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