Class 9 Exam > Class 9 Notes > Mathematics (Maths) Class 9 > RS Aggarwal Solutions: Lines and Angles- 1
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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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FAQs on RS Aggarwal Solutions: Lines and Angles- 1 - Mathematics (Maths) Class 9
| 1. What are lines and angles? |  |
Ans. Lines and angles are fundamental concepts in geometry. A line is a straight path that extends indefinitely in both directions. An angle is formed when two rays share a common endpoint called the vertex. It is measured in degrees and can be acute, right, obtuse, or straight.
| 2. How do you classify angles based on their measures? | |
Ans. Angles can be classified based on their measures as follows:
- Acute Angle: An angle that measures less than 90 degrees.
- Right Angle: An angle that measures exactly 90 degrees.
- Obtuse Angle: An angle that measures more than 90 degrees but less than 180 degrees.
- Straight Angle: An angle that measures exactly 180 degrees.
| 3. What are the different types of angles based on their relationships with each other? | |
Ans. There are several types of angles based on their relationships with each other:
- Adjacent Angles: Angles that share a common vertex and a common side but do not overlap.
- Vertical Angles: Pairs of opposite angles formed by two intersecting lines. They have equal measures.
- Supplementary Angles: Two angles whose measures add up to 180 degrees.
- Complementary Angles: Two angles whose measures add up to 90 degrees.
| 4. How can lines be classified based on their relationships with each other? | |
Ans. Lines can be classified based on their relationships with each other as follows:
- Parallel Lines: Lines that are always the same distance apart and never intersect.
- Intersecting Lines: Lines that cross each other at a point.
- Perpendicular Lines: Lines that intersect at a right angle (90 degrees).
- Skew Lines: Lines that are not in the same plane and do not intersect.
| 5. How can we find the measure of an unknown angle using the given information about other angles? | |
Ans. The measure of an unknown angle can be found using the given information about other angles by applying various angle relationships and properties. Some common methods include using the properties of supplementary angles, complementary angles, or vertically opposite angles. Additionally, the properties of parallel lines and transversals can also be used to find unknown angle measures.
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