myhelper: RS Aggarwal solution class 8 chapter 16 Parallelograms Exercise 16A

Exercise 16A

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Q1 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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Question 1:

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

Answer 1:

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

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

Q2 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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Question 2:

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

Answer 2:

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

Q3 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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Question 3:

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

Answer 3:

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

Q4 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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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 4:

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

Q5 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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Question 5:

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

Answer 5:

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

Q6 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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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 6:

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

Q7 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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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 7:

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

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

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

Q8 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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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 8:

Refer to the figure given in the book.

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

Q9 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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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 9:

Refer to the figure of the book.

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

Q10 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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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 10:

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

Q11 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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Question 11:

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

Answer 11:

Refer to the figure given in the book.

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

Q12 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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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 12:

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

Q13 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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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 13:

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

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Q14 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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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 14:

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

RS Aggarwal solution class 8 chapter 16 Parallelograms Exercise 16A

Exercise 16A

Page-193

Q1 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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Answer 1:

Q2 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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Q3 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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Q4 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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Answer 4:

Q5 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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Answer 5:

Q6 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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Answer 6:

Q7 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

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Answer 7:

Q8 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

Question 8:

Answer 8:

Q9 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

Question 9:

Answer 9:

Q10 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

Question 10:

Answer 10:

Q11 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

Question 11:

Answer 11:

Q12 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

Question 12:

Answer 12:

Q13 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

Question 13:

Answer 13:

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Q14 | Ex-16A | Class 8 | RS AGGARWAL | chapter 16 | Parallelograms

Question 14:

Answer 14:

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