Exercise 4.1

NCERT Solutions for Class 8 Maths Chapter 4 Quadrilaterals - Exercise 4.1

Mastering the classification of polygons and curves is simple with our NCERT Solutions for Class 8 Maths Ch 4: Understanding Quadrilaterals – Exercise 4.1. In this chapter, we explore the world of plane figures, distinguishing between open and closed curves, and identifying the unique properties of various polygons.

Class 8 Math Lessons (Ch 4): Learn to Classify Shapes with Ease! The solutions to Exercise 4.1 will be presented in a step-by-step approach to give students a clear understanding of concave vs. convex polygons and the angle sum property. These solutions are in accordance with the CBSE Guidelines to provide the best path for improving accuracy in geometry. Additionally, these solutions build the foundation needed for more complex constructions and proofs in later exercises.

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You can get our simple NCERT Solutions for Class 8 Maths Chapter 4 Exercise 4.1 in PDF format. Ideal for rapid reference and offline study.

NCERT Solutions Class 8 Maths Chapter 4 Ex 4.1

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2.0Detailed NCERT Class 8 Maths Chapter 4 Solutions of Exercise 4.1

1.Find all the other angles inside the following rectangles.

Sol. (i) The given rectangle is ABCD . D C

We have $∠1 = 3 0^{\circ}$ $∠1 + ∠2 = 9 0^{\circ}$ $∴ ∠2 = 9 0^{\circ} - ∠1 = 9 0^{\circ} - 3 0^{\circ} = 6 0^{\circ}$ $MD = MA$ $\Rightarrow ∠3 = ∠2 = 6 0^{\circ}$ $∠3 + ∠ Z 4 = 9 0^{\circ}$ $∴ ∠4 = 9 0^{\circ} - ∠3 = 9 0^{\circ} - 6 0^{\circ} = 3 0^{\circ}$ $MC = MD$ $\Rightarrow ∠5 = ∠4 = 3 0^{\circ}$ $∠5 + ∠6 = 9 0^{\circ}$ $∴ ∠6 = 9 0^{\circ} - ∠5 = 9 0^{\circ} - 3 0^{\circ} = 6 0^{\circ}$ $MB = MC$ $\Rightarrow ∠7 = ∠6 = 6 0^{\circ}$ $MB = MA$ $∠8 = ∠1 = 3 0^{\circ}$ In $△ AMB$ , we have $∠1 + ∠9 + ∠8 = 18 0^{\circ}$ $∴ 3 0^{\circ} + ∠9 + 3 0^{\circ} = 18 0^{\circ}$ $∴ ∠9 + 18 0^{\circ} - 6 0^{\circ} = 12 0^{\circ}$ $∠11 = ∠9 = 12 0^{\circ}$ (Vertically opposite angles) $∠9 + ∠10 = 18 0^{\circ}$ (Linear angles) $∴ ∠10 = 18 0^{\circ} - 12 0^{\circ} = 6 0^{\circ}$ $∠12 = ∠10 = 6 0^{\circ}$ (Vertically opposite angles) $∴ ∠2 = 6 0^{\circ}, ∠3 = 6 0^{\circ}, ∠4 = 3 0^{\circ}$ , $∠5 = 3 0^{\circ}, ∠6 = 6 0^{\circ}, ∠7 = 6 0^{\circ}$ , $∠8 = 3 0^{\circ}, ∠9 = 12 0^{\circ}, ∠10 = 6 0^{\circ}$ , $∠11 = 12 0^{\circ}$ , and $∠12 = 6 0^{\circ}$ , (ii) The given rectangle is PSRQ.

Q R

We have $∠9 = 11 0^{\circ}$ $∠11 + ∠9 = 11 0^{\circ}$ (Vertically opposite angles) $∠9 + ∠10 = 18 0^{\circ}$ (Linear angles) $∴ ∠10 = 18 0^{\circ} - 11 0^{\circ} = 7 0^{\circ}$ $∠12 = ∠10 = 7 0^{\circ}$ (Vertically opposite angles) $MP = MS$ $\Rightarrow ∠1 = ∠8$ In $△ PMS$ , we have $∠1 + ∠11 + ∠8 = 18 0^{\circ}$ $\Rightarrow ∠1 + 11 0^{\circ} + ∠1 = 18 0^{\circ}$ $\Rightarrow 2∠1 = 18 0^{\circ} - 11 0^{\circ}$ $\Rightarrow 2∠1 = 7 0^{\circ}$ $\Rightarrow ∠1 = 3 5^{\circ}$ $∴ ∠8$ is also $3 5^{\circ}$ . $∴ ∠1 + ∠2 = 9 0^{\circ}$ $\Rightarrow ∠2 = 9 0^{\circ} - ∠1$ $\Rightarrow ∠2 = 9 0^{\circ} - 3 5^{\circ}$ $\Rightarrow ∠2 = 5 5^{\circ}$ $MQ = MP$ $\Rightarrow ∠3 = ∠2 = 5 5^{\circ}$ $∴ ∠3 + ∠4 = 9 0^{\circ}$ $\Rightarrow ∠4 = 9 0^{\circ} - ∠3$ $\Rightarrow ∠4 = 9 0^{\circ} - 5 5^{\circ}$ $\Rightarrow ∠4 = 3 5^{\circ}$ $MR = MQ$ $\Rightarrow ∠5 = ∠4 = 3 5^{\circ}$ $∴ ∠5 + ∠6 = 9 0^{\circ}$ $\Rightarrow ∠6 = 9 0^{\circ} - ∠5$ $\Rightarrow ∠6 = 9 0^{\circ} - 3 5^{\circ}$ $\Rightarrow ∠6 = 5 5^{\circ}$ $MS = MR$ $\Rightarrow ∠7 = ∠6 = 5 5^{\circ}$ $∴ ∠1 = 3 5^{\circ}, ∠2 = 5 5^{\circ}, ∠3 = 5 5^{\circ}$ , $∠4 = 3 5^{\circ}, ∠5 = 3 5^{\circ}, ∠6 = 5 5^{\circ}, ∠7 = 5 5^{\circ}$ , $∠8 = 3 5^{\circ}, ∠10 = 7 0^{\circ}, ∠11 = 11 0^{\circ}$ and $∠12 = 7 0^{\circ}$ . 2. Draw a quadrilateral whose diagonals have equal lengths of 8 cm that bisect each other, and intersect at an angle of (i) $3 0^{\circ}$ (ii) $4 0^{\circ}$ (iii) $9 0^{\circ}$ (iv) $14 0^{\circ}$

Sol. (i) Draw a line AB equal to 8 cm . Take point M on AB such that $AM = BM = 4 cm$ . Using a protractor, draw an angle of $3 0^{\circ}$ at M on MB . On this line, take points C and D such that $MC = MD = 4 cm$ . Join $AD, DB, BC$ , and CA . ABCD is the required quadrilateral.

Since diagonals AB and CD are equal and are bisecting each other at M, ACBD is a rectangle. (ii) Draw a line AB equal to 8 cm . Take point M on AB such that $AM = BM = 4 cm$ .

Using a protractor, draw an angle of $4 0^{\circ}$ at M on MB . On this line, take points C and D such that $MC = MD = 4 cm$ . Join $AD, DB, BC$ , and $CA . ABCD$ is the required quadrilateral.

Since diagonals AB and CD are equal and are bisecting each other at M , ACBD is a rectangle. (iii) Draw a line AB equal to 8 cm . Take a point M on AB such that $AM = BM = 4 cm$ . Using a protractor, draw an angle of $9 0^{\circ}$ at M on MB . On this line, take points C and D such that $MC = MD = 4 cm$ . Join $AD, DB, BC$ , and $CA . ACBD$ is the required square.

Since diagonals AB and CD are equal and are bisecting each other at M , and also the diagonals are perpendicular to each other, ACBD is a square. (iv) Draw a line AB equal to 8 cm . Take a point M on AB such that $AM = BM =$ 4 cm .

Using a protractor, draw an angle of $14 0^{\circ}$ at M on MB.

On this line, take points C and D such that $MC = MD = 4 cm$ .

Join $AD, DB, BC$ , and $CA . ACBD$ is the required quadrilateral.

Since diagonals AB and CD are equal and are bisecting each other at M , ACBD is a rectangle. 3. Consider a circle with centre O . Line segments PL and AM are two perpendicular diameters of the circle. What is the figure APML? Reason and/or experiment to figure this out. Sol. In the figure, PL and AM are two perpendicular diameters of the circle. Let $r$ be the radius of the circle.

Since $PL = PO + OL = r + r = 2 r$ and $AM = AO + OM = r + r = 2 r$ $∴ PL = AM$ ∴ In the quadrilateral APML, diagonals PL and AM are equal and are perpendicular to each other.

Also, $OP = OA = OL = OM = r$ ∴ Diameters PL and AM bisect each other at 0 . ∴ Quadrilateral APML is a square. 4. We have seen how to get $9 0^{\circ}$ using paper folding. Now, suppose we do not have any paper but two sticks of equal length, and a thread. How do we make an exact $9 0^{\circ}$ using these?

Sol. Let AB and CD be two sticks of equal length, say 6 cm .

Mark the midpoints of the sticks using a ruler.

Fix a screw to the sticks at their midpoints. Using a thread, measure distances AD and BD .

Keep on moving the sticks about the screw, so that the distances AD and BD are equal.

In this position, fix the sticks by tightening the screw.

The new positions of the sticks are shown in the figure.

Tie pieces of thread along AD and BD .

Consider $△ AMD$ and $△ BMD$ . We have $AM = BM, AD = BD$ and MD is common. $∴$ By the SSS condition, $Δ AMD$ and $Δ BMD$ are congruent. $∴ ∠ AMD = ∠ BMD$ Also $∠ AMD + ∠ BMD = 18 0^{\circ}$ (Linear angles) $∴ ∠ AMD + ∠ AMD = 18 0^{\circ}$ $\Rightarrow 2∠ AMD = 18 0^{\circ}$ $\Rightarrow ∠ AMD = 9 0^{\circ}$ $∴ ∠ AMD = ∠ BMD = 9 0^{\circ}$ $∴$ Angle between the sticks is $9 0^{\circ}$ . 5. We saw that one of the properties of a rectangle is that its opposite sides are parallel. Can this be chosen as a definition of a rectangle? In other words, is every quadrilateral that has opposite sides parallel and equal, a rectangle?

Sol. Let ABCD be a quadrilateral in which opposite sides are parallel and equal. Here $AB ∥ DC$ and $AD ∥ BC$ . Also, $AB = DC$ and $AD = BC$ .

In the quadrilateral ABCD , opposite sides are equal.

For ABCD to be a rectangle, we require each angle to be $9 0^{\circ}$ .

Given information $AB ∥ DC$ and $AD ∥ BC$ can not help us to prove that each angle of ABCD is $9 0^{\circ}$ .

∴ ABCD

may not be a rectangle. ∴ A rectangle can not be defined as a quadrilateral with equal and parallel opposite sides.

3.0

Multiple choice questions

1.Which of the following is not true? (1) A plane figure formed by joining a number of points without lifting a pencil from the paper and without retracing any portion of the drawing other than single point is called a curve. (2) A simple closed curve made up of only line segments is called a polygon. (3)

is a figure of closed curve (4) None of these

2. A closed figure having three sides is (1) Not a polygon (2) A triangle (3) A quadrilateral (4) None of these

3. A heptagon is a (1) Polygon having 5 sides (2) Polygon having 6 sides (3) Both (1) and (2) (4) None of these

4. Adjacent sides of a polygon are (1) Any two sides of the polygon (2) Any two sides connecting two nonconsecutive vertices of a polygon (3) Any two sides with a common vertex (4) None of these

5. Adjacent vertices are (1) Uncommon vertices of two adjacent sides of a polygon (2) The end points of the same side of a polygon (3) The end points of the diagonal of a polygon (4) None of these

6. Which of the following is not true? (1) A quadrilateral in which one pair of opposite sides is parallel is called a trapezium. (2) A parallelogram has four sides. (3) A quadrilateral in which both the pairs of opposite sides are parallel is called a parallelogram. (4) The sum of all the four angles of a parallelogram is not $36 0^{\circ}$ .

7. Which of the following is not true? (1) Every trapezium is a parallelogram but every parallelogram is not a trapezium. (2) Opposite sides of a parallelogram are not equal. (3) Opposite angles of a parallelogram are equal. (4) Both (1) and (2)

8. Which of the following is not true? (1) A parallelogram having a pair of adjacent sides equal, is called a rhombus (2) The diagonals of a rhombus do not bisect each other (3) If the diagonals of a parallelogram bisect each other at right angles, it is a rhombus (4) A rectangle is a parallelogram.

9. In the figure, PQRS is a parallelogram. If $∠ P = 7 5^{\circ}$ , then $∠ Q =$

(1)

7 5^{\circ}

(2)

9 0^{\circ}

(3)

10 5^{\circ}

(4)

10 0^{\circ}

10. In the given figure, PQRS is a parallelogram. If perimeter of parallelogram PQRS is 40 cm and $PQ = 12 cm$ then PS is equal to

(1) 12 cm (2) 10 cm (3) 8 cm (4) 9 cm

11. In the given figure, PQRS is a parallelogram and diagonal PR and QS intersect each other at A . If $QA = 3 cm$ , $AR = 5 cm$ and $PS = 6 cm$ , then perimeter of $△ A QR$ is

(1) 16 cm (2) 14 cm (3) 12 cm (4) 10 cm

12. In the given figure, ABCD is a parallelogram, diagonals BD and AC intersect each other at E . If $BE + CE = 8 cm$ , then $AC + BD$ is equal to

(1) 16 cm (2) 14 cm (3) 20 cm (4) 24 cm

13. Which of the following is not the property of a rectangle? (1) A rectangle is a parallelogram in which one of the angles is a right angle. (2) Diagonals of a rectangle are equal (3) Opposite sides of a rectangle are equal (4) Diagonals of a rectangle bisect each other at right angle

14. In the given figure, ABCD is a rhombus. Diagonals AC and BD intersect each other at E . If $∠1 = 5 0^{\circ}$ then $∠ BCD =$

(1)

10 0^{\circ}

(2)

9 0^{\circ}

(3)

8 0^{\circ}

(4) None of these

15. Which one has all the properties of a parallelogram and also that of a kite? (1) Trapezium (2) Rhombus (3) Rectangle (4) None of these

16. In a parallelogram if each angle is equal, then it is called (1) Rectangle (2) Square (3) Rhombus (4) None of these

17. Which of these is not necessarily a property of a rhombus? (1) All sides are equal (2) Opposite angles are equal (3) Diagonals are equal (4) Diagonals bisect each other at right angles

18. A square is a special case of which of the following? (1) Parallelogram (2) Rectangle (3) Rhombus (4) All of these

19. Statement 1: In a parallelogram, the adjacent angles are in the ratio $2 : 3$ , so the bigger angle of the parallelogram is $10 8^{\circ}$ .

Statement 2: The sum of adjacent angles of a parallelogram is equal to $18 0^{\circ}$ . (1) Statement-1 is true and Statement-2 is false. (2) Statement-1 is false and Statement-2 is true. (3) Both the statements are true. (4) Both the statements are false.

20. Statement-1: In a kite, the diagonals are equal.

Statement-2: In a kite, the diagonals intersect at right angles. (1) Statement-1 is true and Statement-2 is false. (2) Statement-1 is false and Statement-2 is true. (3) Both the statements are true. (4) Both the statements are false.

True or false

21.The opposite sides of a rectangle are equal.

22. The diagonals of a rectangle are equal.

23. The diagonals of a rectangle are perpendicular to each other.

24. The diagonals of a rectangle bisect each other.

25. The diagonals of a rectangle bisect each other at right angles.

26. The diagonals of a rectangle are equal and bisect each other.

27. If the diagonals of a quadrilateral are perpendicular to each other, it is always a square.

28. If the diagonals of a rhombus are equal it is always a square.

29. The diagonals of a parallelogram are equal and bisect each other.

Crossword puzzle: 30.

Across

b. A parallelogram with four congruent sides. e. Any 4-sided polygon. f. A parallelogram with opposite congruent sides and four right angles. g. A quadrilateral with two distinct pairs of equal adjacent sides.

Down

a. A quadrilateral with exactly one pair of parallel sides. c. A quadrilateral whose opposite sides are parallel and congruent and opposite angles are congruent. d. A parallelogram with four congruent sides and four congruent angles.

ANSWER KEY