NCERT Solution - Quadrilaterals Class 9 Notes | EduRev

Class 9 : NCERT Solution - Quadrilaterals Class 9 Notes | EduRev

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1 
  
  
Class IX  Chapter 8 – Quadrilaterals   
Maths   
   
  
Exercise 8.1 Question 1:   
The angles of quadrilateral are in the ratio 3: 5: 9: 13. Find all the angles of the 
quadrilateral.   
Answer:   
Let the common ratio between the angles be x. Therefore, the angles will be 3x, 5x, 
9x, and 13x respectively.   
As the sum of all interior angles of a quadrilateral is 360º,   
 3x + 5x + 9x + 13x = 360º   
30x = 360º x  
= 12º   
Hence, the angles are   
3x = 3 × 12 = 36º 5x =  
5 × 12 = 60º   
9x = 9 × 12 = 108º 13x =  
13 × 12 = 156º Question 2:   
If the diagonals of a parallelogram are equal, then show that it is a rectangle.   
Answer:   
Page 2


Cbse-spot.blogspot.com  
Cbse-spot.blogspot.com  
1 
  
  
Class IX  Chapter 8 – Quadrilaterals   
Maths   
   
  
Exercise 8.1 Question 1:   
The angles of quadrilateral are in the ratio 3: 5: 9: 13. Find all the angles of the 
quadrilateral.   
Answer:   
Let the common ratio between the angles be x. Therefore, the angles will be 3x, 5x, 
9x, and 13x respectively.   
As the sum of all interior angles of a quadrilateral is 360º,   
 3x + 5x + 9x + 13x = 360º   
30x = 360º x  
= 12º   
Hence, the angles are   
3x = 3 × 12 = 36º 5x =  
5 × 12 = 60º   
9x = 9 × 12 = 108º 13x =  
13 × 12 = 156º Question 2:   
If the diagonals of a parallelogram are equal, then show that it is a rectangle.   
Answer:   
Cbse-spot.blogspot.com  
Cbse-spot.blogspot.com  
2 
  
   
Let ABCD be a parallelogram. To show that ABCD is a rectangle, we have to prove that 
one of its interior angles is 90º.   
In ?ABC and ?DCB,   
AB = DC (Opposite sides of a parallelogram are equal)   
BC = BC (Common)   
AC = DB (Given)   
 ?ABC  ?DCB (By SSS Congruence rule)   
    
 ABC = DCB   
It is known that the sum of the measures of angles on the same side of transversal is 
180º.   
 
ABC + DCB = 180º (AB || CD)   
  ABC +  ABC = 180º   
 2 ABC = 180º   
  ABC = 90º   
Since ABCD is a parallelogram and one of its interior angles is 90º, ABCD is a rectangle.   
Question 3:   
Show that if the diagonals of a quadrilateral bisect each other at right angles, then it 
is a rhombus.   
Answer:   
Page 3


Cbse-spot.blogspot.com  
Cbse-spot.blogspot.com  
1 
  
  
Class IX  Chapter 8 – Quadrilaterals   
Maths   
   
  
Exercise 8.1 Question 1:   
The angles of quadrilateral are in the ratio 3: 5: 9: 13. Find all the angles of the 
quadrilateral.   
Answer:   
Let the common ratio between the angles be x. Therefore, the angles will be 3x, 5x, 
9x, and 13x respectively.   
As the sum of all interior angles of a quadrilateral is 360º,   
 3x + 5x + 9x + 13x = 360º   
30x = 360º x  
= 12º   
Hence, the angles are   
3x = 3 × 12 = 36º 5x =  
5 × 12 = 60º   
9x = 9 × 12 = 108º 13x =  
13 × 12 = 156º Question 2:   
If the diagonals of a parallelogram are equal, then show that it is a rectangle.   
Answer:   
Cbse-spot.blogspot.com  
Cbse-spot.blogspot.com  
2 
  
   
Let ABCD be a parallelogram. To show that ABCD is a rectangle, we have to prove that 
one of its interior angles is 90º.   
In ?ABC and ?DCB,   
AB = DC (Opposite sides of a parallelogram are equal)   
BC = BC (Common)   
AC = DB (Given)   
 ?ABC  ?DCB (By SSS Congruence rule)   
    
 ABC = DCB   
It is known that the sum of the measures of angles on the same side of transversal is 
180º.   
 
ABC + DCB = 180º (AB || CD)   
  ABC +  ABC = 180º   
 2 ABC = 180º   
  ABC = 90º   
Since ABCD is a parallelogram and one of its interior angles is 90º, ABCD is a rectangle.   
Question 3:   
Show that if the diagonals of a quadrilateral bisect each other at right angles, then it 
is a rhombus.   
Answer:   
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Cbse-spot.blogspot.com  
3 
  
   
Let ABCD be a quadrilateral, whose diagonals AC and BD bisect each other at right 
angle i.e., OA = OC, OB = OD, and  AOB =  BOC = COD =  AOD = 90º. To prove 
ABCD a rhombus, we have to prove ABCD is a parallelogram and all the sides of ABCD 
are equal.   
In ?AOD and ?COD,   
OA = OC (Diagonals bisect each other)   
AOD =  COD (Given)   
OD = OD (Common)   
 ?AOD  ?COD (By SAS congruence rule)   
 AD = CD (1)   
Similarly, it can be proved that   
AD = AB and CD = BC (2)   
From equations (1) and (2),   
AB = BC = CD = AD   
Since opposite sides of quadrilateral ABCD are equal, it can be said that ABCD is a 
parallelogram. Since all sides of a parallelogram ABCD are equal, it can be said that 
ABCD is a rhombus.   
Question 4:   
Show that the diagonals of a square are equal and bisect each other at right angles.   
Answer:   
Page 4


Cbse-spot.blogspot.com  
Cbse-spot.blogspot.com  
1 
  
  
Class IX  Chapter 8 – Quadrilaterals   
Maths   
   
  
Exercise 8.1 Question 1:   
The angles of quadrilateral are in the ratio 3: 5: 9: 13. Find all the angles of the 
quadrilateral.   
Answer:   
Let the common ratio between the angles be x. Therefore, the angles will be 3x, 5x, 
9x, and 13x respectively.   
As the sum of all interior angles of a quadrilateral is 360º,   
 3x + 5x + 9x + 13x = 360º   
30x = 360º x  
= 12º   
Hence, the angles are   
3x = 3 × 12 = 36º 5x =  
5 × 12 = 60º   
9x = 9 × 12 = 108º 13x =  
13 × 12 = 156º Question 2:   
If the diagonals of a parallelogram are equal, then show that it is a rectangle.   
Answer:   
Cbse-spot.blogspot.com  
Cbse-spot.blogspot.com  
2 
  
   
Let ABCD be a parallelogram. To show that ABCD is a rectangle, we have to prove that 
one of its interior angles is 90º.   
In ?ABC and ?DCB,   
AB = DC (Opposite sides of a parallelogram are equal)   
BC = BC (Common)   
AC = DB (Given)   
 ?ABC  ?DCB (By SSS Congruence rule)   
    
 ABC = DCB   
It is known that the sum of the measures of angles on the same side of transversal is 
180º.   
 
ABC + DCB = 180º (AB || CD)   
  ABC +  ABC = 180º   
 2 ABC = 180º   
  ABC = 90º   
Since ABCD is a parallelogram and one of its interior angles is 90º, ABCD is a rectangle.   
Question 3:   
Show that if the diagonals of a quadrilateral bisect each other at right angles, then it 
is a rhombus.   
Answer:   
Cbse-spot.blogspot.com  
Cbse-spot.blogspot.com  
3 
  
   
Let ABCD be a quadrilateral, whose diagonals AC and BD bisect each other at right 
angle i.e., OA = OC, OB = OD, and  AOB =  BOC = COD =  AOD = 90º. To prove 
ABCD a rhombus, we have to prove ABCD is a parallelogram and all the sides of ABCD 
are equal.   
In ?AOD and ?COD,   
OA = OC (Diagonals bisect each other)   
AOD =  COD (Given)   
OD = OD (Common)   
 ?AOD  ?COD (By SAS congruence rule)   
 AD = CD (1)   
Similarly, it can be proved that   
AD = AB and CD = BC (2)   
From equations (1) and (2),   
AB = BC = CD = AD   
Since opposite sides of quadrilateral ABCD are equal, it can be said that ABCD is a 
parallelogram. Since all sides of a parallelogram ABCD are equal, it can be said that 
ABCD is a rhombus.   
Question 4:   
Show that the diagonals of a square are equal and bisect each other at right angles.   
Answer:   
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Cbse-spot.blogspot.com  
4 
  
   
Let ABCD be a square. Let the diagonals AC and BD intersect each other at a point O. 
To prove that the diagonals of a square are equal and bisect each other at right angles,  
we have to prove AC = BD, OA = OC, OB = OD, and AOB = 90º.   
In ?ABC and ?DCB,   
AB = DC (Sides of a square are equal to each other)   
ABC = DCB (All interior angles are of 90 )   
BC = CB (Common side)   
 ?ABC  ?DCB (By SAS congruency)   
 AC = DB (By CPCT)   
Hence, the diagonals of a square are equal in length.   
In ?AOB and ?COD,   
AOB =  COD (Vertically opposite angles)   
ABO = CDO (Alternate interior angles)   
AB = CD (Sides of a square are always equal)   
 ?AOB  ?COD (By AAS congruence rule)   
 AO = CO and OB = OD (By CPCT)   
Hence, the diagonals of a square bisect each other.   
In ?AOB and ?COB,   
As we had proved that diagonals bisect each other, therefore,   
AO = CO   
Page 5


Cbse-spot.blogspot.com  
Cbse-spot.blogspot.com  
1 
  
  
Class IX  Chapter 8 – Quadrilaterals   
Maths   
   
  
Exercise 8.1 Question 1:   
The angles of quadrilateral are in the ratio 3: 5: 9: 13. Find all the angles of the 
quadrilateral.   
Answer:   
Let the common ratio between the angles be x. Therefore, the angles will be 3x, 5x, 
9x, and 13x respectively.   
As the sum of all interior angles of a quadrilateral is 360º,   
 3x + 5x + 9x + 13x = 360º   
30x = 360º x  
= 12º   
Hence, the angles are   
3x = 3 × 12 = 36º 5x =  
5 × 12 = 60º   
9x = 9 × 12 = 108º 13x =  
13 × 12 = 156º Question 2:   
If the diagonals of a parallelogram are equal, then show that it is a rectangle.   
Answer:   
Cbse-spot.blogspot.com  
Cbse-spot.blogspot.com  
2 
  
   
Let ABCD be a parallelogram. To show that ABCD is a rectangle, we have to prove that 
one of its interior angles is 90º.   
In ?ABC and ?DCB,   
AB = DC (Opposite sides of a parallelogram are equal)   
BC = BC (Common)   
AC = DB (Given)   
 ?ABC  ?DCB (By SSS Congruence rule)   
    
 ABC = DCB   
It is known that the sum of the measures of angles on the same side of transversal is 
180º.   
 
ABC + DCB = 180º (AB || CD)   
  ABC +  ABC = 180º   
 2 ABC = 180º   
  ABC = 90º   
Since ABCD is a parallelogram and one of its interior angles is 90º, ABCD is a rectangle.   
Question 3:   
Show that if the diagonals of a quadrilateral bisect each other at right angles, then it 
is a rhombus.   
Answer:   
Cbse-spot.blogspot.com  
Cbse-spot.blogspot.com  
3 
  
   
Let ABCD be a quadrilateral, whose diagonals AC and BD bisect each other at right 
angle i.e., OA = OC, OB = OD, and  AOB =  BOC = COD =  AOD = 90º. To prove 
ABCD a rhombus, we have to prove ABCD is a parallelogram and all the sides of ABCD 
are equal.   
In ?AOD and ?COD,   
OA = OC (Diagonals bisect each other)   
AOD =  COD (Given)   
OD = OD (Common)   
 ?AOD  ?COD (By SAS congruence rule)   
 AD = CD (1)   
Similarly, it can be proved that   
AD = AB and CD = BC (2)   
From equations (1) and (2),   
AB = BC = CD = AD   
Since opposite sides of quadrilateral ABCD are equal, it can be said that ABCD is a 
parallelogram. Since all sides of a parallelogram ABCD are equal, it can be said that 
ABCD is a rhombus.   
Question 4:   
Show that the diagonals of a square are equal and bisect each other at right angles.   
Answer:   
Cbse-spot.blogspot.com  
Cbse-spot.blogspot.com  
4 
  
   
Let ABCD be a square. Let the diagonals AC and BD intersect each other at a point O. 
To prove that the diagonals of a square are equal and bisect each other at right angles,  
we have to prove AC = BD, OA = OC, OB = OD, and AOB = 90º.   
In ?ABC and ?DCB,   
AB = DC (Sides of a square are equal to each other)   
ABC = DCB (All interior angles are of 90 )   
BC = CB (Common side)   
 ?ABC  ?DCB (By SAS congruency)   
 AC = DB (By CPCT)   
Hence, the diagonals of a square are equal in length.   
In ?AOB and ?COD,   
AOB =  COD (Vertically opposite angles)   
ABO = CDO (Alternate interior angles)   
AB = CD (Sides of a square are always equal)   
 ?AOB  ?COD (By AAS congruence rule)   
 AO = CO and OB = OD (By CPCT)   
Hence, the diagonals of a square bisect each other.   
In ?AOB and ?COB,   
As we had proved that diagonals bisect each other, therefore,   
AO = CO   
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Cbse-spot.blogspot.com  
5 
  
AB = CB (Sides of a square are equal)   
BO = BO (Common)   
 ?AOB  ?COB (By SSS congruency)   
 AOB = COB (By CPCT)   
However, AOB + COB = 180º (Linear pair)   
AOB = 180º 2   
AOB = 90º   
Hence, the diagonals of a square bisect each other at right angles.   
Question 5:   
Show that if the diagonals of a quadrilateral are equal and bisect each other at right 
angles, then it is a square.   
Answer:   
   
Let us consider a quadrilateral ABCD in which the diagonals AC and BD intersect each 
other at O. It is given that the diagonals of ABCD are equal and bisect each other at  
right angles. Therefore, AC = BD, OA = OC, OB = OD, and AOB = BOC = COD   
AOD = = 90º. To prove ABCD is a square, we have to prove that ABCD is a  
parallelogram, AB = BC = CD = AD, and one of its interior angles is 90º.   
In ?AOB and ?COD,   
AO = CO (Diagonals bisect each other)   
OB = OD (Diagonals bisect each other)   
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