Class 9 Maths - Chapter 6 Question Answers Lines and Angles

 Page 1


 
Q u e s t i o n : 7 4
Define complementary angles.
S o l u t i o n :
Complementary Angles: Two angles, the sum of whose measures is , are called complementary angles.
Thus, angles and are complementary angles, if
Example 1:
Angles of measure and are complementary angles, because
Example 2:
Angles of measure and are complementary angles, because
Q u e s t i o n : 7 5
Define supplementary angles.
S o l u t i o n :
Supplementary Angles: Two angles, the sum of whose measures is , are called supplementary angles.
Thus, angles and are supplementary angles, if
 
Example 1:
Angles of measure and are supplementary angles, because
Example 2:
Angles of measure and are supplementary angles, because
Page 2


 
Q u e s t i o n : 7 4
Define complementary angles.
S o l u t i o n :
Complementary Angles: Two angles, the sum of whose measures is , are called complementary angles.
Thus, angles and are complementary angles, if
Example 1:
Angles of measure and are complementary angles, because
Example 2:
Angles of measure and are complementary angles, because
Q u e s t i o n : 7 5
Define supplementary angles.
S o l u t i o n :
Supplementary Angles: Two angles, the sum of whose measures is , are called supplementary angles.
Thus, angles and are supplementary angles, if
 
Example 1:
Angles of measure and are supplementary angles, because
Example 2:
Angles of measure and are supplementary angles, because
Q u e s t i o n : 7 6
Define adjacent angles.
S o l u t i o n :
Adjacent angles: Two angles are called adjacent angles, if:
i. They have the same vertex,
ii. They have a common arm, and
iii. Uncommon arms are on either side of the common arm.
In the figure above, and have a common vertex .
Also, they have a common arm and the distinct arms and , lies on the opposite sides of the line .
Therefore, and are adjacent angles.
Q u e s t i o n : 7 7
The complement of an acute angle is ..............
S o l u t i o n :
The complement of an acute angle is an acute angle.
Explanation:
As the sum of the complementary angles is .
Let one of the angle measures .
Then, other angle becomes , which is clearly an acute angle.
Q u e s t i o n : 7 8
The supplement of an acute angle is .................
S o l u t i o n :
The supplement of an acute angle is an obtuse angle.
Explanation:
As the sum of the supplementary angles is .
Let one of the angle measures , such that
Let the other angle measures
As the angles are supplementary there sum is .
Then, other angle y is clearly an obtuse angle.
Illustration:
Let the given acute angle be
Page 3


 
Q u e s t i o n : 7 4
Define complementary angles.
S o l u t i o n :
Complementary Angles: Two angles, the sum of whose measures is , are called complementary angles.
Thus, angles and are complementary angles, if
Example 1:
Angles of measure and are complementary angles, because
Example 2:
Angles of measure and are complementary angles, because
Q u e s t i o n : 7 5
Define supplementary angles.
S o l u t i o n :
Supplementary Angles: Two angles, the sum of whose measures is , are called supplementary angles.
Thus, angles and are supplementary angles, if
 
Example 1:
Angles of measure and are supplementary angles, because
Example 2:
Angles of measure and are supplementary angles, because
Q u e s t i o n : 7 6
Define adjacent angles.
S o l u t i o n :
Adjacent angles: Two angles are called adjacent angles, if:
i. They have the same vertex,
ii. They have a common arm, and
iii. Uncommon arms are on either side of the common arm.
In the figure above, and have a common vertex .
Also, they have a common arm and the distinct arms and , lies on the opposite sides of the line .
Therefore, and are adjacent angles.
Q u e s t i o n : 7 7
The complement of an acute angle is ..............
S o l u t i o n :
The complement of an acute angle is an acute angle.
Explanation:
As the sum of the complementary angles is .
Let one of the angle measures .
Then, other angle becomes , which is clearly an acute angle.
Q u e s t i o n : 7 8
The supplement of an acute angle is .................
S o l u t i o n :
The supplement of an acute angle is an obtuse angle.
Explanation:
As the sum of the supplementary angles is .
Let one of the angle measures , such that
Let the other angle measures
As the angles are supplementary there sum is .
Then, other angle y is clearly an obtuse angle.
Illustration:
Let the given acute angle be
Then, the other angle becomes
This is clearly an obtuse angle.
Q u e s t i o n : 7 9
The supplement of a right angle is ..............
S o l u t i o n :
We have to find the supplement of a right angle.
We know that a right angle is equal to .
Let the required angle be .
Since the two angles are supplementary, therefore their sum must be equal to .
Thus, the require angle becomes
Q u e s t i o n : 8 0
Write the complement of an angle of measure x°.
S o l u t i o n :
We have to write the complement of an angle which measures .
Let the other angle be .
We know that the sum of the complementary angles be 90°.
Therefore,
Q u e s t i o n : 8 1
Write the supplement of an angle of measure 2y°.
S o l u t i o n :
Let the required angle measures
It is given that two angles measuring and are supplementary. Therefore, their sum must be equal to .
Or, we can say that:
Hence, the required angle measures .
Q u e s t i o n : 8 2
If a wheel has six spokes equally spaced, then find the measure of the angle between two adjacent spokes.
S o l u t i o n :
It is given that the six spokes are equally spaced, thus, two adjacent spokes subtend equal angle at the centre of the wheel.
Let that angle measures
Also, the six spokes form a complete angle, that is,
Page 4


 
Q u e s t i o n : 7 4
Define complementary angles.
S o l u t i o n :
Complementary Angles: Two angles, the sum of whose measures is , are called complementary angles.
Thus, angles and are complementary angles, if
Example 1:
Angles of measure and are complementary angles, because
Example 2:
Angles of measure and are complementary angles, because
Q u e s t i o n : 7 5
Define supplementary angles.
S o l u t i o n :
Supplementary Angles: Two angles, the sum of whose measures is , are called supplementary angles.
Thus, angles and are supplementary angles, if
 
Example 1:
Angles of measure and are supplementary angles, because
Example 2:
Angles of measure and are supplementary angles, because
Q u e s t i o n : 7 6
Define adjacent angles.
S o l u t i o n :
Adjacent angles: Two angles are called adjacent angles, if:
i. They have the same vertex,
ii. They have a common arm, and
iii. Uncommon arms are on either side of the common arm.
In the figure above, and have a common vertex .
Also, they have a common arm and the distinct arms and , lies on the opposite sides of the line .
Therefore, and are adjacent angles.
Q u e s t i o n : 7 7
The complement of an acute angle is ..............
S o l u t i o n :
The complement of an acute angle is an acute angle.
Explanation:
As the sum of the complementary angles is .
Let one of the angle measures .
Then, other angle becomes , which is clearly an acute angle.
Q u e s t i o n : 7 8
The supplement of an acute angle is .................
S o l u t i o n :
The supplement of an acute angle is an obtuse angle.
Explanation:
As the sum of the supplementary angles is .
Let one of the angle measures , such that
Let the other angle measures
As the angles are supplementary there sum is .
Then, other angle y is clearly an obtuse angle.
Illustration:
Let the given acute angle be
Then, the other angle becomes
This is clearly an obtuse angle.
Q u e s t i o n : 7 9
The supplement of a right angle is ..............
S o l u t i o n :
We have to find the supplement of a right angle.
We know that a right angle is equal to .
Let the required angle be .
Since the two angles are supplementary, therefore their sum must be equal to .
Thus, the require angle becomes
Q u e s t i o n : 8 0
Write the complement of an angle of measure x°.
S o l u t i o n :
We have to write the complement of an angle which measures .
Let the other angle be .
We know that the sum of the complementary angles be 90°.
Therefore,
Q u e s t i o n : 8 1
Write the supplement of an angle of measure 2y°.
S o l u t i o n :
Let the required angle measures
It is given that two angles measuring and are supplementary. Therefore, their sum must be equal to .
Or, we can say that:
Hence, the required angle measures .
Q u e s t i o n : 8 2
If a wheel has six spokes equally spaced, then find the measure of the angle between two adjacent spokes.
S o l u t i o n :
It is given that the six spokes are equally spaced, thus, two adjacent spokes subtend equal angle at the centre of the wheel.
Let that angle measures
Also, the six spokes form a complete angle, that is,
Therefore,
Hence, the measure of the angle between two adjacent spokes measures .
Q u e s t i o n : 8 3
An angle is equal to its supplement. Determine its measure.
S o l u t i o n :
Let the supplement of the angle be
According the given statement, the required angle is equal to its supplement, therefore, the required angle becomes .
Sine both the angles are supplementary, therefore, their sum must be equal to
Or we can say that:
Hence, the required angle measures .
Q u e s t i o n : 8 4
An angle is equal to five times its complement. Determine its measure.
S o l u t i o n :
Let the complement of the required angle measures
Therefore, the required angle becomes
Since, the angles are complementary, thus, their sum must be equal to .
Or we can say that :
Hence, the required angle becomes:
Q u e s t i o n : 8 5
How many pairs of adjacent angles are formed when two lines intersect in a point?
S o l u t i o n :
Let us draw the following diagram showing two lines and  intersecting at a point .
Page 5


 
Q u e s t i o n : 7 4
Define complementary angles.
S o l u t i o n :
Complementary Angles: Two angles, the sum of whose measures is , are called complementary angles.
Thus, angles and are complementary angles, if
Example 1:
Angles of measure and are complementary angles, because
Example 2:
Angles of measure and are complementary angles, because
Q u e s t i o n : 7 5
Define supplementary angles.
S o l u t i o n :
Supplementary Angles: Two angles, the sum of whose measures is , are called supplementary angles.
Thus, angles and are supplementary angles, if
 
Example 1:
Angles of measure and are supplementary angles, because
Example 2:
Angles of measure and are supplementary angles, because
Q u e s t i o n : 7 6
Define adjacent angles.
S o l u t i o n :
Adjacent angles: Two angles are called adjacent angles, if:
i. They have the same vertex,
ii. They have a common arm, and
iii. Uncommon arms are on either side of the common arm.
In the figure above, and have a common vertex .
Also, they have a common arm and the distinct arms and , lies on the opposite sides of the line .
Therefore, and are adjacent angles.
Q u e s t i o n : 7 7
The complement of an acute angle is ..............
S o l u t i o n :
The complement of an acute angle is an acute angle.
Explanation:
As the sum of the complementary angles is .
Let one of the angle measures .
Then, other angle becomes , which is clearly an acute angle.
Q u e s t i o n : 7 8
The supplement of an acute angle is .................
S o l u t i o n :
The supplement of an acute angle is an obtuse angle.
Explanation:
As the sum of the supplementary angles is .
Let one of the angle measures , such that
Let the other angle measures
As the angles are supplementary there sum is .
Then, other angle y is clearly an obtuse angle.
Illustration:
Let the given acute angle be
Then, the other angle becomes
This is clearly an obtuse angle.
Q u e s t i o n : 7 9
The supplement of a right angle is ..............
S o l u t i o n :
We have to find the supplement of a right angle.
We know that a right angle is equal to .
Let the required angle be .
Since the two angles are supplementary, therefore their sum must be equal to .
Thus, the require angle becomes
Q u e s t i o n : 8 0
Write the complement of an angle of measure x°.
S o l u t i o n :
We have to write the complement of an angle which measures .
Let the other angle be .
We know that the sum of the complementary angles be 90°.
Therefore,
Q u e s t i o n : 8 1
Write the supplement of an angle of measure 2y°.
S o l u t i o n :
Let the required angle measures
It is given that two angles measuring and are supplementary. Therefore, their sum must be equal to .
Or, we can say that:
Hence, the required angle measures .
Q u e s t i o n : 8 2
If a wheel has six spokes equally spaced, then find the measure of the angle between two adjacent spokes.
S o l u t i o n :
It is given that the six spokes are equally spaced, thus, two adjacent spokes subtend equal angle at the centre of the wheel.
Let that angle measures
Also, the six spokes form a complete angle, that is,
Therefore,
Hence, the measure of the angle between two adjacent spokes measures .
Q u e s t i o n : 8 3
An angle is equal to its supplement. Determine its measure.
S o l u t i o n :
Let the supplement of the angle be
According the given statement, the required angle is equal to its supplement, therefore, the required angle becomes .
Sine both the angles are supplementary, therefore, their sum must be equal to
Or we can say that:
Hence, the required angle measures .
Q u e s t i o n : 8 4
An angle is equal to five times its complement. Determine its measure.
S o l u t i o n :
Let the complement of the required angle measures
Therefore, the required angle becomes
Since, the angles are complementary, thus, their sum must be equal to .
Or we can say that :
Hence, the required angle becomes:
Q u e s t i o n : 8 5
How many pairs of adjacent angles are formed when two lines intersect in a point?
S o l u t i o n :
Let us draw the following diagram showing two lines and  intersecting at a point .
We have the following pair of adjacent angles, so formed:
 and 
 and 
 and 
 and 
Hence, in total four pair of adjacent angles are formed.