Rs Aggarwal 2020 2021 for Class 8 Maths Chapter 16 - Parallelograms

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Rs Aggarwal 2020 2021 Solutions for Class 8 Maths Chapter 16 Parallelograms are provided here with simple step-by-step explanations. These solutions for Parallelograms are extremely popular among Class 8 students for Maths Parallelograms Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rs Aggarwal 2020 2021 Book of Class 8 Maths Chapter 16 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rs Aggarwal 2020 2021 Solutions. All Rs Aggarwal 2020 2021 Solutions for class Class 8 Maths are prepared by experts and are 100% accurate.

Page No 193:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

Answer:

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

Page No 194:

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:

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:

Answer:

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

Page No 194:

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:

Answer:

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

Page No 194:

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:

Answer:

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

Page No 194:

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:

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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