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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
Two opposite angles of a parallelogram are (3x - 2)° and (50 - x)°. Find the measure of each angle of the parallelogram.
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°.
If an angle of a parallelogram is two-third of its adjacent angle, find the angles of the parallelogram.
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°
Find the measure of all the angles of a parallelogram, if one angle is 24° less than twice the smallest angle.
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°.
The perimeter of a parallelogram is 22 cm. If the longer side measures 6.5 cm what is the measure of the shorter side?
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
In a parallelogram ABCD, ∠D = 135°. Determine the measures of ∠A and ∠B.
In a parallelogram ABCD
So, ∠D + ∠C = 180°
∠C = 180° − 135°
∠C = 45°
In a parallelogram opposite sides are equal.
∠A = ∠C = 45°
∠B = ∠D = 135°
ABCD is a parallelogram in which ∠A = 700. Compute ∠B, ∠C and ∠D.
∠A = 70°
∠A + ∠B = 180° [Since, adjacent angles are supplementary]
70° + ∠B = 180° [∵ ∠A = 70°]
∠B = 1800 − 70°
∠B = 110°
∠A = ∠C = 70°
∠B = ∠D = 110°
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.
AP bisects ∠A
Then, ∠DAP = ∠PAB = 30°
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]
In figure, ABCD is a parallelogram in which ∠DAB = 75° and ∠DBC = 60°. Compute ∠CDB, and ∠ADB.
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°
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.
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.
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.
(i) False
(ii) True
(iii) False
(iv) False
(v) True
(vi) False
(vii) False
(viii) True
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