## Chapter 19 - Quadrilaterals Exercise Ex. 19.1

Question 1
In the following figures, find the remaining angles of the  parallelogram
Solution 1
Question 2
In the following figures, find the remaining angles of the  parallelogram
Solution 2
Question 3
In the following figures, find the remaining angles of the  parallelogram
Solution 3
Question 4
In the following figures, find the remaining angles of the  parallelogram
Solution 4
Question 5
In the following figures, find the remaining angles of the  parallelogram
Solution 5
Question 6
Solution 6

Question 7
The consecutive angles of a parallelogram are in the ratio 3:6. Calculate the measures of all the angles of the parallelogram.
Solution 7

Question 8
Solution 8
Question 9
In the given figure, ABCD is a parallelogram, find the values of x and y.

Solution 9
Question 10
Solution 10

Question 11

Solution 11
Question 12

Solution 12

Question 13
Solution 13
Question 14
Solution 14
Question 15

Solution 15
Question 16
Solution 16
Question 17
In the given figure, MP is the bisector of ∠P and RN is the bisector of ∠R of parallelogram PQRS. Prove that PMRN is a parallelogram.
Solution 17

Construction: Join PR.

Question 18
Solution 18

Question 19
Solution 19

## Chapter 19 - Quadrilaterals Exercise Ex. 19.2

Question 1
Solution 1

Question 2
Solution 2
Question 3
Solution 3

Question 4
Solution 4

Question 5

Solution 5

Question 6
Solution 6

Question 7
Solution 7

Question 8
Solution 8

Question 9
Solution 9

Question 10
Solution 10
Question 11
Solution 11

Question 12
In a parallelogram PQRS, M and N are the midpoints of the opposite sides PQ and RS respectively. Prove that
RN and RM trisect QS.
Solution 12

Question 13
In a parallelogram PQRS, M and N are the midpoints of the opposite sides PQ and RS respectively. Prove that
PMRN is a parallelogram.
Solution 13

Question 14
In a parallelogram PQRS, M and N are the midpoints of the opposite sides PQ and RS respectively. Prove that
MN bisects QS.
Solution 14

Question 15
Solution 15

Question 16
Solution 16

Question 17
In the given figure, PQRS is a parallelogram in which PA = AB = Prove that:
SA ‖ QB and SA = QB.

Solution 17
Construction:
Join BS and AQ.
Join diagonal QS.

Question 18
In the given figure, PQRS is a parallelogram in which PA = AB = Prove that:
SAQB is a parallelogram.
Solution 18
Construction:
Join BS and AQ.
Join diagonal QS.

Question 19
In the given figure, PQRS is a trapezium in which PQ ‖ SR and PS = QR. Prove that:
∠PSR = ∠QRS and ∠SPQ = ∠RQP

Solution 19
Construction:
Draw SM ⊥ PQ and RN ⊥ PQ

Question 20
In a parallelogram ABCD, E is the midpoint of AB and DE bisects angle D. Prove that:
1. BC = BE.
2. CE is the bisector of angle C and angle DEC is a right angle
Solution 20

Question 21
Prove that if the diagonals of a quadrilateral bisect each other at right angles then it is a rhombus.
Solution 21

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 = 90°
To prove ABCD a rhombus, we need to prove ABCD is a parallelogram and all sides of ABCD are equal.
Now, in ΔAOD and DCOD
OA = OC  (Diagonal bisects each other)
∠AOD = ∠COD (Each 90°)
OD = OD   (common)
∴ΔAOD ≅ΔCOD  (By SAS congruence rule)
Similarly, we can prove that
AD = AB and CD = BC ….(ii)
From equations (i) and (ii), we can say that
AB = BC = CD = AD
Since opposite sides of quadrilateral ABCD are equal, so, we can say that ABCD is a parallelogram.
Since all sides of a parallelogram ABCD are equal, so, we can say that ABCD is a rhombus.
Question 22
Prove that the diagonals of a kite intersect each other at right angles.
Solution 22
Consider ABCD is a kite.
Then, AB = AD and BC = DC

Question 23
Prove that the diagonals of a square are equal and perpendicular to each other.
Solution 23