NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Edited By Ramraj Saini | Updated on Feb 29, 2024 06:00 PM IST

Understanding Quadrilaterals Class 8 Questions And Answers provided here. These NCERT Solutions are created by expert team at craeers360 keeping the latest syllabus and pattern of CBSE 2023-23. A geometrical closed shape made by using 4 straight lines and having four connecting points is called Quadrilateral. The sum of the interior angles in a quadrilateral is always 360 degrees. NCERT Solutions for Class 8 Maths chapter 3 Understanding Quadrilaterals are covering the chapter to help you with the answers and conceptual clarity. It carries around 35% of geometry weightage and includes most important types of questions based on concepts of rectangles, squares, rhombus, parallelograms, trapezium, etc.

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Understanding Quadrilaterals Class 8 Questions And Answers PDF Free Download
Understanding Quadrilaterals Class 8 Solutions - Important Formulae
Understanding Quadrilaterals Class 8 NCERT Solutions (Intext Questions and Exercise)
NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals - Topic: Kite
Understanding Quadrilaterals Class 8 Solutions - Topics
NCERT Solutions for Class 8 Maths -Chapter Wise
Key Features of Understanding Quadrilaterals Class 8 NCERT Solutions

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

NCERT solutions for Class 8 Maths discussed all questions and answers of all chapters including chapter 3 understanding quadrilaterals. The subtopics covered under the chapter are polygons, angle sum property, properties of different kinds of quadrilaterals. In this particular chapter, there are 4 exercises consisting of a total of 31 questions. NCERT solutions for Class 8 Maths chapter 3 Understanding Quadrilaterals is covering every question in a comprehensive manner.

Understanding Quadrilaterals Class 8 Questions And Answers PDF Free Download

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Understanding Quadrilaterals Class 8 Solutions - Important Formulae

Polygons are categorised based on the quantity of sides or vertices they possess, as outlined below:

Understanding-Quadrilaterals

Angle Sum Property for Quadrilaterals: The sum of all angles of a quadrilateral is 360°.
Sum of the Measures of Exterior Angles of a Polygon: Regardless of the number of sides in a polygon, the total measure of the exterior angles equals 360 degrees.

>> Categories of Quadrilaterals: Quadrilaterals are grouped based on the measurements of their angles and side lengths. The area represents the total space enclosed by the figure, while the perimeter is the complete distance around its boundaries. Here, you'll find descriptions, area calculations, and perimeter formulas for different types of quadrilaterals:

Quadrilaterals

Free download NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals for CBSE Exam.

Understanding Quadrilaterals Class 8 NCERT Solutions (Intext Questions and Exercise)

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals Exercise: Angle Sum Property

Q1 Take any quadrilateral, say ABCD (Fig 3.4). Divide it into two triangles, by drawing a diagonal. You get six angles 1, 2, 3, 4, 5 and 6. Use the angle-sum property of a triangle and argue how the sum of the measures of and amounts to $180\degree + 180\degree = 360\degree$ .

1643706688938

Answer:

As shown in $\triangle$ ACD,

$\angle 1 + \angle 2+\angle 3 = 180\degree$

As shown in $\triangle$ ABC,

$\angle 4 + \angle 5+\angle 6 = 180\degree$

$\angle A + \angle B +\angle C + \angle D= (\angle 1+\angle 4) +\angle 6 + (\angle 2+\angle 5) + \angle 3$ ( Since, $\angle A = (\angle 1+\angle 4)$ , $\angle B = \angle 6$ , $\angle C = (\angle 2+\angle 5)$ ,

$\angle D = \angle 3$ )

$\angle A + \angle B +\angle C + \angle D= (\angle 1+\angle 2 +\angle 3) + (\angle 4+\angle 5 + \angle 6)$

$= 180\degree + 180\degree$

$= 360\degree$

Hence proved, the sum of the measures of $\angle A,\angle B,\angle C$ and
$\angle D$ amounts to $180\degree + 180\degree = 360\degree$ .

Q2 Take four congruent card-board copies of any quadrilateral ABCD, with angles as shown [Fig 3.5 (i)]. Arrange the copies as shown in the figure, where angles ∠1, ∠2, ∠3, ∠4 meet at a point [Fig 3.5 (ii)].

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What can you say about the sum of the angles ∠1, ∠2, ∠3 and ∠4?

[Note: We denote the angles by ∠1, ∠2, ∠3, etc., and their respective measures by m∠1, m∠2, m∠3, etc.]

The sum of the measures of the four angles of a quadrilateral is___________.
You may arrive at this result in several other ways also.

Answer:

As we know the sum of all four angles of a quadrilateral is $360\degree$ .

$\angle 1,\angle 2,\angle 3,\angle 4$ are four angles of quadrilateral ABCD.

Hence, the sum of these angles is $360\degree$ = $\angle 1+\angle 2+\angle 3+\angle 4$

Angles ∠1, ∠2, ∠3, ∠4 meet at a point and sum of these angles is $360\degree$ = $\angle 1+\angle 2+\angle 3+\angle 4$ .

Q3 As before consider quadrilateral ABCD (Fig 3.6). Let P be any point in its interior. Join P to vertices A, B, C and D. In the figure, consider $\Delta PAB$ . From this we see $x = 180\degree - m\angle 2 - m\angle 3$ ; similarly from $\Delta PBC$ , $y = 180\degree - m\angle 4 - m\angle 5$ , from $\Delta PCD$ , $z = 180\degree - m\angle 6 - m\angle 7$ and from $\Delta PDA$ , $w = 180\degree - m\angle 8 - m\angle 1$ . Use this to find the total measure $m\angle 1 + m\angle 2 + ... + m\angle 8$ , does it help you to arrive at the result? Remember
$\angle x + \angle y + \angle z + \angle w = 360\degree$ .

1643706789031 Answer:

As we know $\angle x + \angle y + \angle z + \angle w = 360\degree$

$(180\degree - m\angle 2 - m\angle 3)$ $(180\degree - m\angle 4 - m\anlge 5)+$ $(180\degree - m\angle 6 - m\angle 7) +$ $( 180\degree - m\angle 8 - m\angle 1)$ $= 360\degree$

$m\angle 1 + m\angle 2 + ... + m\angle 8$ $\left ( 180\degree+180\degree+180\degree+180\degree \right ) - 360\degree$

$m\angle 1 + m\angle 2 + ... + m\angle 8$ $= 720\degree - 360\degree$

$m\angle 1 + m\angle 2 + ... + m\angle 8$ $= 360\degree$

Hence, the sum of

$m\angle 1 + m\angle 2 + ... + m\angle 8$ $= 360\degree$

Q4 These quadrilaterals were convex. What would happen if the quadrilateral is not convex? Consider quadrilateral ABCD. Split it into two triangles and find the sum of the interior angles (Fig 3.7).

1643706874150 Answer:

1643706885559

Draw a line matching points B and D.Line BD divides ABCD in two triangles . $\triangle BCD$ and $\triangle ABD$ .

Sum of angles of $\triangle BCD$ = $\angle DBC + \angle BDC + \angle C = 180\degree$ $\cdot \cdot \cdot \cdot \cdot \cdot \cdot \left ( 1 \right )$

Sum of angles of $\triangle ABD$ = $\angle ADB + \angle ABD + \angle A = 180\degree$ $\cdot \cdot \cdot \cdot \cdot \cdot \cdot \left ( 2\right )$

Here, $\angle ADB + \angle BDC = \angle D$

$\angle DBC + \angle ABD = \angle B$

Adding equation 1 and equation 2,

$(\angle DBC + \angle BDC + \angle C ) +$ $(\angle ADB + \angle ABD + \angle A )$ $=$ $360\degree$ { $\angle ADB + \angle BDC = \angle D$ and

i.e. $\angle A + \angle B +\angle C +\angle D$ = $360\degree$ . $\angle DBC + \angle ABD = \angle B$ }

The sum of the interior angles in a quadilateral,

$\angle A + \angle B +\angle C +\angle D$ = $360\degree$

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals - Exercise: 3.1

Q1 (a) Given here are some figures.

1643706930897

Classify each of them on the basis of the following.

(a) Simple curve

Answer:

(a) Simple curve : The curve which does not cross itself and have only one curve.

Some simple curves are $: 1,2,5,6,7$

Q1 (b) Given here are some figures.

1643706970443

Classify each of them on the basis of the following.

(b) Simple closed curve

Answer:

Simple closed curve : The simple curves which are closed by line segment or curved line.

Some simple closed curve are $:1,2,5,6,7$

Q1 (c) Given here are some figures.

1643706985737

Classify each of them on the basis of the following.

Answer:

A normally closed curve made up of more than 4 line segments is called a polygon.

Some polygons are shown in figures $1,2.$

Q1 (d) Given here are some figures.

1643707002734

Classify each of them on the basis of the following.

(d) Convex polygon

Answer:

(d) Convex polygon :Convex polygon are polygons having all interior angles less than $180\degree$ .

Convex polygon = 2.

Q1 (e) Given here are some figures.

1643707016897

Classify each of them on the basis of the following.

(e) Concave polygon

Answer:

(e) Concave polygon : Concave polygon have one or more interior angles greater than $180\degree$ .

Concave polygon = 1.

Q2 (a) How many diagonals does each of the following have?
A convex quadrilateral

Answer:

(a) A convex quadrilateral :

1643707041585

There are 2 diagonals in a convex quadrilateral.

Q2 (b) How many diagonals does each of the following have?

A regular hexagon

Answer:

(b) A regular hexagon :

1643707076614

Total number of diagonal a regular hexagon have are 9 .

Q2 (c) How many diagonals does each of the following have?

A triangle

Answer:

1643707124642

A triangle does not have any diagonal.

Q3 What is the sum of the measures of the angles of a convex quadrilateral? Will this property
hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!)

Answer:

The sum of the measures of the angles of a convex quadrilateral is $360\degree$ . A convex quadrilateral is made of two triangles and sum of angles of a triangle is $180\degree$ . Hence,sum of angles of two triangles is $360\degree$ which is also sum of angles of a quadrilateral.

1643707165921

Here,as we can see this quadrilateral has two triangles and the sum of angles of these triangles is $180\degree+ 180\degree=360\degree$ = sum of all angles of a quadrilateral.

Q4 (a) Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)

1643708214801

What can you say about the angle sum of a convex polygon with the number of sides?

(a) 7

Answer:

(a) 7

the sum of angles of a convex polygon $= (number of sides - 2)\ast 180\degree$

the sum of angle of a convex polygon with 7 sides $= (7 - 2)\ast 180\degree$

the sum of angles of a convex polygon with 7 sides $= (5)\ast 180\degree$

the sum of angles of a convex polygon with 7 sides $= 900\degree$

Q4 (a) Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)

1643708203466

What can you say about the angle sum of a convex polygon with the number of sides?

(b) 8

Answer:

(b) 8

The sum of the angles of a convex polygon $= (number of sides - 2)\ast 180\degree$

The sum of the angles of a convex polygon with 8 sides $= (8 - 2)\ast 180\degree$

The sum of the angles of a convex polygon with 8 sides $= (6)\ast 180\degree$

The sum of the angles of a convex polygon with 8 sides $= 1080\degree$

Q4 (c) Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)

1643708189452

What can you say about the angle sum of a convex polygon with number of sides?

Answer:

angle sum of a convex polygon $= (number of sides - 2)\ast 180\degree$

angle sum of a convex polygon with 10 sides $= (10 - 2)\ast 180\degree$

angle sum of a convex polygon with 10 sides $= (8)\ast 180\degree$

angle sum of a convex polygon with 10 sides $= 1440\degree$

Q4 (d) Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)

1643708239951

What can you say about the angle sum of a convex polygon with number of sides?

(d) $n$

Answer:

(d) $n$

All the angles of a convex polygon added up to $= (number of sides - 2)\ast 180\degree$

the sum of the angles of a convex polygon having n sides $= (n - 2)\ast 180\degree$

Q5 (i) What is a regular polygon?
State the name of a regular polygon of 3 sides

Answer:

A regular polygon is a polygon which has equal sides and equal angles.

The name of a regular polygon of 3 sides is an equilateral triangle.

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All sides of the equilateral triangle are equal and angles are also equal.

Each angle = $60\degree$

Q5 (ii) What is a regular polygon?
State the name of a regular polygon of 4 sides

Answer:

A regular polygon is a polygon which has equal sides and equal angles.

The name of a regular polygon of 4 sides is square.

1643708267992

Square has all angles of $90\degree$ and all sides are equal.

Q5 What is a regular polygon?
State the name of a regular polygon of 6 sides

Answer:

A regular polygon is a polygon which have equal sides and equal angles.

The name of a regular polygon of 6 sides is a hexagon.

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All angles of the hexagon are $120\degree$ each.

Q6 Find the angle measure $x$ in the following figures.

1643708295069

Answer:

The solution for the above-written question is mentioned below,

Sum of angles of a quadrilateral $= 360\degree$

$130\degree+120\degree+50\degree+x= 360\degree$

$300\degree+x= 360\degree$

$x = 360\degree-300\degree$

$x = 60\degree$

Q6 Find the angle measure $x$ in the following figures.

1643709053623 Answer:

The Solution for the above-written question is mentioned below,

Sum of angles of a quadrilateral $= 360\degree$

$90\degree+60\degree+70\degree + x=360\degree$

$220\degree + x=360\degree$

$x=360\degree-220\degree$

$x=140\degree$

Q6 Find the angle measure $x$ in the following figures.

1643709093629 Answer:

This pentagon has all sides and all angles equal.

Sum of all angles of pentagon is $540\degree$

$x+x+x+x+x=540\degree$

$5\ast x=540\degree$

$x=108\degree$

Q6 Find the angle measure $x$ in the following figures.

1643709125510 Answer:

$a+70\degree=180\degree$ (linear pair) $b+60\degree =180\degree$ (linear pair)

$a =110\degree$ $b =120\degree$

Sum of all angles of pentagon = $540\degree$

$110\degree+120\degree+x+x+30\degree=540\degree$

$260\degree+2x=540\degree$

$2x=540\degree-260\degree$

$2x=280\degree$

$x=140\degree$

Q7 (a) Find $x + y + z$

Answer:

(a) Find

$x + y + z$

$z+30\degree=180\degree$

$z=150\degree$

$x+90\degree=180\degree$

$x=90\degree$

The Sum of all angles of triangle is $180\degree$

the unmarked angle of triangle be A. $A+90\degree+30\degree=180\degree$

$A=180\degree-120\degree$

$A=60\degree$

$A+y=180\degree$

$60\degree+y=180\degree$

$y=180\degree - 60\degree$

$y=120\degree$

$x + y + z$ = $90\degree+120\degree+150\degree$

$x + y + z$ = $360\degree$

Q7 (b) Find $x + y + z + w$

Answer:

Here you will find the detailed solution of the above-written question,

$x+120\degree = 180\degree$

$x=180\degree - 120\degree$

$x=60\degree$

$y+80\degree = 180\degree$

$y=180\degree - 80\degree$

$y=100\degree$

$z+60\degree = 180\degree$

$z=180\degree - 60\degree$

$z=120\degree$

Sum of all angles of quadrilateral = $360\degree$

Let the unmarked angle be A.

$A+120\degree+60\degree+80\degree=360\degree$

$A=100\degree$

$A+w=180\degree$

$w= 80\degree$

$x + y + z + w$ = $60\degree+100\degree+120\degree+80\degree$

$x + y + z + w$ = $360\degree$

NCERT solutions for class 8 maths chapter 3 Understanding Quadrilaterals Excercise: Sum of the Measures of the Exterior Angles of a Polygon

Take a regular hexagon Fig 3.10.
Q1 What is the sum of the measures of its exterior angles $x, y, z, p, q, r$

1643709272718

Answer:

the sum of all its exterior angles will be equal to $360\degree$ .

x,y,z,p,q,r are all exterior angles.

Hence, $x+y+z+p+q+r=360\degree$

Q2 Take a regular hexagon Fig 3.10.

Is $x = y = z = p = q = r$ . Why?

1643709315173

Answer:

It is a hexagon with all sides equal.

All interior angles are also equal.

$a+r=a+x=a+y=a+z=a+p=a+q=180\degree$ (linear pairs)

$r=x=y=z=p=q=180\degree - a$

Hence, $x = y = z = p = q = r$

because it is a hexagon with all sides and angles equal.

Q3 (i) What is the measure of each? exterior angle

1643709354128

Answer:

It is a hexagon with all sides equal.

All interior angles are also equal.

Sum of all angles of hexagon is $720\degree$ .

$6a = 720\degree$

$a = 120\degree$

$a+r=a+x=a+y=a+z=a+p=a+q=180\degree$ (linear pairs)

$r=x=y=z=p=q=180\degree - a$

Each exterior angle = $r=x=y=z=p=q=180\degree - a$ = $180\degree-120\degree$ = $60\degree$

Q3 (ii) What is the measure of each? interior angle

1643709384347

Answer:

It is a hexagon with all sides equal.

All interior angles are also equal.

Sum of all angles of the hexagon is $720\degree$ .

$6a = 720\degree$

$a = 120\degree$

Q4 (i) Repeat this activity for the cases of a regular octagon

Answer:

(i) a regular octagon: It has all 8 angles equal and sum of all eight angles is 1080. Interior angles are equal so exterior angles are also equal.

Let interior angle be A.

$8\ast A= 1080\degree$

$A= 135\degree$ .

All exterior angles be B.

$B=180\degree - 135\degree$

$B=45\degree$

Q4 (ii) Repeat this activity for the cases of a regular 20-gon

Answer:

(ii) a regular 20-gon: It has all 20 angles equal and sum of all eight angles is 3240.Interior angles are equal so exterior angles are also equal.

Let interior angle be A.

$20\ast A= 3240\degree$

$A= 162\degree$ .

All exterior angles be B.

$B=180\degree - 162\degree$

$B=18\degree$

Class 8 maths chapter 3 question answer - Exercise: 3.2

Q1 (a) Find $x$ in the following figures.

1643709424495

Answer:

Sum of all exterior angles of a polygon is $360\degree.$

$x+125\degree+125\degree=360\degree.$

$x=360\degree-250\degree.$

$x=110\degree.$

Q1 (a) Find $x$ in the following figures.

1643709470866

Answer:

Sum of all exterior angles of polygon is $360\degree$

$x+90\degree+60\degree+90\degree+70\degree=360\degree$

$x=360\degree-310\degree$

$x=50\degree$

Q2 (i) Find the measure of each exterior angle of a regular polygon of 9 sides

Answer:

A regular polygon of 9 sides has all sides, interior angles and exterior angles equal.

Sum of exterior angles of a polygon = $360\degree$

Let the interior angle be A.

Sum of exterior angles of 9 sided polygon = $9 \ast A =$ $360\degree$

Exterior angles of 9 sided polygon $= A= 360\degree \div 9$

$A= 40\degree$

Hence, the measure of each exterior angle of a regular polygon of 9 sides is $40\degree$

Q2 (ii) Find the measure of each exterior angle of a regular polygon of 15 sides

Answer:

A regular polygon of 15 sides has all sides, interior angles and exterior angles equal.

Sum of exterior angles of a polygon = $360\degree$

Let the interior angle be A.

Sum of exterior angles of 15 sided polygon = $15 \ast A =$ $360\degree$

Exterior angles of 15 sided polygon $= A= 360\degree \div 15$

$A= 24\degree$

Hence, the measure of each exterior angle of a regular polygon of 15 sides is $24\degree$

Q3 How many sides does a regular polygon have if the measure of an exterior angle is 24°?

Answer:

The measure of an exterior angle is 24°

A regular polygon has all exterior angles equal.

Sum of exterior angles of a polygon = $360\degree$

Let a number of sides be X.

Sum of exterior angles of a polygon = $X \ast 24\degree =$ $360\degree$

Exterior angles of 15 sided polygon $= X= 360\degree \div 24\degree$

$X= 15$

Hence, $15$ sided regular polygon have measure of an exterior angle 24°

Q4 How many sides does a regular polygon have if each of its interior angles is 165°?

Answer:

The measure of each interior angle is 165°

So, measure of each exterior angle = 180°-165° = 15°

Regular polygon has all exterior angles equal.

Let number of sides of polygon = n

Sum of Exterior angles of a polygon = $360\degree$

$\left ( n \right )\ast 15\degree= 360\degree$

$n = 24$

Hence,regular polygon having each of its interior angles is 165° has 24 sides.

Q5 (a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?

Answer:

The measure of an exterior angle is 22°

A regular polygon has all exterior angles equal.

Sum of exterior angles of a polygon = $360\degree$

Let the number of sides be X.

Sum of exterior angles of a polygon = $X \ast 22\degree =$ $360\degree$

Exterior angles of 15 sided polygon $= X= 360\degree \div 22\degree$

$X= 16.36$

Hence, the side of a polygon should be an integer but as shown above, the side is not an integer. So,it is not possible to have a regular polygon with the measure of each exterior angle as 22^o.

Q5 (b) Can it be an interior angle of a regular polygon? Why?

Answer:

The measure of an interior angle is 22°

A regular polygon has all interior angles equal.

Let the number of sides and number of interior angles be n.

Sum of interior angles of a polygon = $\left ( n-2 \right )\ast 180\degree$

Sum of interior angles of a polygon = $\left ( n-2 \right )\ast 180\degree$ = $\left ( n \right )\ast 22\degree$

$\left ( n \right )\ast 180\degree-\left ( 2 \right )\ast 180\degree=\left ( n \right )\ast 22\degree$

$\left ( n \right )\ast 180\degree-\left ( n \right )\ast 22\degree= 360\degree$

$\left ( n \right )\ast 158\degree= 360\degree$

$n = 2.28$

The number of sides of a polygon should be an integer but since it is not an integer.So, it cannot be a regular polygon with interior angle as 22^o

Q6 (a) What is the minimum interior angle possible for a regular polygon? Why?

Answer:

Consider a polygon with the lowest number of sides i.e. 3.

Sum of interior angles of 3 sided polygon = $\left ( 3-2 \right )\ast 180\degree=180\degree$

Interior angles of a regular polygon are equal = A .

$\therefore$ $A+A+A=180\degree$

$3\ast A=180\degree$

$A=60\degree$

Hence,the minimum interior angle possible for a regular polygon is $60\degree$ .

Q6 (b) What is the maximum exterior angle possible for a regular polygon?

Answer:

Let there be a polygon with the minimum number of sides i.e. 3.

Exterior angles an equilateral triangle has a maximum measure.

Sum of exterior angles of polygon = $360\degree$

Let the exterior angle be A.

$\therefore$ $A+A+A=360\degree$

$3\ast A=360\degree$

$A=120\degree$

Hence, the maximum exterior angle possible for a regular polygon is $120\degree$ .

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals - Topic: Kinds of Quadrilaterals

Q1 Take identical cut-outs of congruent triangles of sides 3 cm, 4 cm, 5 cm. Arrange them as shown (Fig 3.11).

1643709649772 You get a trapezium. (Check it!) Which are the parallel sides here? Should the
non-parallel sides be equal?
You can get two more trapeziums using the same set of triangles. Find them out and
discuss their shapes.

Answer:

1643709682641

AB and CD are parallel sides.BC and AD are nonparallel sides. Non-parallel sides need not be equal.

We can get two more trapeziums using the same set of triangles-

1596039255438 and 1596039263553

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals - Topic: Kite

Q1 Take a thick white sheet.
Fold the paper once.
Draw two line segments of different lengths as shown in Fig 3.12.
Cut along the line segments and open up.
You have the shape of a kite (Fig 3.13).
Has the kite any line symmetry

Fold both the diagonals of the kite. Use the set-square to check if they cut at

1643709778477

1643709810087

right angles. Are the diagonals equal in length? Verify (by paper-folding or measurement) if the diagonals bisect each other. By folding an angle of the kite on its opposite, check for angles of equal measure.
Observe the diagonal folds; do they indicate any diagonal being an angle bisector? Share your findings with others and list them. A summary of these results are given elsewhere in the chapter for your reference. Show that triangle ABC and triangle ADC
are congruent. What do we infer from this?

Answer :

1643709829785

Kite has symmetry along AC diagonal. $\triangleABC$ $\triangle ABC$ and $\triangle ACD$ are congruent and equal triangles.

Diagonals AC and BD are of different lengths.

Diagonals bisect each other.

The two diagonals AC and BD bisect $\angle A,\angle B,\angle C,\angle D$ .

NCERT Solutions for Class 8 Maths 3 Understanding Quadrilaterals Excercise: Elements of Parallelogram

Q1 Take cut-outs of two identical parallelograms, say ABCD and A′B′C′D′ (Fig 3.19).

1643712023541

Here $\overline{AB}$ is same as $\overline{A'B'}$ except for the name. Similarly the other corresponding
sides are equal too.
Place $\overline{A'B'}$ over $\overline{DC}$ . Do they coincide? What can you now say about the lengths
$\overline{AB}$ and $\overline{DC}$ ?
Similarly examine the lengths $\overline{AD}$ and $\overline{BC}$ . What do you find?
You may also arrive at this result by measuring $\overline{AB}$ and $\overline{DC}$ .

Answer:

1643712042895

Take cut-outs of two identical parallelograms, say ABCD and A′B′C′D′ (Fig 3.19).

Here $\overline{AB}$ is same as $\overline{A'B'}$ .

Also, $\overline{CD}$ is same as $\overline{C'D'}$

Place $\overline{A'B'}$ over $\overline{DC}$ . They coincide with each other.

The lengths $\overline{AB}$ and $\overline{DC}$ are equal and parallel lines.

The lengths $\overline{AD}$ and $\overline{BC}$ are equal and parallel lines.

Q2 Take two identical set squares with angles $30\degree - 60\degree - 90\degree$ and place them adjacently to form a parallelogram as shown in Fig 3.20. Does this help you to verify the above property?

Property: The opposite sides of a parallelogram are of equal length.

1643712076840

Answer:

Property : The opposite sides of a parallelogram are of equal length.

As we can see in the figure above, the opposite sides of figure are equal.

The figure above is a rectangle which is part of parallelogram.

Hence, the opposite sides of a parallelogram are of equal length.

Q3 Take two identical 30° – 60° – 90° set-squares and form a parallelogram as before. Does the figure obtained help you to confirm the above property?

Property: The opposite angles of a parallelogram are of equal measure.

Answer:

1643712575591

As shown in the above figure opposites angles are equal and are equal to $90 \degree$ .

Hence,the figure obtained help you to confirm the property: The opposite angles of a parallelogram are of equal measure.

Q4 Take a cut-out of a parallelogram, say,

1643712545744

ABCD (Fig 3.29). Let its diagonals $\overline{AC}$ and $\overline{DB}$ meet at O.
Find the mid point of $\overline{AC}$ by a fold, placing C on A. Is the
mid-point same as O?
Does this show that diagonal $\overline{DB}$ bisects the diagonal $\overline{AC}$ at the point O? Discuss it
with your friends. Repeat the activity to find where the mid point of $\overline{DB}$ could lie.

Answer:

1643712554894

ABCD . Let its diagonals $\overline{AC}$ and $\overline{DB}$ meet at O.
The mid point of $\overline{AC}$ is at point O.Hence, the
mid-point is same as O.

This show that diagonal $\overline{DB}$ bisects the diagonal $\overline{AC}$ at the point O.

The mid point of $\overline{DB}$ is point O.

Class 8 maths chapter 3 NCERT Solutions - exercise: 3.3

Q1 (i) Given a parallelogram ABCD. Complete each statement along with the definition or property used.

1643712594104

$AD =$ ......

Answer:

In a parallelogram, opposite sides are equal in lengths.

Hence,

$AD = BC$

Q1 (ii) Given a parallelogram ABCD. Complete each statement along with the definition or property used.

1643712617369

$\angle DCB =$ ......

Answer:

In a parallelogram ,opposite angles are equal.

(ii) $\angle DCB =$ $\angle BAD.$

Q1 (iii) Given a parallelogram ABCD. Complete each statement along with the definition or property used.

1643712639173

$OC =$ ......

Answer:

In a parallelogram, both diagonals bisect each other.

$OC =$ $OA$

Q1 (iv) Given a parallelogram ABCD. Complete each statement along with the definition or property used.

1643712654425

$m \angle DAB + m\angle CDA =$ .....

Answer:

In a parallelogram, adjacent angles are supplementary to each other.

(iv) $m \angle DAB + m\angle CDA =$ $180\degree$

Q2 (i) Consider the following parallelograms. Find the values of the unknowns $x, y, z$ .

1643712721769

Answer:

In a parallelogram, adjacent angles are supplementary to each other.

$\angle B+\angle C = 180\degree$

$100\degree+ x = 180\degree$

$x = 80\degree$

Opposite angles are equal.

Hence, z = $x = 80\degree$

and y = 100

Q2 (ii) Consider the following parallelograms. Find the values of the unknowns $x, y, z$ .

1643712782976

Answer:

$50\degree+x=180\degree$ ( Two adjacent angles are supplementary to each other)

$x=130\degree$

x=y= $130\degree$ (opposite angles are equal)

z=x= $130\degree$ ( corresponding angles are equal)

Q2 (iii) Consider the following parallelograms. Find the values of the unknowns $x, y, z$ .

1643712805599

Answer:

x= $90\degree$ (vertically opposite angles)

$x+y+30\degree = 180\degree$ (sum of angles of a triangle is $180\degree$ )

$90\degree+y+30\degree = 180\degree$

$y = 60\degree$

y=z= $60\degree$ (alternate interior angles)

Q2 (iv) Consider the following parallelograms. Find the values of the unknowns $x, y, z$ .

1643712837010

Answer:

$x+80\degree=180\degree$ (adjacent angles are supplementary)

$x=100\degree$

y = $80\degree$ ( opposite angles are equal)

z= $80\degree$ (corresponding angles are equal)

Q2 (v) Consider the following parallelograms. Find the values of the unknowns $x, y, z$ .

1596039614290

Answer:

y = $112 \degree$ (opposite angles are equal)

$z+40\degree+112 \degree=180\degree$ (adjacent angles are supplementary)

$z=180\degree -152\degree$

$z=28\degree$

x = z = $28\degree$ (alternate angles are equal)

Q3 (i) Can a quadrilateral ABCD be a parallelogram if $\angle D + \angle B = 180\degree$

Answer:

(i) $\angle D + \angle B = 180\degree$

Opposite angles should be equal and adjacent angles should be supplementary to each other.

$\angle B,\angle D$ are opposite angles.

Hence, a quadrilateral ABCD can be a parallelogram but it is not confirmed.

Q3 (ii) Can a quadrilateral ABCD be a parallelogram if

$AB = DC = 8 cm, AD = 4 cm$ and $BC = 4.4 cm$ ?

Answer:

Opposite sides of a parallelogram are equal in length.

Since, $AD = 4 cm$ and $BC = 4.4 cm$ are opposite sides and have different lengths.

No, it is not a parallelogram.

Q3 (iii) Can a quadrilateral ABCD be a parallelogram if $\angle A = 70 \degree$ and $\angle C = 65 \degree$ ?

Answer:

Opposite angles of a parallelogram are equal

Since here $\angle A = 70 \degree$ and $\angle C = 65 \degree$ are different.

So, it is not a parallelogram.

Q4 Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles
of equal measure.

Answer:

1643712912618

The above-shown figure shows two opposite angles equal. $\angle B=\angle D$ .

But, it's not a parallelogram because the other two angles are different i.e. $\angle A\neq \angle C$ .

Q5 The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.

Answer:

The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2.

Sum of adjacent angles is $180\degree$ .

$\therefore$ $3\times x+2\times x = 180\degree$

$5\times x = 180\degree$

$x = 36\degree$

Hence, angles are $2\times 36\degree=72\degree$ and $3\times 36\degree=108\degree$ .

Let there be a parallelogram ABCD then, $\angle A=\angle C=108\degree$ and $\angle B=\angle D=72\degree$ . (Opposite angles are equal).

Q6 Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.

Answer:

Given: Two adjacent angles of a parallelogram have equal measure = $\angle A= \angle B$ .

$\angle A+ \angle B =180\degree$ ( adjacent angles of a parallelogram are supplementary)

$2\times \angle A =180\degree$

$\angle A =90\degree$

$\therefore$ $\angle A=\angle B =90\degree$

$\angle A=\angle C =90\degree$ and $\angle B=\angle D =90\degree$ ( Opposite angles of a parallelogram are equal)

Hence, $\angle A=\angle B=\angle C=\angle D =90\degree$

Q7 The adjacent figure HOPE is a parallelogram. Find the angle measures $x, y$ and $z$ . State the properties you use to find them.

1643712941421

Answer:

The adjacent figure HOPE is a parallelogram.

$\angle HOP +70\degree = 180\degree$ (linear pairs)

$\angle HOP = 110\degree$

$x= \angle HOP = 110\degree$ (opposite angles of a parallelogram are equal)

$\angle HOP +\angle EHO = 180\degree$ ( adjacent angles are supplementary )

$110\degree +(40\degree+z) = 180\degree$

$z = 180\degree - 150\degree$

$z = 30\degree$

y= $40\degree$ (Alternate interior angles are equal)

Q8 (i) The following figures GUNS and RUNS are parallelograms. Find $x$ and $y$ . (Lengths are in cm)

1596039860919

Answer:

GUNS is a parallelogram, so opposite sides are equal in length

$SG\:=\:UN$

$3\times x=18$

$x=18\div 3$

$x=6$

$UG\:=\:NS$

$3\times Y-1=26$

$3\times Y=27$

$Y=9$

Hence, $x=6$ cm and $Y=9$ cm.

Q8 (ii) The following figures GUNS and RUNS are parallelograms. Find $x$ and $y$ . (Lengths are in cm)

1643712998474

Answer:

Diagonals of a parallelogram intersect each other.

$y+7=20$

$y=20-7$

$y=13$

$x+y=16$

$x=16-13$

$x=3$

Hence, $x=3$ cm and $y=13$ cm.

Q9 In the above figure both RISK and CLUE are parallelograms. Find the value of $x$ .

Answer:

$\angle SKR +\angle KSI = 180\degree$ ( adjacent angles are supplemantary)

$\angle KSI = 180\degree-120\degree$

$\angle KSI = 60\degree$

$\angle CLU=\angle UEC = 70\degree$ (oppsite angles are equal)

$x+ \angle UEC+\angle KSI = 180\degree$ (sum of angles of a triangle is $180\degree$ )

$x+ 70\degree+60\degree = 180\degree$

$x=180\degree -130\degree$

$x=50\degree$

Q10 Explain how this figure is a trapezium. Which of its two sides are parallel? (Fig 3.32)

1643713112417

Answer:

Given, $\angle M+\angle L=100\degree+80\degree$ = $180\degree$ .

A transverse line is intersecting two lines such that the sum of angles on the same side of the transversal line is $180\degree$ .

And hence, lines KL and MN are parallel to each other.

Quadrilateral KLMN has a pair of parallel lines so it is a trapezium.

Q11 Find $m\angle C$ in Fig 3.33 if $\overline{AB} || \overline{DC}$ .

1643713142883

Answer:

Given , $\overline{AB} || \overline{DC}$

$\angle B+\angle C=180\degree$ (Angles on same side of transversal)

$120\degree+\angle C=180\degree$

$\angle C=180\degree - 120\degree$

$\angle C=60\degree$

Hence, $m\angle C=60\degree$ .

Q12 Find the measure of $\angle P$ and $\angle S$ if $\overline {SP} || \overline {RQ}$ in Fig 3.34.
(If you find $m\angle R$ , is there more than one method to find $m\angle P$ ?)

Screenshot%20(74)

Answer:

Given, $\overline {SP} || \overline {RQ}$

$\angle P+\angle Q=180\degree$ (angles on the same side of transversal)

$\angle P=180\degree-130\degree$

$\angle P=50\degree$

$\angle R+\angle S=180\degree$ (angles on the same side of transversal)

$\angle S=180\degree-90\degree$

$\angle S=90\degree$

Yes, to find $m\angle P$ there is more than one method.

PQRS is quadrilateral so the sum of all angles is 360

$\angle P+\angle Q+\angle R+\angle S=360\degree$

and we know $\angle Q,\:\angle R ,\:\angle S$

so put values of $\angle Q,\:\angle R ,\:\angle S$ and we get a measurement of $\angle P$

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals - Topic: Squares

Q1 Take a square sheet, say PQRS (Fig 3.42). Fold along both the diagonals. Are their mid-points the same? Check if the angle at O is 90° by using a set-square. This verifies the property stated.

Property: The diagonals of a square are perpendicular bisectors of each other.

1643713324883

Answer:

PO = OR ( Square is a type of parallelogram)

By SSS congruency rule, $\triangle POS$ and $\triangle ROS$ are congruent triangles.

$\triangle POS\cong \triangle ROS$

$\therefore \angle POS = \angle ROS$

& $\angle POS + \angle ROS = 180\degree$

$2\times \angle ROS = 180\degree$

$\angle ROS = 90\degree$

Hence, $\angle O = 90\degree$ .

Thus,we can say that the diagonals of a square are perpendicular bisectors of each other

NCERT Solutions for Class 8 Maths 3 Understanding Quadrilaterals Exercise: 3.4

Q1 (a) State whether True or False. All rectangles are squares

Answer:

(a). False, All squares are rectangles but all rectangles cannot be square.

Q1 (b) State whether True or False. All rhombuses are parallelograms

Answer:

True. The opposite sides of the rhombus are parallel and equal.

Q1 (c) State whether True or False. All squares are rhombuses and also rectangles.

Answer:

True. All squares are rhombus because rhombus has opposite sides parallel and equal and same square has.

Also, all squares are rectangles because they have all interior angles as $90\degree$ .

Q1 (d) State whether True or False. All squares are not parallelograms.

Answer:

False.

All squares have their opposite sides equal and parallel. Hence, they are parallelograms.

Q1 (e) State whether True or False. All kites are rhombuses.

Answer:

False,

Kites do not have all sides equal so they are not rhombus.

Q1 (f) State whether True or False. All rhombuses are kites.

Answer:

True, all rhombuses are kites because they have two adjacent sides equal.

Q1 (g) State whether True or False. All parallelograms are trapeziums.

Answer:

True, all parallelograms are trapezium because they have a pair of parallel sides.

Q1 (h) State whether True or False. All squares are trapeziums.

Answer:

True, All squares are trapeziums because all squares have pairs of parallel sides.

Q2 (a) Identify all the quadrilaterals that have. four sides of equal length

Answer:

The quadrilateral having four sides of equal length are square and rhombus.

Q2 (b) Identify all the quadrilaterals that have. four right angles

Answer:

All the quadrilaterals that have four right angles are rectangle and square

Q3 (i) Explain how a square is a quadrilateral

Answer:

A square is a quadrilateral because a square has four sides.

Q3 (ii) Explain how a square is a parallelogram

Answer:

A square is a parallelogram because the square has opposite sides parallel to each other.

Q3 (iii) Explain how a square is a rhombus

Answer:

A square is a rhombus because a square has four sides equal.

Q3 (iv) Explain how a square is.a rectangle

Answer:

A square is a rectangle since it has all interior angles of $90\degree$ .

Q4 (i) Name the quadrilaterals whose diagonals bisect each other

Answer:

The quadrilaterals whose diagonals bisect each other are square, rectangle, parallelogram and rhombus.

Q4 (ii) Name the quadrilaterals whose diagonals are perpendicular bisectors of each other

Answer:

The quadrilaterals whose diagonals are perpendicular bisectors of each other are rhombus and square.

Q4 (iii) Name the quadrilaterals whose diagonals are equal

Answer:

The quadrilaterals whose diagonals are equal are squares and rectangles.

Q5 Explain why a rectangle is a convex quadrilateral.

Answer:

A rectangle is a convex quadrilateral because it has two diagonals and both lie in the interior of the rectangle.

Q6 ABC is a right-angled triangle and O is the midpoint of the side opposite to the right angle. Explain why O is equidistant from A, B and C. (The dotted lines are drawn additionally to help you).

1643713424260

Answer:

Draw line AD and DC such that $AB \parallel CD$ and $AD \parallel BC$ .

$AD= BC$ and $AB= CD$

ABCD is a rectangle as it has opposite sides equal and parallel.

All angles of the rectangle are $90\degree$ and a rectangle has two diagonals equal and bisect each other.

Hence, AO = BO = CO = DO

$\therefore$ O is equidistant from A,B,C,D.

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals - Topic: Rectangle

Q1 A mason has made a concrete slab. He needs it to be rectangular. In what different ways can he make sure that it is rectangular?

Answer:

(1).All the properties of a parallelogram.

(2) Each of the angles is a right angle.

(3) Diagonals are equal

Class 8 understanding quadrilaterals NCERT Solutions - Topic: Square

Q2 A square was defined as a rectangle with all sides equal. Can we define it as

rhombus with equal angles? Explore this idea.

Answer:

Properties of the rectangle are :

(1) All the properties of a parallelogram.

(2) Each of the angles is a right angle.

(3) Diagonals are equal.

A square satisfies all the properties of rectangles so a square can be defined as a rectangle with all sides equal.

Properties of a rhombus are :

(1) All the properties of a parallelogram.

(2) Diagonals are perpendicular to each other.

A square satisfies all the properties of the rhombus so we can define it as
rhombus with equal angles.

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals - Topic: Trapezium

Q3 Can a trapezium have all angles equal? Can it have all sides equal? Explain.

Answer:

Trapezium has two sides parallel and the other two sides are non-parallel. Parallel sides may be equal or unequal but we cannot have a trapezium with all sides and angles equal.

Understanding Quadrilaterals Class 8 Solutions - Topics

Polygons
Sum of the Measures of the Exterior Angles of a Polygon
Kinds of Quadrilaterals
Some Special Parallelograms

NCERT Solutions for Class 8 Maths -Chapter Wise

Chapter -1	Rational Numbers
Chapter -2	Linear Equations in One Variable
Chapter-3	Understanding Quadrilaterals
Chapter-4	Practical Geometry
Chapter-5	Data Handling
Chapter-6	Squares and Square Roots
Chapter-7	Cubes and Cube Roots
Chapter-8	Comparing Quantities
Chapter-9	Algebraic Expressions and Identities
Chapter-10	Visualizing Solid Shapes
Chapter-11	Mensuration
Chapter-12	Exponents and Powers
Chapter-13	Direct and Inverse Proportions
Chapter-14	Factorization
Chapter-15	Introduction to Graphs
Chapter-16	Playing with Numbers

Key Features of Understanding Quadrilaterals Class 8 NCERT Solutions

Clarity of Concepts: In this maths chapter 3 class 8 we are ocusing on clarifying the fundamental properties and properties of different types of quadrilaterals.

Real-Life Applications: Explaining how the concepts of quadrilaterals are applied in practical situations and in geometry.

Practice Questions: Additional practice problems to reinforce learning and problem-solving skills. Students can practice class 8 maths ch 3 question answer using the pdf.

NCERT Solutions for Class 8: Subject-Wise

How to use NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals?

Read about different types of quadrilaterals and their properties.
Learn how to use their properties in specific questions using the solved examples.
Practice the problems given in the NCERT textbook.
During the practice, if you find a problem in solving any of the questions you can use NCERT solutions for Class 8 Maths chapter 3 Understanding Quadrilaterals.

Keep working hard and happy learning!

Also Check NCERT Books and NCERT Syllabus here:

Frequently Asked Question (FAQs)

1. What are the important topics of chapter Understanding Quadrilaterals ?

Constructing a Quadrilateral

when the lengths of four sides and a diagonal are given, when two diagonals and three sides are given
When two adjacent sides and three angles are known
When three sides and two included angles are given

are the important topics of this chapter.

2. How many chapters are there in the CBSE class 8 maths ?

There are 16 chapters starting from rational number to playing with numbers in the CBSE class 8 maths.

3. Does CBSE provides the solutions of NCERT for class 8 ?

No, CBSE doesn’t provide NCERT solutions for any class or subject.

4. Where can I find the complete solutions of NCERT for class 8 ?

Here you will get the detailed NCERT solutions for class 8 by clicking on the link.

5. How does the NCERT solutions are helpful ?

NCERT solutions are provided in a very detailed manner which will give the conceptual clarity to the students. Also, students can take help from these solutions when they are not able to solve them on their own.

Also Read

NCERT solutions for Class 8 Maths Chapter 16 Playing with Numbers

NCERT Solutions for Class 8 Maths Chapter 15 Introduction to Graphs

NCERT Solutions for Class 8 Maths Chapter 14 Factorization

NCERT Solutions for Class 8 Maths Chapter 13 Direct and Inverse Proportions

NCERT Solutions for Class 8 Maths Chapter 12 Exponents and Powers

NCERT Solutions For Class 8 Maths Chapter 11 Mensuration

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Option 3) $zero\;$	Option 4) $K/4$

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NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Understanding Quadrilaterals Class 8 Questions And Answers PDF Free Download

Understanding Quadrilaterals Class 8 Solutions - Important Formulae

Understanding Quadrilaterals Class 8 NCERT Solutions (Intext Questions and Exercise)

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals - Topic: Kite

Understanding Quadrilaterals Class 8 Solutions - Topics

NCERT Solutions for Class 8 Maths -Chapter Wise

Key Features of Understanding Quadrilaterals Class 8 NCERT Solutions

NCERT Solutions for Class 8: Subject-Wise

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