior of an angle
iii
Obtuse angle
iv
Reflex angle
v
Complementary angles
vi
Supplementary angles
Solution:
i
Two rays OA and OB, with a common end-point O, form an angle AOB that is represented as ?AOB
.
ii
The interior of an angle is the set of all points in its plane, which lie on the same side of OA as B and also on the same side of OB as A.
iii
An angle greater than 90°
but less than 180°
is called an obtuse angle.
iv
An angle greater than 180°
but less than 360°
is called a reflex angle.
v
Two angles are said to be complementary if the sum of their measures is 90°
.
vi
Two angles are said to be supplementary if the sum of their measures is 180°
.
Question:2
Find the complement of each of the following angles.
i
55°
ii
16°
iii
iv
2
3
of a right angle
Solution:
Two angles whose sum is 90° are called complementary angles.
i
Complement of 55° = 90° - 55° = 35°
ii
Complement of 16° = (90 -16)°
= 74°
90°
iii
Complement of 90° = 90° - 90° = 0°
iv
2
3
of a right angle ? 90 ×
2
3
° = 60°
Complement of
2
3
of a right angle = (90 -60)° = 30°
Question:3
Find the supplement of each of the following angles.
i
42°
ii
90°
iii
124°
iv
3
5
of a right angle
Solution:
Two angles whose sum is 180° are called supplementary angles.
i
Supplement of 42° = 180° - 42° = 138°
ii
Supplement of 90° = 180° - 90° = 90°
iii
Supplement of 124° = 180° - 124° = 56°
iv
3
5
of a right angle ?
3
5
×90 = 54°
Supplement of
3
5
of a right angle = (180 -54)° = 126°
Question:4
Find the measure of an angle which is
i
equal to its complement
ii
equal to its supplement
Solution:
i
Let the measure of the required angle be x°
.
Then, in case of complementary angles:
x +x = 90° ? 2x = 90° ? x = 45°
Hence, measure of the angle that is equal to its complement is 45°
.
ii
Let the measure of the required angle be x°
?.
Then, in case of supplementary angles:
x +x = 180° ? 2x = 180° ? x = 90°
Hence, measure of the angle that is equal to its supplement is 90°
.
Question:5
Find the measure of an angle which is 36° more than its complement.
Solution:
Let the measure of the required angle be x°
.
Then, measure of its complement = (90 -x)°
.
Therefore,
x -(90°-x) = 36° ? 2x = 126° ? x = 63°
Hence, the measure of the required angle is 63°
.
Question:6
Find the measure of an angle which is 30° less than its supplement.
Solution:
Let the measure of the angle be x°.
? Supplement of x° = 180° - x°
It is given that,
( )
( )
(180° - x°) - x° = 30°
? 180° - 2x°= 30°
? 2x° = 180° - 30° = 150°
? x° = 75°
Thus, the measure of the angle is 75°.
Question:7
Find the angle which is four times its complement.
Solution:
Let the measure of the required angle be x
.
Then, measure of its complement = (90°-x)
.
Therefore,
x = (90°-x)4 ? x = 360°-4x ? 5x = 360° ? x = 72°
Hence, the measure of the required angle is 72°
.
Question:8
Find the angle which is five times its supplement.
Solution:
Let the measure of the required angle be x
.
Then, measure of its supplement = (180°-x)
.
Therefore,
x = (180°-x)5 ? x = 900°-5x ? 6x = 900° ? x = 150°
Hence, the measure of the required angle is 150°
.
Question:9
Find the angle whose supplement is four times its complement.
Solution:
Let the measure of the required angle be x°
.
Then, measure of its complement = (90 -x)°
.
And, measure of its supplement = (180 -x)°
.
Therefore,
(180 -x) = 4(90 -x) ? 180 -x = 360 -4x ? 3x = 180 ? x = 60
Hence, the measure of the required angle is 60°
.
Question:10
Find the angle whose complement is one-third of its supplement.
Solution:
Let the measure of the required angle be x°
.
Then, the measure of its complement = (90 -x)°
.
And the measure of its supplement = (180 -x)°
.
Therefore,
(90 -x) =
1
3
(180 -x) ? 3(90 -x) = (180 -x) ? 270 -3x = 180 -x ? 2x = 90 ? x = 45
Hence, the measure of the required angle is 45°
.
Question:11
Two complementary angles are in the ratio 4 : 5. Find the angles.
Solution:
Let the two angles be 4x and 5x, respectively.
Then,
4x +5x = 90 ? 9x = 90 ? x = 10°
Hence, the two angles are 4x = 4 ×10° = 40° and 5x = 5 ×10° = 50°
.
Question:12
Find the value of x for which the angles (2x – 5)° and (x – 10)° are the complementary angles.
Solution:
Two angles whose sum is 90° are called complementary angles.
It is given that the angles (2x – 5)° and (x – 10)° are the complementary angles.
? (2x – 5)° + (x – 10)° = 90°
? 3x° – 15° = 90°
? 3x° = 90° + 15° = 105°
? x° =
105°
3
= 35°
Thus, the value of x is 35.
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