NCERT Solutions Class 8 Maths Chapter 3 Understanding Quadrilaterals

• Exercise 3.1 introduces polygons, covering their basic definition and classification based on the number of sides, such as triangles, quadrilaterals, pentagons, etc. It also distinguishes between convex and concave polygons, helping students understand the differences between these types of polygons.

• Exercise 3.2 delves into the properties of quadrilaterals, exploring various types like trapeziums, kites, and parallelograms. This exercise helps students learn to identify different quadrilaterals and understand their specific properties.

• Exercise 3.3 examines the properties of parallelograms, such as opposite sides being equal and parallel, opposite angles being equal, and diagonals bisecting each other. It includes problems for identifying parallelograms based on these properties and proving certain properties using theorems.

• Exercise 3.4 looks at special parallelograms like rhombuses, rectangles, and squares. It highlights their unique properties, such as all sides being equal in a rhombus and all angles being 90 degrees in a rectangle. This exercise helps students understand and differentiate between these specific types of parallelograms.

List of Formulas

There are two major kinds of formulas related to quadrilaterals - Area and Perimeter. The following tables depict the formulas related to the areas and perimeters of different kinds of quadrilaterals.

Area of Quadrilaterals

Perimeter of Quadrilaterals

Perimeter of any quadrilateral is equal to the sum of all its sides, that is, AB + BC + CD + AD.

Access NCERT Solutions for Class 8 Maths Chapter 3 – Understanding Quadrilaterals

Exercise 3.1

1. Given here are some figures.

Classify each of them on the basis of following.

1. Simple Curve

Ans: Given: the figures $(1)$to $(8)$

We need to classify the given figures as simple curves.

We know that a curve that does not cross itself is referred to as a simple curve.

Therefore, simple curves are $1,2,5,6,7$.

2. Simple Closed Curve

Ans: Given: the figures $(1)$to $(8)$

We need to classify the given figures as simple closed curves.

We know that a simple closed curve is one that begins and ends at the same point without crossing itself.

Therefore, simple closed curves are $1,2,5,6,7$.

3. Polygon

Ans: Given: the figures $(1)$to $(8)$

We need to classify the given figures as polygon.

We know that any closed curve consisting of a set of sides joined in such a way that no two segments

cross is known as a polygon.

Therefore, the polygons are $1,2$.

4. Convex Polygon

Ans: Given: the figures $(1)$to $(8)$

We need to classify the given figures as convex polygon.

We know that a closed shape with no vertices pointing inward is called a convex polygon.

Therefore, the convex polygon is $2$.

5. Concave Polygon

Ans: Given: the figures $(1)$to $(8)$

We need to classify the given figures as concave polygon.

We know that a polygon with at least one interior angle greater than 180 degrees is called a concave

polygon.

Therefore, the concave polygon is $1$.

2. What is a regular polygon?

State the name of a regular polygon of

(i) 3 sides (ii) 4 sides (iii) 6 sides

Solution: A regular polygon is a flat shape with all sides of equal length and all interior 5angles equal in measure.  In simpler terms, all the sides are the same size and all the corners look the same.

Here are the names of regular polygons based on the number of sides:

(i) 3 sides -  Equilateral Triangle (all three angles are also 60 degrees each)

(ii) 4 sides - Square

(iii) 6 sides - Hexagon

Exercise-3.2

1. Find ${\text{x}}$in the following figures.

Ans:

Given:

We need to find the value of ${\text{x}}{\text{.}}$

We know that the sum of all exterior angles of a polygon is ${360^ \circ }.$

Thus.

$  {\text{x}} + {125^ \circ } + {125^ \circ } = {360^ \circ } $

$   \Rightarrow {\text{x}} + {250^ \circ } = {360^ \circ } $

$   \Rightarrow {\text{x}} = {360^ \circ } - {250^ \circ } $

$   \Rightarrow {\text{x}} = {110^ \circ } $ 

Ans:

Given:

We need to find the value of ${\text{x}}{\text{.}}$

We know that the sum of all exterior angles of a polygon is ${360^ \circ }.$

Thus,

$  {\text{x}} + {90^ \circ } + {60^ \circ } + {90^ \circ } + {70^ \circ } = {360^ \circ } $

$  \Rightarrow {\text{x}} + {310^ \circ } = {360^ \circ } $

$   \Rightarrow {\text{x}} = {360^ \circ } - {310^ \circ } $

 $  \Rightarrow {\text{x}} = {50^ \circ } $ 

2. Find the measure of each exterior angle of a regular polygon of 

1. 9 Sides

Ans:

Given: a regular polygon with $9$ sides

We need to find the measure of each exterior angle of the given polygon.

We know that all the exterior angles of a regular polygon are equal.

The sum of all exterior angle of a polygon is ${360^ \circ }$.

Formula Used: ${\text{Exterior}}\;{\text{angle}} = \dfrac{{{{360}^ \circ }}}{{{\text{Number}}\;{\text{of}}\;{\text{sides}}}}$

Therefore,

Sum of all angles of given regular polygon $ = {360^ \circ }$

Number of sides $ = 9$

Therefore, measure of each exterior angle will be

$   = \dfrac{{{{360}^ \circ }}}{9} $

 $  = {40^ \circ } $ 

2. 15 Sides

Ans:

Given: a regular polygon with $15$ sides

We need to find the measure of each exterior angle of the given polygon.

We know that all the exterior angles of a regular polygon are equal.

The sum of all exterior angle of a polygon is ${360^ \circ }$.

Therefore,

Sum of all angles of given regular polygon $ = {360^ \circ }$

Number of sides $ = 15$

Formula Used: ${\text{Exterior}}\;{\text{angle}} = \dfrac{{{{360}^ \circ }}}{{{\text{Number}}\;{\text{of}}\;{\text{sides}}}}$

Therefore, measure of each exterior angle will be

$   = \dfrac{{{{360}^ \circ }}}{{15}} $

$ = {24^ \circ } $ 

3. How many sides does a regular polygon have if the measure of an exterior angle is ${24^ \circ }$?

Ans: Given: A regular polygon with each exterior angle ${24^ \circ }$

We need to find the number of sides of given polygon.

We know that sum of all exterior angle of a polygon is ${360^ \circ }$.

Formula Used: ${\text{Number}}\;{\text{of}}\;{\text{sides}} = \dfrac{{{{360}^ \circ }}}{{{\text{Exterior}}\;{\text{angle}}}}$

Thus,

Sum of all angles of given regular polygon $ = {360^ \circ }$

Each angle measure $ = {24^ \circ }$

Therefore, number of sides of given polygon will be

$   = \dfrac{{{{360}^ \circ }}}{{{{24}^ \circ }}} $

 $  = 15 $ 

4. How many sides does a regular polygon have if each of its interior angles is ${165^ \circ }$?

Ans: Given: A regular polygon with each interior angle ${165^ \circ }$

We need to find the sides of the given regular polygon.

We know that sum of all exterior angle of a polygon is ${360^ \circ }$.

Formula Used: ${\text{Number}}\;{\text{of}}\;{\text{sides}} = \dfrac{{{{360}^ \circ }}}{{{\text{Exterior}}\;{\text{angle}}}}$

${\text{Exterior}}\;{\text{angle}} = {180^ \circ } - {\text{Interior}}\;{\text{angle}}$

Thus,

Each interior angle $ = {165^ \circ }$

So, measure of each exterior angle will be

$   = {180^ \circ } - {165^ \circ } $

$   = {15^ \circ } $ 

Therefore, number of sides of polygon will be

$   = \dfrac{{{{360}^ \circ }}}{{{{15}^ \circ }}} $

$   = 24 $ 

5. 

1. Is it possible to have a regular polygon with measure of each exterior angle as ${22^ \circ }$?

Ans:

Given: A regular polygon with each exterior angle ${22^ \circ }$

We need to find if it is possible to have a regular polygon with given angle measure.

We know that sum of all exterior angle of a polygon is ${360^ \circ }$. The polygon will be possible if ${360^ \circ }$ is a perfect multiple of exterior angle.

Thus,

$\dfrac{{{{360}^ \circ }}}{{{{22}^ \circ }}}$ does not give a perfect quotient. 

Thus, ${360^ \circ }$ is not a perfect multiple of exterior angle. So, the polygon will not be possible.

2. Can it be an interior angle of a regular polygon? Why?

Ans: Given: Interior angle of a regular polygon $ = {22^ \circ }$

We need to state if it can be the interior angle of a regular polygon.

We know that sum of all exterior angle of a polygon is ${360^ \circ }$. The polygon will be possible if ${360^ \circ }$ is a perfect multiple of exterior angle.

And, ${\text{Exterior}}\;{\text{angle}} = {180^ \circ } - {\text{Interior}}\;{\text{angle}}$

Thus, Exterior angle will be

$   = {180^ \circ } - {22^ \circ } $

 $  = {158^ \circ } $ 

$\dfrac{{{{158}^ \circ }}}{{{{22}^ \circ }}}$ does not give a perfect quotient. 

Thus, ${158^ \circ }$ is not a perfect multiple of exterior angle. So, the polygon will not be possible.

6. 

1. What is the minimum interior angle possible for a regular polygon?

Ans:  Given: A regular polygon

We need to find the minimum interior angle possible for a regular polygon.

A polygon with minimum number of sides is an equilateral triangle.

So, number of sides $ = 3$

We know that sum of all exterior angle of a polygon is ${360^ \circ }$.

And, 

${\text{Exterior}}\;{\text{angle}} = \dfrac{{{{360}^ \circ }}}{{{\text{Number}}\;{\text{of}}\;{\text{sides}}}}$

Thus, Maximum Exterior angle will be

$   = \dfrac{{{{360}^ \circ }}}{3} $

$   = {120^ \circ } $ 

We know, ${\text{Interior}}\;{\text{angle}} = {180^ \circ } - {\text{Exterior}}\;{\text{angle}}$

Therefore, minimum interior angle will be

$   = {180^ \circ } - {120^ \circ } $

$   = {60^ \circ } $ 

2. What is the maximum exterior angel possible for a regular polygon?

Ans: Given: A regular polygon

We need to find the maximum exterior angle possible for a regular polygon.

A polygon with minimum number of sides is an equilateral triangle.

So, number of sides $ = 3$

We know that sum of all exterior angle of a polygon is ${360^ \circ }$.

And, 

${\text{Exterior}}\;{\text{angle}} = \dfrac{{{{360}^ \circ }}}{{{\text{Number}}\;{\text{of}}\;{\text{sides}}}}$

Therefore, Maximum Exterior angle possible will be

$   = \dfrac{{{{360}^ \circ }}}{3} $

$ = {120^ \circ } $

 Exercise 3.3

1. Given a parallelogram ABCD. Complete each statement along with the definition or property used.

1. $\;{\text{AD}}$ = $...$

Ans:

Given: A parallelogram ${\text{ABCD}}$ 

We need to complete each statement along with the definition or property used.

We know that opposite sides of a parallelogram are equal.

Hence, ${\text{AD}}$ = ${\text{BC}}$ 

2. $\;\angle {\text{DCB }} = $ $...$

Ans:

Given: A parallelogram ${\text{ABCD}}$.

We need to complete each statement along with the definition or property used.

${\text{ABCD}}$ is a parallelogram, and we know that opposite angles of a parallelogram are equal.

Hence, $\angle {\text{DCB   =  }}\angle {\text{DAB}}$

3. ${\text{OC}} = ...$ 

Ans:

Given: A parallelogram ${\text{ABCD}}$.

We need to complete each statement along with the definition or property used.

${\text{ABCD}}$ is a parallelogram, and we know that diagonals of parallelogram bisect each other.

Hence, ${\text{OC  =  OA}}$

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

Ans:

Given: A parallelogram ${\text{ABCD}}$.

We need to complete each statement along with the definition or property used.

${\text{ABCD}}$ is a parallelogram, and we know that adjacent angles of a parallelogram are supplementary to each other.

Hence, $m\angle DAB\; + \;m\angle CDA\; = \;180^\circ $

 2. Consider the following parallelograms. Find the values of the unknowns x, y, z.

(i)

Ans:

Given: A parallelogram ${\text{ABCD}}$

We need to find the unknowns ${\text{x,y,z}}$

The adjacent angles of a parallelogram are supplementary.

Therefore, ${\text{x} + 100^\circ  = 180^\circ }$

${\text{x} = 80^\circ }$ 

Also, the opposite angles of a parallelogram are equal.

Hence, ${\text{z}} = {\text{x}} = 80^\circ $ and ${\text{y}} = 100^\circ $

(ii) 

Ans:

Given: A parallelogram.

We need to find the values of ${\text{x,y,z}}$

The adjacent pairs of a parallelogram are supplementary.

Hence, $50^\circ  + {\text{y}} = 180^\circ $

${\text{y}} = 130^\circ $

Also, ${\text{x}} = {\text{y}} = 130^\circ $(opposite angles of a parallelogram are equal)

And, ${\text{z}} = {\text{x}} = 130^\circ $ (corresponding angles)

(iii)  

Ans:

Given: A parallelogram 

We need to find the values of ${\text{x,y,z}}$

${\text{x}} = 90^\circ $(Vertically opposite angles)

Also, by angle sum property of triangles

${\text{x}} + {\text{y}} + 30^\circ  = 180^\circ $

${\text{y}} = 60^\circ $

Also,${\text{z}} = {\text{y}} = 60^\circ $(alternate interior angles)

(iv)

Ans:

Given: A parallelogram

We need to find the values of ${\text{x,y,z}}$

Corresponding angles between two parallel lines are equal.

Hence, ${\text{z}} = 80^\circ $
Also,${\text{y}} = 80^\circ $ (opposite angles of parallelogram are equal)

In a parallelogram, adjacent angles are supplementary

Hence,${\text{x}} + {\text{y}} = 180^\circ $

$  {\text{x}} = 180^\circ  - 80^\circ  $

$  {\text{x}} = 100^\circ  $ 

(v) 

Ans:

Given: A parallelogram

We need to find the values of ${\text{x,y,z}}$

As the opposite angles of a parallelogram are equal, therefore,${\text{y}} = 112^\circ $ 

Also, by using angle sum property of triangles

$  {\text{x}} + {\text{y}} + 40^\circ  = 180^\circ  $

$  {\text{x}} + 152^\circ  = 180^\circ  $

$  {\text{x}} = 28^\circ  $ 

And ${\text{z}} = {\text{x}} = 28^\circ $(alternate interior angles)

3. Can a quadrilateral ${\text{ABCD}}$be a parallelogram if 

(i) $\angle {\text{D}}\;{\text{ + }}\angle {\text{B}} = 180^\circ ?$

Given: A quadrilateral ${\text{ABCD}}$

We need to find whether the given quadrilateral is a parallelogram.

For the given condition, quadrilateral ${\text{ABCD}}$ may or may not be a parallelogram.

For a quadrilateral to be parallelogram, the sum of measures of adjacent angles should be $180^\circ $ and the opposite angles should be of same measures.

(ii) ${\text{AB}} = {\text{DC}} = 8\;{\text{cm}},\;{\text{AD}} = 4\;{\text{cm}}\;$and ${\text{BC}} = 4.4\;{\text{cm}}$

Ans:

Given: A quadrilateral ${\text{ABCD}}$

We need to find whether the given quadrilateral is a parallelogram.

As, the opposite sides ${\text{AD}}$and ${\text{BC}}$are of different lengths, hence the given quadrilateral is not a parallelogram.

(iii) $\angle {\text{A}} = 70^\circ $and $\angle {\text{C}} = 65^\circ $

Ans:

Given: A quadrilateral ${\text{ABCD}}$

We need to find whether the given quadrilateral is a parallelogram.

As, the opposite angles have different measures, hence, the given quadrilateral is a parallelogram.

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

Ans:

Given: A quadrilateral.

We need to draw a rough figure of a quadrilateral that is not a paralleloghram but has exactly two opposite angles of equal measure.

A kite is a figure which has two of its interior angles, $\angle {\text{B}}$and $\angle {\text{D}}$of same measures. But the quadrilateral ${\text{ABCD}}$is not a parallelogram as the measures of the remaining pair of opposite angles are not equal.

5. 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.

Ans: Given: A parallelogram with adjacent angles in the ratio $3:2$

We need to find the measure of each of the angles of the parallelogram.

Let the angles be $\angle {\text{A}} = 3{\text{x}}$and $\angle {\text{B}} = 2{\text{x}}$

As the sum of measures of adjacent angles is $180^\circ $ for a parallelogram.

$  \angle {\text{A}} + \angle {\text{B}} = 180^\circ  $

 $ 3{\text{x}} + 2{\text{x}} = 180^\circ  $

 $ 5{\text{x}} = 180^\circ  $

 $ {\text{x}} = 36^\circ  $ 

$~\angle A=$ $\angle {\text{C}}$ $= 3{\text{x}} = 108^\circ$and $~\angle B=$ $\angle {\text{D}}$ $= 2{\text{x}} = 72^\circ$(Opposite angles of a parallelogram are equal).

Hence, the angles of a parallelogram are $108^\circ ,72^\circ ,108^\circ $and $72^\circ $.

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

Ans:

Given: A parallelogram with two equal adjacent angles.

We need to find the measure of each of the angles of the parallelogram.

The sum of adjacent angles of a parallelogram are supplementary.

$  \angle {\text{A}} + \;\angle {\text{B}} = 180^\circ  $

$  2\angle {\text{A}}\;{\text{ =  180}}^\circ  $

$  \angle {\text{A}}\;{\text{ = }}\;{\text{90}}^\circ  $

$  \angle {\text{B}}\;{\text{ = }}\angle {\text{A}}\;{\text{ = }}\;{\text{90}}^\circ  $

Also, opposite angles of a parallelogram are equal

Therefore,

$  \angle {\text{C}} = \angle {\text{A}} = 90^\circ  $

$  \angle {\text{D}} = \angle {\text{B}} = 90^\circ  $ 

Hence, each angle of the parallelogram measures $90^\circ $.

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

Ans:

Given: A parallelogram ${\text{HOPE}}$.

We need to find the measures of angles ${\text{x,y,z}}$and also state the properties used to find these angles.

$\angle {\text{y}} = 40^\circ $(Alternate interior angles)

And $\angle {\text{z}} + 40^\circ  = 70^\circ $(corresponding angles are equal)

$\angle {\text{z}} = 30^\circ $

Also, ${\text{x}} + {\text{z}} + 40^\circ  = 180^\circ $(adjacent pair of angles)

${\text{x}} = 110^\circ $

8. The following figures ${\text{GUNS}}$and ${\text{RUNS}}$are parallelograms. Find ${\text{x}}$and${\text{y}}$. (Lengths are in cm).

(i) 

Ans:

Given: Parallelogram ${\text{GUNS}}$.

We need to find the measures of ${\text{x}}$and ${\text{y}}$.

${\text{GU = SN}}$(Opposite sides of a parallelogram are equal).

$  3{\text{y }} - {\text{ }}1{\text{ }} = {\text{ }}26{\text{ }} $

$  3{\text{y }} = {\text{ }}27{\text{ }} $

$  {\text{y }} = {\text{ }}9{\text{ }} $ 

Also,${\text{SG = NU}}$

Therefore, 

$  3{\text{x}} = 18 $

$  {\text{x}} = 3 $ 

(ii) 

Ans:

Given: Parallelogram ${\text{RUNS}}$

We need to find the value of ${\text{x}}$and ${\text{y}}{\text{.}}$

The diagonals of a parallelogram bisect each other, therefore, 

$  {\text{y }} + {\text{ }}7{\text{ }} = {\text{ }}20{\text{ }} $

$  {\text{y }} = {\text{ }}13 $

 $ {\text{x }} + {\text{ y }} = {\text{ }}16 $

$  {\text{x }} + {\text{ }}13{\text{ }} = {\text{ }}16 $

 $ {\text{x }} = {\text{ }}3{\text{ }} $ 

9. In the above figure both ${\text{RISK}}$and ${\text{CLUE}}$are parallelograms. Find the value of ${\text{x}}{\text{.}}$

Ans:

Given: Parallelograms ${\text{RISK}}$and ${\text{CLUE}}$

We need to find the value of ${\text{x}}{\text{.}}$

As we know that the adjacent angles of a parallelogram are supplementary, therefore, 

In parallelogram ${\text{RISK}}$

$  \angle {\text{RKS  + }}\angle {\text{ISK}} = 180^\circ  $

 $ 120^\circ  + \angle {\text{ISK}} = 180^\circ  $ 

As the opposite angles of a parallelogram are equal, therefore,

In parallelogram ${\text{CLUE}}$,

$\angle {\text{ULC}} = \angle {\text{CEU}} = 70^\circ $

Also, the sum of all the interior angles of a triangle is $180^\circ $

Therefore,

$  {\text{x }} + {\text{ }}60^\circ {\text{ }} + {\text{ }}70^\circ {\text{ }} = {\text{ }}180^\circ  $

$  {\text{x }} = {\text{ }}50^\circ  $ 

10. Explain how this figure is a trapezium. Which of its two sides are parallel?

Ans:

Given:

We need to explain how the given figure is a trapezium and find its two sides that are parallel.

If a transversal line intersects two specified lines in such a way that the sum of the angles on the same side of the transversal equals $180^\circ $, the two lines will be parallel to each other.

Here, $\angle {\text{NML}} = \angle {\text{MLK}} = 180^\circ $

Hence, ${\text{NM}}||{\text{LK}}$

Hence, the given figure is a trapezium.

11. Find ${\text{m}}\angle {\text{C}}$in the following figure if ${\text{AB}}\parallel {\text{CD}}$${\text{AB}}\parallel {\text{CD}}$.

Ans:

Given: ${\text{AB}}\parallel {\text{CD}}$ and quadrilateral

We need to find the measure of $\angle {\text{C}}$

$\angle {\text{B}} + \angle {\text{C}} = 180^\circ $(Angles on the same side of transversal).

$  120^\circ  + \angle {\text{C}} = 180^\circ  $

$  \angle {\text{C}} = 60^\circ  $ 

12. Find the measure of $\angle {\text{P}}$and$\angle {\text{S}}$, if ${\text{SP}}\parallel {\text{RQ}}$in the following figure. (If you find${\text{m}}\angle {\text{R}}$, is there more than one method to find${\text{m}}\angle {\text{P}}$?)

Ans:

Given: ${\text{SP}}\parallel {\text{RQ}}$and 

We need to find the measure of $\angle {\text{P}}$and $\angle {\text{S}}$.

The sum of angles on the same side of transversal is $180^\circ .$

$\angle {\text{P}} + \angle {\text{Q}} = 180^\circ $

$  \angle {\text{P}} + 130^\circ  = 180^\circ  $

$  \angle {\text{P}} = 50^\circ  

Also,

 $\angle {\text{R }} + {\text{ }}\angle {\text{S }} = {\text{ }}180^\circ {\text{ }} $

$  {\text{ }}90^\circ {\text{ }} + {\text{ }}\angle {\text{S }} = {\text{ }}180^\circ  $

  ${\text{ }}\angle {\text{S }} = {\text{ }}90^\circ {\text{ }} $ 

Yes, we can find the measure of ${\text{m}}\angle {\text{P}}$ by using one more method.

In the question,${\text{m}}\angle {\text{R}}$and ${\text{m}}\angle {\text{Q}}$are given. After finding ${\text{m}}\angle {\text{S}}$ we can find ${\text{m}}\angle {\text{P}}$ by using angle sum property.

Exercise 3.4

1. State whether True or False.

(a) All rectangles are squares.

(b) All rhombuses are parallelograms.

(c) All squares are rhombuses and also rectangles.

(d) All squares are not parallelograms.

(e) All kites are rhombuses.

(f) All rhombuses are kites.

(g) All parallelograms are trapeziums.

(h) All squares are trapeziums.

Solution:

(a) False

Every square is indeed a type of rectangle, not every rectangle can be called a square.

(b) True

(c) True

(d) False

It's correct to say that all squares can be classified as parallelograms due to their shared characteristic of having opposite sides that are parallel and opposite angles that are equal.

(e) False.

Because, a kite shape is that its adjacent sides are not necessarily equal in length, unlike those of a square.

(f) True

(g) True

(h) True

2. Identify all the quadrilaterals that have.

(a) four sides of equal length

(b) four right angles

Solution:

(a) Rhombus and square have all four sides of equal length.

(b) Square and rectangles have four right angles. 

3. Explain how a square is

(i) a quadrilateral

(ii) a parallelogram

(iii) a rhombus

(iv) a rectangle

Solution:

(i) Square is a quadrilateral because it has four sides.

(ii) A square is a parallelogram because its opposite sides are parallel and opposite angles are equal.

(iii) Square is a rhombus because all four sides are of equal length and diagonals bisect at right angles.

(iv)Square is a rectangle because each interior angle, of the square, is 90° 

4. Name the quadrilaterals whose diagonals.

(i) bisect each other 

(ii) are perpendicular bisectors of each other 

(iii) are equal

Solution:

(i) Parallelogram, Rhombus, Square and Rectangle

(ii) Rhombus and Square

(iii)Rectangle and Square 

5. Explain why a rectangle is a convex quadrilateral.

Solution:

A rectangle is a convex quadrilateral because both of its diagonals lie inside the rectangle.

6. ABC is a right-angled triangle and O is the mid-point 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).

Solution:

AD and DC are drawn so that AD || BC and AB || DC

AD = BC and AB = DC

ABCD is a rectangle as opposite sides are equal and parallel to each other and all the interior angles are of 90°.

In a rectangle, diagonals are of equal length and also bisect each other.

Hence, AO = OC = BO = OD

Thus, O is equidistant from A, B and C.

Overview of Deleted Syllabus for CBSE Class 8 Maths Understanding Quadrilaterals

Class 8 Maths Chapter 3: Exercises Breakdown

Conclusion

In conclusion, NCERT Solutions for Class 8 Maths Chapter 3 - Understanding Quadrilaterals provides a comprehensive and detailed understanding of the properties and characteristics of various types of quadrilaterals. By studying this chapter and using the NCERT solutions, students can enhance their knowledge of quadrilaterals and develop their problem-solving abilities. The chapter starts by introducing quadrilaterals and their diverse types, including parallelograms, rectangles, squares, rhombuses, and trapeziums. It goes on to explain each type, detailing their characteristic properties like side lengths, angles, diagonals, and symmetry. Students that practice these kinds of questions will gain confidence and perform well on tests.

Other Study Material for CBSE Class 8 Maths Chapter 3

Chapter-Specific NCERT Solutions for Class 8 Maths

Given below are the chapter-wise NCERT Solutions for Class 8 Maths. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.