## 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

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