Chapter 14: Quadrilaterals Exercise – 14.2

Question: 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^°

Therefore, (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^°.

Question: 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.

Therefore, 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^°, 180^°, 72^°

Question: 3

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

Solution:

x + 2x - 24 = 180^°

⟹ 3x - 24 = 180^°

⟹ 3x = 108^° + 24

⟹ 3x = 204^°

⟹ x = 204/3 = 68°

⟹ x = 68^°

⟹ 2x - 24^° = 2*68^° - 24^° = 112^°

Hence, four angles are 68^°, 112^°, 68^°, 112^°.

Question: 4

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

Solution:

Let the shorter side be 'x'.

Therefore, perimeter = x + 6.5 + 6.5 + x [Sum of all sides]

22 = 2(x + 6.5)

11 = x + 6.5

⟹ x = 11 - 6.5 = 4.5 cm

Therefore, shorter side = 4.5 cm

Question: 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°

∠C = 180° − 135°

∠C = 45°

In a parallelogram opposite sides are equal.

∠A = ∠C = 45°

∠B = ∠D = 135°

Question: 6

ABCD is a parallelogram in which ∠A = 700. Compute ∠B, ∠C and ∠D.

Solution:

In a parallelogram ABCD

∠A = 70°

∠A + ∠B = 180°        [Since, adjacent angles are supplementary]

70° + ∠B = 180°        [∵ ∠A = 70°]

∠B = 1800 − 70°

∠B = 110°

In a parallelogram opposite sides are equal.

∠A = ∠C = 70°

∠B = ∠D = 110°

Question: 7

In Figure, 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, ∠DAP = ∠PAB = 30°

Adjacent angles are supplementary

Then, ∠A + ∠B = 180°

∠B + 600 = 180°

∠B = 180° − 60°

∠B = 120°

BP bisects ∠B

Then, ∠PBA = ∠PBC = 30°

∠PAB = ∠APD = 30° [Alternate interior angles]

Therefore, AD = DP   [Sides opposite to equal angles are in equal length]

Similarly

∠PBA = ∠BPC = 60° [Alternate interior angles]

Therefore, PC = BC

DC = DP + PC

DC = AD + BC   [Since, DP = AD and PC = BC]

DC = 2AD    [Since, AD = BC, opposite sides of a parallelogram are equal]

Question: 8

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

Solution:

To find ∠CDB and ∠ADB

∠CBD = ∠ABD = 60°      [Alternate interior angle. AD∥ BC and BD is the transversal]

In ∠BDC

∠CBD + ∠C + ∠CDB = 180°      [Angle sum property]

⇒ 60° + 75° + ∠CDB = 180°

⇒ ∠CDB = 180° − (60° + 75°)

⇒ ∠CDB = 45°

Hence, ∠CDB = 45°, ∠ADB = 60°

Question: 9

In figure, 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                     [Since, E is the mid-point of BC]

∠EBF = ∠ECD            [Since, Alternate interior angles are equal]

∴  ΔBEF ≅ ΔCED         [ASA congruence]

∴  BF = CD [CPCT]

AF = AB + AF

AF = AB + AB

AF = 2AB.

Hence proved.

Question: 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