Chapter 14 Quadrilaterals RD Sharma Solutions Exercise 14.2 Class 9 Maths

Chapter 14 Quadrilaterals RD Sharma Solutions Exercise 14.2 Class 9 Maths

Chapter Name

RD Sharma Chapter 14 Quadrilaterals Exercise 14.2

Book Name

RD Sharma Mathematics for Class 10

Other Exercises

  • Exercise 14.1
  • Exercise 14.3
  • Exercise 14.4

Related Study

NCERT Solutions for Class 10 Maths

Exercise 14.2 Solutions

1. Two opposite angles of a parallelogram are (3x - 2)° and (50 - x)° . Find the measure of each angle of the parallelogram.

Solution

We know that 
Opposite sides of a parallelogram are equal
∴ 3x  2 = 50 - x 
⇒ 3x + x = 50 + 2 
⇒ 4x = 52 
⇒ x = 13°
∴ (3x - 2)° = (3 × 13 - 2) = 37°
(50 - x)° = (50 - 13°) = 37°
Adjacent angles of a parallelogram are supplementary 
∴ x + 37 = 180°
∴ x = 180° - 37° = 143° 
Hence, four angles are : 37°, 143°, 37° , 143°


2. If an angle of a parallelogram is two - third of its adjacent angle, find the angles of the parallelogram. 

Solution

Let the measure of the angle be x 
∴ The measure of the angle adjacent is 2x/3 
We know that the adjacent angle of a parallelogram is supplementary 
Hence x + 2x/3 = 180°
2x + 3x = 540°
⇒ 5x = 540°
⇒ x = 108°
Adjacent angles are supplementary 
⇒ x + 108° = 180°
⇒ x = 180° - 108° = 72°
⇒ x = 72° 
Hence, four angles are :  180° , 72°, 108°, 72°


3. Find the measure of all the angles of a parallelogram, if one angle is 24° less than twice the smallest angle. 

Solution

Let the smallest angle be x 
Then, the other angle is (3x - 24)
Now, x + 2x - 24 = 180°
3x - 24 = 180°
⇒ 3x = 180° + 24
⇒ 3x = 204°
⇒ x = 204/3 = 68°
⇒ x = 68°
⇒ 2x -24° = 2×68° - 24° = 136° - 24° = 112°
Hence four angles are 68°, 112°, 68°, 112°


4. The perimeter of a parallelogram is 22cm. If the longer side measures 6.5 cm what is the measure of the shorter side ?

Solution

Let the shorter side be x
∴Perimeter = x + 6.5 + 6.5 + x 
22 = 2(x + 6.5)
11 = x + 6.5 
⇒ x = 11 - 6.5 = 4.5 cm
∴ Shorter side  = 4.5 cm


5. In a parallelogram ABCD, ∠D = 135°, determine the measures of ∠A and ∠B.

Solution

In a parallelogram ABCD
Adjacent angles are supplementary
So, ∠D + ∠C = 180°
135 + ∠C = 180°
⇒ ∠C = 180° - 135°
⇒ ∠C = 45°
In a parallelogram opposite sides are equal
∠A = ∠C = 45°
∠B = ∠D = 135°


6. ABCD is a parallelogram in which  ∠A = 70°. Compute ∠B, ∠C and ∠D.

Solution

In a parallelogram ABCD 
∠A = 70°  [∵ Adjacent angles supplementary]
∠A = ∠B = 180°
⇒ 70° + ∠B = 180°  [∵ ∠A = 70°]
⇒ ∠B = 180° - 70° = 110°
In a parallelogram opposite sides are equal
∠A = ∠C = 70°
∠B = ∠D = 110°


7. In fig., below, ABCD is a parallelogram in which  ∠A = 60 . If the bisectors of ∠A and ∠B 
meet at P, prove that AD = DP, PC = BC and DC = 2AD. 

Solution

AP bisects ∠A 
Then, ∠AP  = ∠PAB = 30° 
Adjacent angles are  supplementary 
Then, ∠A + ∠B = 180° 
∠B + 60° = 180°  [∠A = 60°]
⇒ ∠B = 180° - 60°
⇒ ∠B = 120°
BP bisects ∠B 
Then, ∠PBA + ∠PBC = 30° 
∠PAB = ∠APD = 30°  [Alternative interior angles]
∴ AD = DP  [∵ Sides opposite to equal  angles are in equal length]
Similarly 
∠PBA = ∠BPC = 60°  [Alternative interior angle]
∴ PC = BC 
DC = DP + PC 
⇒ DC = AD + BC  [∵ DP = AD, PC = BC]
⇒ DC = 2AD         
[∵ AD = BC Opposite sides of a parallelogram are equal].

8. In fig. below, ABCD is a parallelogram in which ∠DAB  = 75 and ∠DBC = 60° . Compute ∠CDB and ∠ADB.
Solution

9. In below fig. ABCD is a parallelogram and E is the mid - point of side BC. If DE and AB when produced meet at F, prove that AF = 2AB. 
Solution
In ΔBEF and ΔCED
∠BEF = ∠CED  [Verified opposite angle]
BE = CE  [∵ E is the mid - point of BC] 
∠EBF = ∠ECD   [∴ Alternate interior angles are equal]
∵ ΔBEF ≅ ΔCED   [Angle side angle congruence]
∵ BF = CD  [Corresponding Parts of Congruent Triangles]
AF = AB + AF 
AF = AB + AB 
AF = 2AB 

10. Which of the following statements are true (T) and which are false (F)?
(i) In a parallelogram, the diagonals are equal.
(ii) In a parallelogram, the diagonals bisect each other.
(iii) In a parallelogram, the diagonals intersect each other at right angles.
(iv) In any quadrilateral, if a pair of opposite sides is equal, it is a parallelogram.
(v) If all the angles of a quadrilateral are equal, it is a parallelogram.
(vi) If three sides of a quadrilateral are equal, it is a parallelogram.
(vii) If three angles of a quadrilateral are equal, it is a parallelogram.
(viii) If all the sides of a quadrilateral are equal it is a parallelogram.
Solution
(i) False
(ii) True
(iii) False
(iv) False
(v) True
(vi) False
(vii) False
(viii) True
Previous Post Next Post