Class 9 Maths Chapter 8 Question Answers - Quadrilaterals

 Page 1


        
  
 
 
Q u e s t i o n : 4 4
In a parallelogram ABCD, write the sum of angles A and B.
S o l u t i o n :
In Parallelogram ABCD, and  are adjacent angles.
Thus, .
Then, we have and as consecutive interior angles which must be supplementary.
Hence, the sum of and  is .
Q u e s t i o n : 4 5
In a parallelogram ABCD, if ?D = 115°, then write the measure of ?A.
S o l u t i o n :
In Parallelogram ABCD , and  are Adjacent angles.
We know that in a parallelogram, adjacent angles are supplementary.
Now, ?A + ?D = 180° ? ?A +115° = 180° ? ?A = 180°-115° ? ?A = 65°So, measure of ?A is 65°.
 
Q u e s t i o n : 4 6
PQRS is a square such that PR and SQ intersect at O. State the measure of ?POQ.
S o l u t i o n :
PQRS is a square given as:
Since the diagonals of a square intersect at right angle.
Therefore, the measure of is .
Q u e s t i o n : 4 7
If PQRS is a square, then write the measure of ?SRP.
S o l u t i o n :
Page 2


        
  
 
 
Q u e s t i o n : 4 4
In a parallelogram ABCD, write the sum of angles A and B.
S o l u t i o n :
In Parallelogram ABCD, and  are adjacent angles.
Thus, .
Then, we have and as consecutive interior angles which must be supplementary.
Hence, the sum of and  is .
Q u e s t i o n : 4 5
In a parallelogram ABCD, if ?D = 115°, then write the measure of ?A.
S o l u t i o n :
In Parallelogram ABCD , and  are Adjacent angles.
We know that in a parallelogram, adjacent angles are supplementary.
Now, ?A + ?D = 180° ? ?A +115° = 180° ? ?A = 180°-115° ? ?A = 65°So, measure of ?A is 65°.
 
Q u e s t i o n : 4 6
PQRS is a square such that PR and SQ intersect at O. State the measure of ?POQ.
S o l u t i o n :
PQRS is a square given as:
Since the diagonals of a square intersect at right angle.
Therefore, the measure of is .
Q u e s t i o n : 4 7
If PQRS is a square, then write the measure of ?SRP.
S o l u t i o n :
The square PQRS is given as:
Since PQRS is a square.
Therefore,
and 
Now, in , we have
That is, Anglesoppositetoequalsidesareequal
By angle sum property of a triangle.
 ( )
Hence, the measure of is .
Q u e s t i o n : 4 8
If ABCD is a rhombus with ?ABC = 56°, find the measure of ?ACD.
S o l u t i o n :
The figure is given as follows:
ABCD is a rhombus.
Therefore,
ABCD is a parallelogram.
Thus,
 [ Given
]
 [ ]
Now in ,we have:
Page 3


        
  
 
 
Q u e s t i o n : 4 4
In a parallelogram ABCD, write the sum of angles A and B.
S o l u t i o n :
In Parallelogram ABCD, and  are adjacent angles.
Thus, .
Then, we have and as consecutive interior angles which must be supplementary.
Hence, the sum of and  is .
Q u e s t i o n : 4 5
In a parallelogram ABCD, if ?D = 115°, then write the measure of ?A.
S o l u t i o n :
In Parallelogram ABCD , and  are Adjacent angles.
We know that in a parallelogram, adjacent angles are supplementary.
Now, ?A + ?D = 180° ? ?A +115° = 180° ? ?A = 180°-115° ? ?A = 65°So, measure of ?A is 65°.
 
Q u e s t i o n : 4 6
PQRS is a square such that PR and SQ intersect at O. State the measure of ?POQ.
S o l u t i o n :
PQRS is a square given as:
Since the diagonals of a square intersect at right angle.
Therefore, the measure of is .
Q u e s t i o n : 4 7
If PQRS is a square, then write the measure of ?SRP.
S o l u t i o n :
The square PQRS is given as:
Since PQRS is a square.
Therefore,
and 
Now, in , we have
That is, Anglesoppositetoequalsidesareequal
By angle sum property of a triangle.
 ( )
Hence, the measure of is .
Q u e s t i o n : 4 8
If ABCD is a rhombus with ?ABC = 56°, find the measure of ?ACD.
S o l u t i o n :
The figure is given as follows:
ABCD is a rhombus.
Therefore,
ABCD is a parallelogram.
Thus,
 [ Given
]
 [ ]
Now in ,we have:
Hence the measure of  is .
Q u e s t i o n : 4 9
The perimeter of a parallelogram is 22 cm. If the longer side measures 6.5 cm, what is the measure of shorter side?
S o l u t i o n :
Let the shorter side of the parallelogram be cm.
The longer side is given as cm.
Perimeter of the parallelogram is given as 22 cm
Therefore,
Hence, the measure of the shorter side is  cm.
Q u e s t i o n : 5 0
If the angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13, then find the measure of the smallest angle.
S o l u t i o n :
We have, .
So, let  ,
,
and 
By angle sum property of a quadrilateral, we get:
Smallest angle is :
Hence, the smallest angle measures .
Page 4


        
  
 
 
Q u e s t i o n : 4 4
In a parallelogram ABCD, write the sum of angles A and B.
S o l u t i o n :
In Parallelogram ABCD, and  are adjacent angles.
Thus, .
Then, we have and as consecutive interior angles which must be supplementary.
Hence, the sum of and  is .
Q u e s t i o n : 4 5
In a parallelogram ABCD, if ?D = 115°, then write the measure of ?A.
S o l u t i o n :
In Parallelogram ABCD , and  are Adjacent angles.
We know that in a parallelogram, adjacent angles are supplementary.
Now, ?A + ?D = 180° ? ?A +115° = 180° ? ?A = 180°-115° ? ?A = 65°So, measure of ?A is 65°.
 
Q u e s t i o n : 4 6
PQRS is a square such that PR and SQ intersect at O. State the measure of ?POQ.
S o l u t i o n :
PQRS is a square given as:
Since the diagonals of a square intersect at right angle.
Therefore, the measure of is .
Q u e s t i o n : 4 7
If PQRS is a square, then write the measure of ?SRP.
S o l u t i o n :
The square PQRS is given as:
Since PQRS is a square.
Therefore,
and 
Now, in , we have
That is, Anglesoppositetoequalsidesareequal
By angle sum property of a triangle.
 ( )
Hence, the measure of is .
Q u e s t i o n : 4 8
If ABCD is a rhombus with ?ABC = 56°, find the measure of ?ACD.
S o l u t i o n :
The figure is given as follows:
ABCD is a rhombus.
Therefore,
ABCD is a parallelogram.
Thus,
 [ Given
]
 [ ]
Now in ,we have:
Hence the measure of  is .
Q u e s t i o n : 4 9
The perimeter of a parallelogram is 22 cm. If the longer side measures 6.5 cm, what is the measure of shorter side?
S o l u t i o n :
Let the shorter side of the parallelogram be cm.
The longer side is given as cm.
Perimeter of the parallelogram is given as 22 cm
Therefore,
Hence, the measure of the shorter side is  cm.
Q u e s t i o n : 5 0
If the angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13, then find the measure of the smallest angle.
S o l u t i o n :
We have, .
So, let  ,
,
and 
By angle sum property of a quadrilateral, we get:
Smallest angle is :
Hence, the smallest angle measures .
Q u e s t i o n : 5 1
In a parallelogram ABCD, if ?A = (3x - 20)°, ?B = (y + 15)°, ?C = (x + 40)°, then find the values of x and y.
S o l u t i o n :
In parallelogram ABCD, and  are opposite angles.
We know that in a parallelogram, the opposite angles are equal.
Therefore,
We have  and 
Therefore,
Therefore,
Similarly,
Also, 
Therefore,
By angle sum property of a quadrilateral, we have:
Hence the required values for x and y are  and  respectively.
Q u e s t i o n : 5 2
If measures opposite angles of a parallelogram are 60 -x
° and 3x -4
°, then find the measures of angles of the parallelogram.
S o l u t i o n :
Let ABCD be a parallelogram, with  and .
We know that in a parallelogram, the opposite angles are equal.
Page 5


        
  
 
 
Q u e s t i o n : 4 4
In a parallelogram ABCD, write the sum of angles A and B.
S o l u t i o n :
In Parallelogram ABCD, and  are adjacent angles.
Thus, .
Then, we have and as consecutive interior angles which must be supplementary.
Hence, the sum of and  is .
Q u e s t i o n : 4 5
In a parallelogram ABCD, if ?D = 115°, then write the measure of ?A.
S o l u t i o n :
In Parallelogram ABCD , and  are Adjacent angles.
We know that in a parallelogram, adjacent angles are supplementary.
Now, ?A + ?D = 180° ? ?A +115° = 180° ? ?A = 180°-115° ? ?A = 65°So, measure of ?A is 65°.
 
Q u e s t i o n : 4 6
PQRS is a square such that PR and SQ intersect at O. State the measure of ?POQ.
S o l u t i o n :
PQRS is a square given as:
Since the diagonals of a square intersect at right angle.
Therefore, the measure of is .
Q u e s t i o n : 4 7
If PQRS is a square, then write the measure of ?SRP.
S o l u t i o n :
The square PQRS is given as:
Since PQRS is a square.
Therefore,
and 
Now, in , we have
That is, Anglesoppositetoequalsidesareequal
By angle sum property of a triangle.
 ( )
Hence, the measure of is .
Q u e s t i o n : 4 8
If ABCD is a rhombus with ?ABC = 56°, find the measure of ?ACD.
S o l u t i o n :
The figure is given as follows:
ABCD is a rhombus.
Therefore,
ABCD is a parallelogram.
Thus,
 [ Given
]
 [ ]
Now in ,we have:
Hence the measure of  is .
Q u e s t i o n : 4 9
The perimeter of a parallelogram is 22 cm. If the longer side measures 6.5 cm, what is the measure of shorter side?
S o l u t i o n :
Let the shorter side of the parallelogram be cm.
The longer side is given as cm.
Perimeter of the parallelogram is given as 22 cm
Therefore,
Hence, the measure of the shorter side is  cm.
Q u e s t i o n : 5 0
If the angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13, then find the measure of the smallest angle.
S o l u t i o n :
We have, .
So, let  ,
,
and 
By angle sum property of a quadrilateral, we get:
Smallest angle is :
Hence, the smallest angle measures .
Q u e s t i o n : 5 1
In a parallelogram ABCD, if ?A = (3x - 20)°, ?B = (y + 15)°, ?C = (x + 40)°, then find the values of x and y.
S o l u t i o n :
In parallelogram ABCD, and  are opposite angles.
We know that in a parallelogram, the opposite angles are equal.
Therefore,
We have  and 
Therefore,
Therefore,
Similarly,
Also, 
Therefore,
By angle sum property of a quadrilateral, we have:
Hence the required values for x and y are  and  respectively.
Q u e s t i o n : 5 2
If measures opposite angles of a parallelogram are 60 -x
° and 3x -4
°, then find the measures of angles of the parallelogram.
S o l u t i o n :
Let ABCD be a parallelogram, with  and .
We know that in a parallelogram, the opposite angles are equal.
Therefore,
Thus, the given angles become
Similarly,
Also, adjacent angles in a parallelogram form the consecutive interior angles of parallel lines, which must be
supplementary.
Therefore,
Similarly,
Thus, the angles of a parallelogram are , , and .
Q u e s t i o n : 5 3
In a parallelogram ABCD, the bisector of ?A also bisects BC at X. Find AB : AD.
S o l u t i o n :
Parallelogram ABCD is given as follows:
We have AX bisects  bisecting BC at X.
That is, 
We need to find 
Since, AX is the bisector 
That is,
 …… i
Also, ABCD is a parallelogram