RS Aggarwal Class 8 Math Sixteenth Chapter Parallelograms Exercise 16A Solution

RS Aggarwal Class 8 Math Sixteenth Chapter Parallelograms Exercise 16A Solution

EXERCISE 16A

(1) ABCD is parallelogram in which ∠A = 110^o. 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 = 110^o. Since, the sum of any two adjacent angles of a parallelogram is 180^o, we have

∠A + ∠B = 180^o

⇒ ∠B = 180^o – 110^o

⇒ ∠B = 70^o

Also, ∠B + ∠C = 180^o

⇒ ∠C = 180^o – 70^o

⇒ ∠C = 110^o

Further, ∠C + ∠D = 180^o

⇒ ∠D = 180^o – 110^o

⇒ ∠D = 70^o

∴ ∠B = 70^o , ∠C = 110^o and ∠D = 70^o.

(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 x^o.

∴ x + x = 180^o

⇒ 2x = 180

⇒ x = 90

Hence, the measure of each angle is 90^o.

(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 = 80^o

∠B = 5 × 20 = 100^o

∠B + ∠C = 180^o

⇒ ∠C = 180^o – 100^o = 80^o

∠C + ∠D = 180^o

⇒ ∠D = 180^o – 80^o = 100^o

(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) = 80^o

∠B = (3× 28 + 16) = 100^o

(5) The sum of two opposite angles of a parallelogram is 130^o. 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 = 90^o

∠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 = x^o

Then, x^o + x^o + 90^o = 180^o

⇒ 2x^o = 180^o – 90^o

⇒ 2x^o = 90^o

⇒ x^o = 45^o

(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.

Ans: Rectangle.

(ii) The diagonals are equal and the adjacent sides are equal.

Ans: Square.

(iii) The diagonals are unequal and the adjacent sides are equal.

Ans: Rhombus.

(iv) All the sides are equal and one angle is 60^o.

Ans: Rhombus.

(v) All the sides are equal and angle is 90^o.

Ans: Square.

(vi) All the angles are equal and the adjacent sides are unequal.

Ans: Rectangle.

(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

 

For more exercise solution, Click below –

• Exercise – 16B