RS Aggarwal 2017 Solutions for Class 8 Math Chapter 16 - Parallelograms

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RS Aggarwal 2017 Solutions for Class 8 Math Chapter 16 Parallelograms are provided here with simple step-by-step explanations. These solutions for Parallelograms are extremely popular among class 8 students for Math Parallelograms Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the RS Aggarwal 2017 Book of class 8 Math Chapter 16 are provided here for you for free. You will also love the ad-free experience on Meritnation’s RS Aggarwal 2017 Solutions. All RS Aggarwal 2017 Solutions for class 8 Math are prepared by experts and are 100% accurate.

Page No 193:

Question 1:

ABCD is a parallelogram in which ∠A = 110°. Find the measure of each of the angles ∠B, ∠C and ∠D.

Answer:

$It is given that A B C D is a parallelogram in which ∠ A is equal to 110 ° . Sum of t h e adjacent angles of a parallelogram is 180 ° . ∴ ∠ A + ∠ B = 180 ° \Rightarrow 110 ° + ∠ B = 180 ° \Rightarrow ∠ B = (180 ° - 110 °) \Rightarrow ∠ B = 70 ° ∴ ∠ B = 70 °$

$Also, ∠ B + ∠ C = 180 ° \Rightarrow 70 ° + ∠ C = 180 ° \Rightarrow ∠ C = (180 ° - 70 °) \Rightarrow ∠ C = 110 ° ∴ ∠ C = 110 ° Further, ∠ C + ∠ D = 180 ° \Rightarrow 110 ° + ∠ D = 180 ° \Rightarrow ∠ D = (180 ° - 110 °) \Rightarrow ∠ D = 70 ° ∴ ∠ D = 70 °$

Page No 193:

Question 2:

Two adjacent angles of a parallelogram are equal. What is the measure of each of these angles?

Answer:

$Let the required angle be x ° . As the adjacent angles are equal, we have : x + x = 180 (s ince the sum of adjacent angles of a parallelogram is 180 °) \Rightarrow 2 x = 180 \Rightarrow x = \frac{180}{2} \Rightarrow x = 90 ° Hence, the measure of each of the angles is 90 ° .$

Page No 193:

Question 3:

Two adjacent angles of a parallelogram are in the ratio 4 : 5. Find the measure of each of its angles.

Answer:

$Let A B C D be the parallelogram . Then, ∠ A and ∠ B are its adjacent angles . Let ∠ A = (4 x) ° ∠ B = (5 x) ° ∴ ∠ A + ∠ B = 180 ° [s ince sum of t h e adjacent angles of a parallelogram is 180 °] \Rightarrow 4 x + 5 x = 180 \Rightarrow 9 x = 180 \Rightarrow x = \frac{180}{9} \Rightarrow x = 20 ∴ ∠ A = (4 \times 20) ° = 80 ° ∠ B = (5 \times 20) ° = 100 ° O pposite angles of parallelogram are equal . ∴ ∠ C = ∠ A = 80 ° ∠ D = ∠ B = 100 °$

Page No 193:

Question 4:

Two adjacent angles of a parallelogram are (3x − 4)° and (3x + 16)°. Find the value of x and hence find the measure of each of its angles.

Answer:

$Let A B C D be a parallelogram . L et ∠ A = (3 x - 4) ° ∠ B = (3 x + 16) ° ∴ ∠ A + ∠ B = 180 ° [s ince the sum of adjacent angles of a parallelogram is 180 °] \Rightarrow 3 x - 4 + 3 x + 16 = 180 \Rightarrow 3 x - 4 + 3 x + 16 = 180 \Rightarrow 6 x + 12 = 180 \Rightarrow 6 x = 168 \Rightarrow x = \frac{168}{6} \Rightarrow x = 28 ∴ ∠ A = (3 \times 28 - 4) ° = (84 - 4) ° = 80 ° ∠ B = ((3 \times 28) + 16) ° = (84 + 16) ° = 100 ° The opposite angles of a paralleleogram are equal . ∴ ∠ C = ∠ A = 80 ° ∠ D = ∠ B = 100 °$

Page No 193:

Question 5:

The sum of two opposite angles of a parallelogram is 130°. Find the measure of each of its angles.

Answer:

$Let A B C D be a parallelogram and let the sum of its opposite angles be 130 ° . ∠ A + ∠ C = 130 ° T he opposite angles are equal in a parallelogram . ∴ ∠ A = ∠ C = x ° \Rightarrow x + x = 130 \Rightarrow 2 x = 130 \Rightarrow x = \frac{130}{2} \Rightarrow x = 65 ∴ ∠ A = 65 ° a n d ∠ C = 65 ° ∠ A + ∠ B = 180 ° [s ince the sum of adjacent angles of a parallelogram is 180 °] \Rightarrow 65 ° + ∠ B = 180 ° \Rightarrow ∠ B = (180 - 65) ° \Rightarrow ∠ B = 115 ° ∠ D = ∠ B = 115 ° [o pposite angles of parallelogram are equal]$

Page No 193:

Question 6:

Two sides of a parallelogram are in the ratio 5 : 3. If its perimeter is 64 cm, find the lengths of its sides.

Answer:

$Let the lengths of two sides of the parallelogram be 5 x cm and 3 x cm, respectively . Then, its perimeter = 2 (5 x + 3 x) cm$
$= 16 x cm$
$∴ 16 x = 64 \Rightarrow x = \frac{64}{16} \Rightarrow x = 4 ∴ One side \Rightarrow (5 \times 4) cm = 20 cm Other side \Rightarrow (3 \times 4) cm = 12 cm$

Page No 193:

Question 7:

The perimeter of a parallelogram is 140 cm. If one of the sides is longer than the other by 10 cm, find the length of each of its sides.

Answer:

$Let the lengths of two sides of the parallelogram be x cm and (x + 10) cm, respectively . Then, its perimeter = 2 [x + (x + 10)] cm$

$= 2 [x + x + 10] cm = 2 [2 x + 10] cm = 4 x + 20 cm$

$4 x + 20 = 140 \Rightarrow 4 x = 140 - 20 \Rightarrow 4 x = 120 \Rightarrow x = \frac{120}{4} \Rightarrow x = 30 Length of one side = 30 cm Length of the other side \Rightarrow (30 + 10) cm = 40 cm$

Page No 193:

Question 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?

Answer:

Refer to the figure given in the book.

$In ∆ B M C and ∆ D N A : ∠ D N A = ∠ B M C = 90 ° ∠ B C M = ∠ D A N (alternate angles) B C = D A (opposite sides) B y AAS congruency criteria : ∆ B M C ≅ ∆ D N A Yes, it is true that B M i s e q u a l t o D N . (b y corresponding parts of congruent triangles B M C and D N A)$

Page No 193:

Question 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.

Answer:

Refer to the figure of the book.

$∠ A = ∠ C (opposite angles of a parallelogram are equal) \Rightarrow \frac{1}{2} ∠ A = \frac{1}{2} ∠ C = > ∠ E A D = ∠ F C B (A E and C F bisect the angles A and C, respectively) In ∆ A D E and ∆ C B F : ∠ B = ∠ D (o pposite angles of a parallelogram are equal) ∠ E A D = ∠ F C B (p roved above) A D = B C (o pposite sides of a parallelogram are equal) B y AAS concruency criteria : ∆ A D E ≅ ∆ B C F D E = B F (c orresponding parts of congruent triangles) C D = A B (o pposite sides of a parallelogram are equal) Also, C D - D E = A B - B F \Rightarrow C E = A F A B C D is a paralleleogram . ∴ C D ∥ A B (o pposite sides of a parallelogram are parallel) = > C E ∥ A F If one pair of sides of a quadrilateral is parallel and equal, then it is a parallelogram . Therefore, AECF is a parallelogram . ∴ A E ∥ C F$

Page No 193:

Question 10:

The lengths of the diagonals of a rhombus are 16 cm and 12 cm respectively. Find the length of each of its sides.

Answer:

$Let A B C D be a rhombus . L et A C and B D be the diagonals of the rhombus intersecting at a point O . Let A C = 16 cm BD = 12 cm We know that the diagonals of a rhombus bisect each other at right angles . ∴ A O = \frac{1}{2} A C = (\frac{1}{2} \times 16) cm = 8 cm B O = \frac{1}{2} B D = (\frac{1}{2} \times 12) cm = 6 cm From the right ∆ A O B : A B^{2} = A O^{2} + B O^{2} = \{{(8)}^{2} + {(6)}^{2}\} {cm}^{2} = (64 + 36) {cm}^{2} = 100 {cm}^{2} \Rightarrow A B = \sqrt{100} cm = 10 cm Hence, the length of the side A B i s 10 cm . A B = B C = C D = D A = 10 cm (all sides of a rhombus are equal)$

Page No 193:

Question 11:

In the given figure ABCD is a square. Find the measure of ∠CAD.

Answer:

Refer to the figure given in the book.

$In ∆ A D C : D A = D C (all sides of a square are equal) \Rightarrow ∠ A C D = ∠ C A D Let ∠ A C D = ∠ C A D = x ° [Angle opposite to the equal sides are equal] x + x + 90 = 180 [since the sum of the angles of a triangle is 180 °] \Rightarrow 2 x + 90 = 180 \Rightarrow 2 x = 90 \Rightarrow x = \frac{90}{2} \Rightarrow x = 45 ∴ ∠ C A D = 45 °$

Page No 193:

Question 12:

The sides of a rectangle are in the ratio 5 : 4 and its perimeter is 90 cm. Find its length and breadth.

Answer:

$Let the length of two sides of the rectangle be 5 x cm and 4 x cm, respectively . Then, its perimeter = 2 (5 x + 4 x) cm$
$= 18 x cm$
$∴ 18 x = 90 \Rightarrow x = \frac{90}{18} \Rightarrow x = 5 Length of one side \Rightarrow (5 \times 5) cm = 25 cm Length of the other side \Rightarrow (4 \times 5) cm = 20 cm ∴ Length of the rectangle = 25 cm Breadth = 20 cm$

Page No 193:

Question 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 60°.
(v) All the sides are equal and one angle is 90°.
(vi) All the angles are equal and the adjacent sides are unequal.

Answer:

$(i) The diagonals are equal and the adjacent sides are unequal . Hence, the given parallelogram is a rectangle . (ii) The diagonals are equal and the adjacent sides are equal . Hence, the given parallelogram is a square . (iii) The diagonals are unequal and the adjacent sides are equal . Hence, the given parallelogram is a rhombus . (iv) All the sides are equal and one angle is 60 ° . Hence, the given parallelogram is a rhombus . (v) All the sides are equal and one angle is 90 ° . Hence, the given parallelogram is a square . (vi) All the angles are equal and the adjacent sides are unequal . Hence, the given parallelogram is a rectangle .$

Page No 194:

Question 14:

Which of the following statements are true and which are false?
(i) The diagonals of a parallelogram are equal.
(ii) The diagonals of a rectangle are perpendicular to each other.
(iii) The diagonals of a rhombus are equal.
(iv) Every rhombus is a kite.
(v) Every rectangle is a square.
(vi) Every square is a a parallelogram.
(vii) Every square is a rhombus.
(viii) Every rectangle is a parallelogram.
(ix) Every parallelogram is a rectangle.
(x) Every rhombus is a parallelogram.

Answer:

$(i) The given statement is false . The diagonals of a parallelogram bisect each other, but they are not equal in length . (ii) The given statement is false . The diagonals of a rectangle are equal and bisect each other, but they are not perpendicular . (iii) The given statement is false . All the sides of a rhombus are equal, but the diagonals are not equal . (iv) The given statement is true . (v) The given statement is false . Every square is a rectangle, but every rectangle is not a square . (vi) The given statement is true . (vii) The given statement is true . (viii) The given statement is true . (ix) The given statement is false . A rectangle is a special type of parallelogram, but every parallelogram is not a rectangle . (x) The given statement is true .$

Page No 194:

Question 1:

Tick (✓) the correct answer
The two diagonals are not necessarily equal in a
(a) rectangle
(b) square
(c) rhombus
(d) isosceles trapezium

Answer:

$(c) rhombus In a rhombus, the two diagonals are not necessarily equal .$

Page No 194:

Question 2:

Tick (✓) the correct answer
The lengths of the diagonals of a rhombus are 16 cm and 12 cm. The length of each side of the rhombus is
(a) 8 cm
(b) 9 cm
(c) 10 cm
(d) 12 cm

Answer:

$(c) 10 cm$

$Let A B C D be a rhombus . L et A C and B D be the diagonals of the rhombus intersecting at a point O . A C = 16 cm B D = 12 cm We know that the diagonals of a rhombus bisect each other at right angles . ∴ A O = \frac{1}{2} A C = (\frac{1}{2} \times 16) cm = 8 cm B O = \frac{1}{2} B D = (\frac{1}{2} \times 12) cm = 6 cm From the right ∆ A O B : A B^{2} = A O^{2} + B O^{2} = \{{(8)}^{2} + {(6)}^{2}\} {cm}^{2} = (64 + 36) {cm}^{2} = 100 {cm}^{2} \Rightarrow A B = \sqrt{100} cm = 10 cm Hence, the length of the side A B i s 10 cm . Therefore, the length of each side of the rhombus is 10 cm because all t h e sides of a rhombus are equal .$

Page No 194:

Question 3:

Tick (✓) the correct answer
Two adjacent angles of a parallelogram are (2x + 25)° and (3x − 5)°. The value of x is
(a) 28
(b) 32
(c) 36
(d) 42

Answer:

$(b) 32 We know that the sum of adjacent angles of a parallelogram i s 180 ° . \Rightarrow 2 x + 25 + 3 x - 5 = 180 \Rightarrow 5 x + 20 = 180 \Rightarrow 5 x = 180 - 20 \Rightarrow 5 x = 160 \Rightarrow x = \frac{160}{5} \Rightarrow x = 32 Therefore, the value of x is 32 .$

Page No 194:

Question 4:

Tick (✓) the correct answer:
The diagonals do not necessarily intersect at right angles in a
(a) parallelogram
(b) rectangle
(c) rhombus
(d) kite

Answer:

$(a) parallelogram In a parallelogram, the diagonals do not necessarily intersect at right angles .$

Page No 194:

Question 5:

Tick (✓) the correct answer:
The length and breadth of a rectangle are in the ratio 4 : 3. If the diagonal measures 25 cm then the perimeter of the rectangle is
(a) 56 cm
(b) 60 cm
(c) 70 cm
(d) 80 cm

Answer:

(c) 70 cm

$Let A B C D be a rectangle and let the diagonal A C b e 25 cm, length A B b e 4 x cm and breadth B C b e 3 x cm . Each angle of a rectangle is a right angle . ∴ ∠ A B C = 90 ° From the right ∆ A B C : A C^{2} = A B^{2} + B C^{2} \Rightarrow {(25)}^{2} = {(4 x)}^{2} + {(3 x)}^{2} \Rightarrow 625 = 16 x^{2} + 9 x^{2} \Rightarrow 625 = 25 x^{2}$

$x^{2} = \frac{625}{25} = 25 \Rightarrow x = 5 ∴ L ength = 4 \times 5 = 20 cm Breadth = 3 \times 5 = 15 cm$

∴ Perimeter of the rectangle = 2(20+15) cm
$= 70 c m$

Page No 194:

Question 6:

Tick (✓) the correct answer:
The bisectors of any two adjacent angles of a parallelogram intersect at
(a) 30°
(b) 45°
(c) 60°
(d) 90°

Answer:

$(d) 90 ° The bisectors of any two adjacent angles of a parallelogram intersect at 90 ° .$

Page No 194:

Question 7:

Tick (✓) the correct answer:
If an angle of a parallelogram is two-thirds of its adjacent angle, the smallest angle of the parallelogram is
(a) 54°
(b) 72°
(c) 81°
(d) 108°

Answer:

$(b) 72 ° Let x ° be the angle of the parallelogram . Sum of the adjacent angles of a parallelogram is 180 ° . ∴ x + (\frac{2}{3} \times x) = 180 \Rightarrow x + \frac{2 x}{3} = 180 \Rightarrow (x + \frac{2 x}{3}) = 180 \Rightarrow \frac{5 x}{3} = 180 \Rightarrow x = (180 \times \frac{3}{5}) \Rightarrow x = 108 Hence, one angle of the parallelogram is 108 ° . Its adjacent angle = (180 - 108) ° = 72 ° Therefore, the smallest angle of the parallelogram is 72 ° .$

Page No 194:

Question 8:

Tick (✓) the correct answer:
The diagonals do not necessarily bisect the interior angles at the vertices in a
(a) rectangle
(b) square
(c) rhombus
(d) all of these

Answer:

$(a) r ectangle In a rectangle, the diagonals do not necessarily bisect the interior angles at the vertices .$

Page No 194:

Question 9:

Tick (✓) the correct answer:
In a square ABCD, AB = (2x + 3) cm and BC = (3x − 5) cm. Then, the value of x is
(a) 4
(b) 5
(c) 6
(d) 8

Answer:

$(d) 8 A ll the sides of a square are equal . ∴ A B = B C \Rightarrow 2 x + 3 = 3 x - 5 \Rightarrow 3 + 5 = 3 x - 2 x \Rightarrow 8 = x Therefore, the value of x is 8 .$

Page No 194:

Question 10:

Tick (✓) the correct answer:
If one angle of a parallelogram is 24° less than twice the smallest angle then the largest angle of the parallelogram is
(a) 68°
(b) 102°
(c) 112°
(d) 176°

Answer:

$(c) 112 ° Let x ° be the smallest angle of the parallelogram . T he sum of adjacent angles of a parallelogram is 180 ° . ∴ x + 2 x - 24 = 180 \Rightarrow 3 x - 24 = 180 \Rightarrow 3 x = 180 + 24 \Rightarrow 3 x = 204 \Rightarrow x = \frac{204}{3} \Rightarrow x = 68 ∴ Smallest angle = 68 ° L argest angle = (180 - 68) ° = 112 °$

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