RS Aggarwal Class 8 Math Sixteenth Chapter Parallelograms Exercise 16A Solution
(1) ABCD is parallelogram in which ∠A = 110o. Find the measure of each of the angles ∠B, ∠C and ∠D.
Solution: It is a given that ABCD is a parallelogram in which ∠A = 110o. Since, the sum of any two adjacent angles of a parallelogram is 180o, we have
∠A + ∠B = 180o
⇒ ∠B = 180o – 110o
⇒ ∠B = 70o
Also, ∠B + ∠C = 180o
⇒ ∠C = 180o – 70o
⇒ ∠C = 110o
Further, ∠C + ∠D = 180o
⇒ ∠D = 180o – 110o
⇒ ∠D = 70o
∴ ∠B = 70o , ∠C = 110o and ∠D = 70o.
(2) Two adjacent angles of a parallelogram are equal. What is the measure of each of these angles?
Solution: Let the measure of each equal angle be xo.
∴ x + x = 180o
⇒ 2x = 180
⇒ x = 90
Hence, the measure of each angle is 90o.
(3) Two adjacent angles of a parallelogram are in the ratio 4 : 5. Find the measure of each of its angles.
Solution: Let the measure of the adjacent angles be 4x and 5x
∴ 4x + 5x = 180
⇒ 9x = 180
⇒ x = 20
Therefore the measure of the required angle is
∠A = 4 × 20 = 80o
∠B = 5 × 20 = 100o
∠B + ∠C = 180o
⇒ ∠C = 180o – 100o = 80o
∠C + ∠D = 180o
⇒ ∠D = 180o – 80o = 100o
(4) Two adjacent angles of a parallelogram are (3x – 4)o and (3x + 16)o. Find the value of x and hence find the measure of each angles.
Solution: (3x – 4) + (3x + 16) = 180
⇒ 3x – 4 + 3x + 16 = 180
⇒ 6x + 12 = 180
⇒ 6x = 180 – 12
⇒ 6x = 168
⇒ x = 28
Therefore, the measure of the each angle is
∠A = (3× 28 – 4) = 80o
∠B = (3× 28 + 16) = 100o
(5) The sum of two opposite angles of a parallelogram is 130o. Find the measure of each of its angles.
Solution: ∠A + ∠C = 130
Let the measure of ∠A = ∠C = x
∴ 2x = 130
⇒ x = 65
Therefore, ∠A = 65
∴ ∠A + ∠B = 180
⇒ ∠B = 180 – 65
⇒ ∠B = 115
∠C = 65
∴ ∠C + ∠D = 180
⇒ ∠D = 180 – 65
⇒ ∠D = 115.
(6) Two sides of a parallelogram are in the ratio 5 : 3. If its perimeter is 64 cm, find the lengths of its sides.
Solution: Let the measure of the sides be 5x and 3x.
Its perimeter = 2(5x + 3x)
∴ 2(5x + 3x) = 64
⇒ 16x = 64
⇒ x = 4
Therefore, one side = 5 × 4 = 20
Other side = 3 × 4 = 12
(7) The perimeter of a parallelogram is 140 cm. If one of the sides is longer than the other by 10cm, find the length of each of its sides.
Solution: Let the length of one side be x cm and other is (x + 10) cm.
∴ 2(x + x + 10) = 140
⇒ 4x + 20 = 140
⇒ 4x = 140 – 20
⇒ 4x = 120
⇒ x = 30
Length of one side is 30 cm and other side = (30 + 10) = 40 cm.
(8) In the adjacent figure, ABCD is a rectangle. If BM and DN are perpendiculars from B and D on AC, prove that ∆BMC ≅ ∆DNA. Is it true that BM = DN?
Solution: In ∆BMC and ∆DNA:
∠DNA = ∠BMC = 90o
∠BCM = ∠DAN (alternative angles)
BC = DA (opposite sides)
By AAs congruency criteria:
∆BMC ≅ ∆DNA (proved)
So, we can write BM = DN.
(9) In the adjacent figure, ABCD is a parallelogram and line segments AE and CF bisect the angles A and C respectively. Show that AE ∥ CF.
Solution: In ∆ADE and ∆CBF,
We have AD = BC, ∠B = ∠D and ∠DAE = ∠BCF
∵ ∠A = ∠C
And therefore, CD – DE = AB – BF
So, CE = AF
∴ AECF is a parallelogram.
Hence, AE ∥ CF.
(10) The lengths of the diagonals of a rhombus are 16 cm and 12 cm respectively. Find the length of each of its sides.
Solution: We know that the diagonals of a rhombus bisect each other at right angles. Ac and BD are intersecting at a point O.
Therefore, length of each side is 10 cm. Because all sides of a rhombus are equal.
(11) In the given figure ABCD is square. Find the measure of ∠CAD.
Solution: In ∆ADC,
DA = DC
⇒ ∠ACD = ∠DAC = xo
Then, xo + xo + 90o = 180o
⇒ 2xo = 180o – 90o
⇒ 2xo = 90o
⇒ xo = 45o
(12) The sides of a rectangle are in the ratio 5 : 4 and its perimeter is 90 cm. Find its length and breadth.
Solution: Let the length and breadth of the rectangle be 5x and 4x respectively.
∴ 2(5x + 4x) = 90
⇒ 18x = 90
⇒ x = 5
Length of the rectangle is (5× 5) = 25 cm and breadth = (4 × 5) = 20 cm.
(13) Name each of the following parallelograms.
(i) The diagonals are equal and the adjacent sides are unequal.
(ii) The diagonals are equal and the adjacent sides are equal.
(iii) The diagonals are unequal and the adjacent sides are equal.
(iv) All the sides are equal and one angle is 60o.
(v) All the sides are equal and angle is 90o.
(vi) All the angles are equal and the adjacent sides are unequal.
(14) Which of the following statements are true and which are false?
(i) The diagonals of a parallelogram are equal. ⇒ False
(ii) The diagonals of a rectangle are perpendicular to each other. ⇒ False
(iii) The diagonals of a rhombus are equal. ⇒ False
(iv) Every rhombus is a kite. ⇒ False
(v) Every rectangle is square. ⇒ False
(vi) Every square is parallelogram. ⇒ True
(vii) Every square is rhombus. ⇒ True
(viii) Every rectangle is a parallelogram. ⇒ True
(ix) Every parallelogram is rectangle. ⇒ False
(x) Every rhombus is a parallelogram. ⇒ True