Important Questions for CBSE Class 8 Maths 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

So, you have solved all types of problems related to quadrilaterals. Using these problems, you can solve your textbook problems easily. But before going to your textbook questions, let us solve these problems to evaluate your understanding of Quadrilaterals.

Practice Problems Based on the Chapter Quadrilaterals Class 9

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

2. Lengths of the two adjacent sides of a parallelogram are  12 cm and 8 cm. Find its perimeter.

3. Find the sum of all internal angles of a regular pentagon.

4. Three angles of a quadrilateral are 45°, 90° and 120°. Find the measure of the fourth angle. 

To solve these problems, you must have a clear understanding of the chapter Quadrilateral and its topics. Some of its topics are given below.

Topics Discuss in the Chapter Quadrilateral

• Introduction to shapes

• Polygons

•  Classification of polygons

• Diagonals

• Convex and concave polygons 

• Regular and irregular polygons 

• Angle sum property

• Sum of the measures of the exterior angles of a polygon

• Kinds of quadrilaterals: Trapezium, Kite, Parallelogram

• Elements of a parallelogram 

• Angles of a parallelogram

• Diagonals of a parallelogram

• Some special parallelograms: Rhombus, Rectangle, Square

Important Points to Remember

• A quadrilateral is made up of four line segments with four vertices normally named A, B, C and D.

• In a quadrilateral figure, four sides are the four line segments between the vertices, eg. AB, BC, CD and DA.

• Two sides of a quadrilateral figure which have a common endpoint i.e. intersect each other, are called consecutive sides. 

• Two sides of a quadrilateral figure that do not have a common endpoint are called opposite sides of a quadrilateral.

• A diagonal of a quadrilateral is the joining of the opposite vertices by a line segment. 

• If the two angles of a quadrilateral do not have a common arm, then the angles are called opposite angles. 

• If the two angles of a quadrilateral have a common arm, then the angles are called consecutive or adjacent angles. 

• The sum of the four angles of a quadrilateral measures 360°. 

Different Types of Quadrilaterals

Following are the different types of quadrilaterals that you will learn in this chapter:

1. Trapezium: A quadrilateral in which at least one pair of opposite sides are parallel is called a trapezium. Please note that if two non-parallel sides of a trapezium are equal, then it is called an isosceles triangle. 

2. Parallelogram: A quadrilateral is a parallelogram if both pairs of opposite sides are parallel, and it is written as ||gm.  

 A quadrilateral is a parallelogram if any of the condition satisfy

1. Its opposite angles are equal.

2. Its opposite sides are equal.

3. Diagonals bisect each other. 

Some Important Points to Keep in Mind About Parallelogram

• A parallelogram is a trapezium but a trapezium is not a parallelogram. This is because, in a trapezium, only one pair of opposite sides is parallel whereas and in a parallelogram, both pairs of opposite sides are parallel. 

• The adjacent angles of a parallelogram sum to 180°.

• Rhombus: A rhombus is a type of parallelogram in which all sides are equal. In other words, when all the four sides in a quadrilateral are equal and both pairs of opposite sides are parallel, it is called a rhombus.

• Rectangle: A parallelogram in which one of its angles is the right angle is called a rectangle. In other words, a quadrilateral in which opposite sides are parallel and equal and one angle is 90° is called a rectangle. 

• Kite: A kite is a quadrilateral that has two pairs of equal adjacent congruent sides. Note that a kite is not a parallelogram. 

Important Facts to Remember

1. Every square, rectangle and rhombus is a parallelogram. 

2. Every square can be both rectangle and rhombus. 

3. A kite is not a parallelogram. 

4. In trapezium only one opposite pair is parallel, so it is not a parallelogram.

5. Every rectangle or rhombus is not a square. 

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

7. The diagonals of a square bisect each other to form right angles. 

8. The diagonals of a square bisect each other at right angles and are of equal lengths. 

9. The angles of a triangle opposite to equal sides of the triangle are equal.

10. The Diagonals of a kite bisect each other to form right angles. 

Interior and Exterior of a Quadrilateral

The sum of interior angles in a quadrilateral and the sum of exterior angles of a quadrilateral are always 360°. 

Convex and Concave Quadrilateral

Convex Quadrilateral is a quadrilateral in which every internal angle measures less than 180°. A Concave Quadrilateral is a quadrilateral in which the internal angles measure greater than 180°. 

Properties of a Parallelogram

1. In a parallelogram, opposite sides are equal. 

2. In a parallelogram, opposite angles are equal.

3. The diagonals of a parallelogram bisect each other.

Properties of a Rhombus

1. In a Rhombus, all sides measure equal.

2. The diagonals of a rhombus bisect each other.

Properties of a Rectangle

1. Each of the angles in a rectangle is a right angle.

2. The diagonals of a rectangle bisect each other.

3. The diagonals of a rectangle are equal. 

Properties of a Square

1. Each of the angles of a square measures 90.

2. The diagonals of a square are of the same length.

3. The diagonals of a square bisect each other to form right angles. 

Conclusion 

Chapter 3 Understanding Quadrilaterals for Class 8 goes over the quadrilateral concepts you learned in previous grades and introduces the angle sum property of a quadrilateral. The essential questions have been developed based on the subjects covered in this chapter. 

The chapter reference notes provided above will assist you in answering the Crucial Questions Of Chapter 3 of Mathematics for Class 8 Knowledge of Quadrilaterals. The chapter's crucial questions and reference notes will not only help you comprehend the idea better, but will also help you answer the questions effectively. The chapter reference notes provided above will assist you in answering the Crucial Questions Of Chapter 3 of Mathematics for Class 8 Knowledge of Quadrilaterals. The chapter's crucial questions and reference notes will not only help you comprehend the idea better, but will also help you answer the questions effectively.

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