CBSE Class 8 Maths Important Questions Chapter 3 - Understanding Quadrilaterals

Question (1 – 7) 1 – Mark:

1. Regular polygon have all sides______

Ans: Equal    

2. Sum of all internal angles of a quadrilateral is _____

Ans: 360°

3. Diagonals of Rectangle are ____ and ____ each other.

 Ans: Equal, Bisect

4.  A quadrilateral with one pair of sides parallel is _____ 

Ans: Trapezium

5. Diagonals of _____ bisect each other at 90°

Ans: Rhombus and square

6. A parallelogram with one angle 90° is _____ 

Ans: Rectangle 

7. The number of Diagonals in triangle is ______ 

Ans: Zero 

Question (8 – 13) 2 – Marks:

8. Find x in the figure

Ans: Sum of all internal angles of quadrilateral = 360

\[30 + 120 + 90 + x = 360 \\240 + x = 360 \\x = 120  \]

9. Find the sum of all internal angle of a pentagon that is regular. 

Ans:  Sum of interior angles of a n sided polygon is $n - 2 \times 180$ 

For pentagon, $n = 5$ 

Thus for a pentagon,

Sum of interior angles= $5 - 2 \times 180 = 540$ 

10. State true or false 

1. All squares are rectangles 

Ans: True

Opposite sides in a square are equal. Therefore, all squares are rectangle. 

2. All Rhombus are kites 

Ans: True

Kite shape has equal adjacent sides and in rhombus all sides are equal which means adjacent sides will also be equal. Therefore, all rhombus are kites. 

11. In the following figure, given a parallelogram ABCD. Find x and y

Ans: Since in Parallelogram ABCD opposite sides are parallel and equal

${\text{AB = CD}} \\2y - 1 = 21 \\2y = 22 \\y = 11 \\{\text{AD = CB}} \\3x = 15 \\x = 5  $ 

12. Length of two adjacent sides of parallelogram are 8cm and 5cm. Find its perimeter

Ans: Let the parallelogram be ABCD

Let,

 $AB = 8,BC = 6 \\AB = CD \\CD = 8cm  $

Likewise,

$BC = AD \\AD = 6cm  $ 

Perimeter = sum of length of all sides 

$= 2(AB = CD) \\= 2(6 + 8) \\= 2(14) \\= 28 $

13. Find sum of angles of a regular pentagon(internal angles).

Ans: Using the formula - Sum of all angles=

 $n(180) - 2 \times 180 \\({\text{here}}\;n = 5)\\= 5(180) - 360 \\= 900 - 360 \\= 540  $

Question (14 – 18) 3 – Marks:

14. The measure of two adjacent angles of a parallelogram is in the ratio of 3 : 7. Find the measure of each angles of the parallelogram.

Ans: Let the parallelogram be ABCD

Let,

$\angle A:\angle B = 3:7 \\\angle A = 3x \text{and } \angle B = 7x \\ $ 

ABCD is a parallelogram and AD is parallel to BC.

$\angle A + \angle B = 180\\3x + 7x = 180 \\10x = 180 \\\angle A = \angle C = 3x \\\angle A = \angle C = 54 \\ \angle B = \angle D = 7x \\\angle B = \angle D = 126 \\  $

15. In the given figure ABCD is a rectangle and its diagonal meet at 0. Find x, if \[{\mathbf{OA}}{\text{ }} = {\text{ }}{\mathbf{2x}}{\text{ }}{\mathbf{and}}{\text{ }}{\mathbf{OD}}{\text{ }} = {\text{ }}{\mathbf{6x}}{\text{ }}--{\text{ }}{\mathbf{8}}{\text{ }}.\] Also find BD.

Ans:

Given, \[OA{\text{ }} = {\text{ }}2x{\text{ and }}OD{\text{ }} = {\text{ }}6x{\text{ }}--{\text{ }}8\]

Since diagonals of Rectangle are equal

AC = BD…… (i)

Since Rectangle is a parallelogram and diagonals of parallelogram bisect each other.

OD = OB and OA = OC…..(ii)

AC = OA + OC (Hence, AC = 2(OA) using (ii))

BD = BO + OD (Hence, BD = 2(OD) using (ii))

$AC = BD \\2(OA) = 2(OD) \\OA = OD \\2x = 6x - 8 \\4x = 8 \\x = 2 \\OD = 6x - 8 = 6(2) - 8 = 4 \\BD = 8 $

16. The diagonal AC of Rhombus ABCD is equal to one of its side BC. Find all the angles of Rhombus.

Ans:

Let ABCD be the Rhombus and according to the question AC = AB = BC [given]

In ∆ABC, AD = AC = BC

∆ABC is a equilateral triangle

$\angle ABC = 60\\\angle ADC = 60 \\\angle D + \angle A = 180 \\\angle A = 180 - 160 \\ \angle A = 120 \\\angle A = \angle C = 120 $

17. Find the values of x , y and z. Where ABCD is a Parallelogram.

Ans:

$\angle A = 50 \\ \angle A + \angle D = 180\\\angle A + y = 180 \\y = 180 - 50 \\y = 130 \\\angle y = \angle x \\y = x = 130 \\ \angle DCB + z = 180 \\z = 180 - \angle DCB \\z = 180 - 50\\z = 130 \\  $

18. Explain how square is

1. Quadrilateral 

Ans: A Quadrilateral is a closed polygon of 4 sides and square is following the definition of a quadrilateral. Hence square is a quadrilateral.

2. Rhombus 

Ans: A Rhombus is a parallelogram whose adjacent sides are equal. Similarly square have the same property. Hence a square is a Rhombus.

3. Rectangle

Ans: A Rectangle is a parallelogram with one angle 90°. Square has all angles 90°. Square has all the properties of a rectangle. Hence square is a Rectangle.

Question (19 – 22) 4 – 5 Marks:

19. Find x + y + z in the following figure.

Ans: Let’s name it as A, B, C, D, E, F

Linear pair:

$\angle ACB + z = 180 \\50 + z = 180 \\z = 130 $ 

Linear pair:

$\angle FCB + \angle ACB = 180 \\x + 90 = 180 \\x = 90 $ 

Applying the exterior angle property

$y = \angle BAC + \angle BCA \\y = 50 + 90 \\y = 140 $ 

hence,

$x + y + z = 140 + 90 + 130 = 360$ 

20.

1. Find x in the figure

Ans:

$\angle ABE = 90 \\ \angle ABE + \angle ABC = 180 \\\angle ABC = 180 - 90 \\\angle ABC = 90 \\\angle ABC + \angle BCD + \angle D + \angle A = 360 \\90 + 60 + 110 + x = 360 \\260 + x = 360 \\x = 100  $ 

2. Figure HEYA shown below is a parallelogram its given OE = 3cm and HY is 7 more than AE. Find OH

Ans:

 (b) HY = 7 more than AE (given)

\[\begin{array}{*{20}{l}}{HY{\text{ }} = {\text{ }}7{\text{ }} + {\text{ }}AE} \\  AE{\text{ }} = {\text{ }}2\left( {OE} \right) \\OE{\text{ }} = {\text{ }}3cm{\text{ }} \\ \end{array}\] 

 (O is the bisector of the diagonals)

\[\begin{array}{*{20}{l}}{AE{\text{ }} = {\text{ }}2{\text{ }}\left( 3 \right)} \\ {AE{\text{ }} = {\text{ }}6cm} \\ HY{\text{ }} = {\text{ }}OH{\text{ }} + {\text{ }}OY \\HY{\text{ }} = {\text{ }}7{\text{ }} + {\text{ }}6 \\ {HY{\text{ }} = {\text{ }}13cm} \end{array} \] 

(since OH = OY as diagonals of parallelogram bisect each other)

HY = 2 (OH)

13 = 2 (OH)

$OH = 13/2 = 6.5cm$ 

21.

1. Name the quadrilateral with exactly one pair of sides parallel.

Ans:

Ans: Trapezium AB||CD

2. Find the length of BD in the given Rectangle

Ans:  In triangle ABC

$AB = 4cm \\BC = 3cm \\\angle ABC = 90  $ 

Using the Pythagoras theorem :

 ${(4)^2} + {(3)^2} = A{C^2} \\A{C^2} = 25 \\AC = 5cm $ 

Diagonals of rectangle are equal

AC=BD

$BD = 5cm$ 

22. Choose the quadrilaterals with their properties

Ans:

a) Parallelogram- 

(i) Opposite sides equal

(ii) Opposite angles equal

(iii) diagonals bisect each other

(b) Rhombus-

(i) Opposite sides equal

(ii) Opposite angles equal

 (iii) diagonals bisect each other

(iv) diagonals are perpendicular to each

other

(c) Rectangle-

(i) Opposite sides equal

(ii) Opposite angles equal

(iii) diagonals bisect each other

(v) each angle is a right angle

(vi) diagonals are equal

(d) Square

(i) Opposite sides equal

(ii) Opposite angles equal

(iii) diagonals bisect each other

(iv) diagonals are perpendicular to each Other

(v) each angle is a right angle

(vi) diagonals are equal

(e) Kite

(iv) diagonals are perpendicular to each other

 (vi) diagonals are equal

Important Topics of Class 8 Maths Chapter 3 You Shouldn’t Miss!

Benefits of Class 8 Chapter 3 Maths Understanding Quadrilaterals Important Questions

• Regular practice improves your problem-solving skills by allowing you to work on different types of questions related to quadrilaterals. This is key for developing critical thinking.

• Focusing on these questions ensures you cover the important concepts that may appear in your exams. This helps you prepare effectively.

• Consistent practice builds your confidence in handling questions about quadrilaterals. You will feel more ready for tests when you practice regularly.

• Working through these questions helps you identify areas where you might struggle. This way, you can focus your study efforts on improving those topics.

• Important questions give you a structured way to learn about quadrilaterals. This helps you organize your study sessions and track your progress.

• Practising important questions also helps you manage your time better. This skill is crucial for completing exams within the given time.

• A solid understanding of quadrilaterals gives you a strong foundation for more complex topics in geometry. This makes learning future topics easier.

• Engaging with these questions makes learning more interesting and enjoyable. This keeps you motivated to study.

Conclusion

Working on important questions from Class 8 Chapter 3 on Understanding Quadrilaterals is essential for mastering the topic. These questions help reinforce key concepts and improve your problem-solving skills, making it easier to tackle similar problems in exams. By practising regularly, you can build confidence and gain a clearer understanding of the properties and types of quadrilaterals. With a strong foundation in this chapter, you will be better prepared for more advanced topics in geometry. Keep practising, and remember that consistent effort will lead to success in your studies!

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