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. How many diagonals does each of the following have? 

1. A Convex Quadrilateral

Ans: Given: a convex quadrilateral

We need to find the number of diagonals in the given quadrilateral

We know that a four-sided closed shape with no vertices pointing inward is called a convex quadrilateral.

Consider, a convex quadrilateral

Now, make diagonals

Therefore, a convex quadrilateral has 2 diagonals.

2. A Regular Hexagon

Ans: Given: A regular hexagon

We need to find the number of diagonals of a regular hexagon.

We know that a regular hexagon is a closed curve with six equal sides.

Consider, a regular hexagon

Therefore, a regular hexagon has $9$ diagonals.

3. A Triangle

Ans:  Given: A triangle

We need to find the number of diagonals of a triangle.

We know that a triangle is a closed curve having three sides.

Consider, a triangle

Therefore, a triangle does not have any diagonal.

3. 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!)

Ans: Given: A convex quadrilateral

We need to find the sum of the measures of the angles of a convex quadrilateral. Will this property hold if the quadrilateral is not convex?

Consider, a convex quadrilateral ABCD and then make a diagonal AD.

We know that the sum of angles of a triangle is ${180^ \circ }$.

So, In $\vartriangle {\text{ACD}}$

Sum of angles of $\vartriangle {\text{ACD}}$ is ${180^ \circ }$

Now, In $\vartriangle {\text{ABD}}$ 

Sum of angles of $\vartriangle {\text{ABD}}$ is ${180^ \circ }$.

Therefore, sum of angles of a convex quadrilateral will be sum of angles of $\vartriangle {\text{ACD}}$ and $\vartriangle {\text{ABD}}$

$= {180^ \circ } + {180^ \circ } $

$= {360^ \circ } $ 

Now, consider a concave quadrilateral ABCD, and then make a diagonal AC. The quadrilateral ABCD is made of two triangles, $\vartriangle {\text{ACD}}$ and $\vartriangle {\text{ABC}}$.

Consider, $\vartriangle {\text{ACD}}$

The sum of angles of the triangle are ${180^ \circ }$.

Now, consider $\vartriangle {\text{ABC}}$

The sum of the angles of triangle are ${180^ \circ }$.

Therefore, sum of the angles of quadrilateral ABCD will be

$ = {180^ \circ } + {180^ \circ } $

$   = {360^ \circ } $ 

Thus, we can say that the property hold true for a quadrilateral which is not convex because a quadrilateral can be divided into two triangles.

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

Figure
Side	3	4	5	6
Angle sum	${180^ \circ }$	$   2 \times {180^ \circ } $ $   = (4 - 2) \times {180^ \circ } $	$   3 \times {180^ \circ } $   $ = (5 - 2) \times {180^ \circ } $	$   4 \times {180^ \circ } $ $   = (6 - 2) \times {180^ \circ } $

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

1. 7

Ans:

Given: the table

We need to observe table and make a statement about the angle sum of a convex polygon with number of sides.

From table, we can observe that the angle sum of a convex polygon with ${\text{n}}$sides is $({\text{n}} - 2) \times {180^ \circ }.$

Therefore, the angle sum of a convex polygon with $7$ number of sides will be

$   = (7 - 2) \times {180^ \circ } $

$   = 5 \times {180^ \circ } $

$ = {900^ \circ } $ 

2. 8

Ans: Given: the table

We need to observe table and make a statement about the angle sum of a convex polygon with number of sides.

From table, we can observe that the angle sum of a convex polygon with ${\text{n}}$sides is $({\text{n}} - 2) \times {180^ \circ }.$

Therefore, the angle sum of a convex polygon with $8$ number of sides will be

$   = (8 - 2) \times {180^ \circ } $

$   = 6 \times {180^ \circ } $

$   = {1080^ \circ } $ 

3. 10

Ans: Given: the table

We need to observe table and make a statement about the angle sum of a convex polygon with number of sides.

From table, we can observe that the angle sum of a convex polygon with ${\text{n}}$sides is $({\text{n}} - 2) \times {180^ \circ }.$

Therefore, the angle sum of a convex polygon with $10$ number of sides will be

$ = (10 - 2) \times {180^ \circ } $

$   = 8 \times {180^ \circ } $

$   = {1440^ \circ } $ 

4. n

Ans: Given: the table

We need to observe table and make a statement about the angle sum of a convex polygon with number of sides.

From table, we can observe that the angle sum of a convex polygon with ${\text{n}}$sides will be

$ = ({\text{n}} - 2) \times {180^ \circ }$

5. What is a regular polygon?
State the name of a regular polygon of 

1. 3sides

Ans:

Given: $3$ sides

We need to write the statement of a regular polygon and then state the name of the regular polygon with given number of sides.

A regular polygon is a polygon having all angles equal and all sides equal.

We know that a polygon with three equal sides and each ${60^ \circ }$ angle is a triangle.

So, it will be an equilateral triangle. The diagram will be

2. 4 Sides

Ans:Given: $4$ sides

We need to write the statement of a regular polygon and then state the name of the regular polygon with given number of sides.

A regular polygon is a polygon having all angles equal and all sides equal.

We know that a polygon with four equal sides and each ${90^ \circ }$ angle is called a square.

So, the diagram will be

3. 6 Sides

Ans: Given: $6$ sides

We need to write the statement of a regular polygon and then state the name of the regular polygon with given number of sides.

A regular polygon is a polygon having all angles equal and all sides equal.

We know that a polygon with six equal sides and each ${120^ \circ }$ angle is called a regular hexagon.

So, the diagram will be

6.  Find the angle measure ${\text{'x'}}$in the following figures.

Ans:

Given: A quadrilateral with angles ${50^ \circ },{130^ \circ },{120^ \circ },{\text{x}}$

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

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

Thus,

${50^ \circ } + {130^ \circ } + {120^ \circ }{\text{ + x}} = {360^ \circ } $

 $  \Rightarrow {300^ \circ } + {\text{x}} = {360^ \circ } $

 $  \Rightarrow {\text{x}} = {360^ \circ } - {300^ \circ } $

 $  \Rightarrow {\text{x}} = {60^ \circ } $ 

Ans:

Given: A quadrilateral with angles ${70^ \circ },{60^ \circ },{\text{x}}$

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

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

From given figure, 

$  {90^ \circ } + y = {180^ \circ } $

 $  \Rightarrow y = {180^ \circ } - {90^ \circ } $

 $  \Rightarrow y = {90^ \circ } $ 

Now, the quadrilateral has angles, ${70^ \circ },{60^ \circ },{90^ \circ }{\text{,x}}$

We know that sum of all interior angles of a quadrilateral is ${360^ \circ }.$

Thus, 

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

$   \Rightarrow {220^ \circ } + {\text{x}} = {360^ \circ } $

 $  \Rightarrow {\text{x}} = {360^ \circ } - {220^ \circ } $

$   \Rightarrow {\text{x}} = {140^ \circ } $ 

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

From given figure,

$  {70^ \circ } + {\text{a}} = {180^ \circ } $

 $ \Rightarrow {\text{a}} = {180^ \circ } - {70^ \circ } $

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

And, 

$  {60^ \circ } + {\text{b}} = {180^ \circ } $

$   \Rightarrow {\text{b}} = {180^ \circ } - {60^ \circ } $

 $  \Rightarrow {\text{b}} = {120^ \circ } $ 

Therefore, the angles of the pentagon are ${30^ \circ }{\text{,x,}}{110^ \circ },{120^ \circ }{\text{,x}}$

We know that the sum of all interior angles of a pentagon is ${540^ \circ }.$

Thus,

  ${30^ \circ } + {\text{x}} + {110^ \circ } + {120^ \circ } + {\text{x}} = {540^ \circ } $

  $ \Rightarrow {260^ \circ } + 2{\text{x}} = {540^ \circ } $

  $ \Rightarrow 2{\text{x}} = {540^ \circ } - {260^ \circ } $

 $  \Rightarrow 2{\text{x}} = {280^ \circ } $

  $ \Rightarrow {\text{x}} = \dfrac{{{{280}^ \circ }}}{2} $

 $  \Rightarrow {\text{x}} = {140^ \circ } $ 

Ans: Given: a regular pentagon with angle ${\text{x}}{\text{.}}$

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

We know that the sum of all interior angles of a pentagon is ${540^ \circ }.$

Thus, 

$ 5{\text{x}} = {540^ \circ } $

 $  \Rightarrow {\text{x}} = \dfrac{{{{540}^ \circ }}}{5} $

$   \Rightarrow {\text{x}} = {108^ \circ } $ 

7. 

1. Find ${\text{x}} + {\text{y}} + {\text{z}}$

Ans:

Given: 

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

Property Used:

• A linear pair can be defined as two adjacent angles that add up to ${180^ \circ },$ or two angles that combine to form a line or right angle.

• Exterior angle theorem: If a polygon is convex, the total of the exterior angle measures, one at each vertex, equals${360^ \circ }$.

Using Linear pair,

$  {\text{z}} + {30^ \circ } = {180^ \circ } $

$   \Rightarrow {\text{z}} = {180^ \circ } - {30^ \circ } $

 $  \Rightarrow {\text{z}} = {150^ \circ } $ 

Again, using Linear pair

$  {\text{x}} + {90^ \circ } = {180^ \circ } $

 $  \Rightarrow {\text{x}} = {180^ \circ } - {90^ \circ } $

$   \Rightarrow {\text{x}} = {90^ \circ } $ 

Using Exterior Angle Theorem,

$  {\text{y}} = {90^ \circ } + {30^ \circ } $

$   \Rightarrow {\text{y}} = {120^ \circ } $ 

Therefore,

$  {\text{x}} + {\text{y}} + {\text{z}} = {90^ \circ } + {120^ \circ } + {150^ \circ } $

$   = {360^ \circ } $ 

Find ${\text{x}} + {\text{y}} + {\text{z}} + {\text{w}}$

Ans:

Given: 

We need to find the measure of ${\text{x + y + z + w}}$.

Property Used:

• A linear pair can be defined as two adjacent angles that add up to ${180^ \circ },$ or two angles that combine to form a line or right angle.

• Sum of all the interior angles of a quadrilateral is ${360^ \circ }$.

Thus,

$  {\text{a}} + {60^ \circ } + {80^ \circ } + {120^ \circ } = {360^ \circ } $

$   \Rightarrow {\text{a}} + {260^ \circ } = {360^ \circ } $

$   \Rightarrow {\text{a}} = {360^ \circ } - {260^ \circ } $

$   \Rightarrow {\text{a}} = {100^ \circ } $ 

Using Linear pair,

$  {\text{a}} + {\text{w}} = {180^ \circ } $

 $  \Rightarrow {100^ \circ } + {\text{w}} = {180^ \circ } $

 $  \Rightarrow {\text{w}} = {180^ \circ } - {100^ \circ } $

$   \Rightarrow {\text{w}} = {80^ \circ } $ 

Again, using Linear pair

$  {\text{x}} + {120^ \circ } = {180^ \circ } $

$   \Rightarrow {\text{x}} = {180^ \circ } - {120^ \circ } $

$   \Rightarrow {\text{x}} = {60^ \circ } $ 

Again, using Linear pair

$  {\text{y}} + {80^ \circ } = {180^ \circ } $

$   \Rightarrow {\text{y}} = {180^ \circ } - {80^ \circ } $

 $  \Rightarrow {\text{y}} = {100^ \circ } $ 

Again, using Linear pair

$  {\text{z}} + {60^ \circ } = {180^ \circ } $

 $  \Rightarrow {\text{z}} = {180^ \circ } - {60^ \circ } $

  $ \Rightarrow {\text{z}} = {120^ \circ } $ 

Therefore,

$  {\text{x}} + {\text{y}} + {\text{z}} + {\text{w}} = {60^ \circ } + {100^ \circ } + {120^ \circ } + {80^ \circ } $

$   = {360^ \circ } $ 

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.

Different Types of Polygons, Their Sides, and Angle Sum

Polygons are closed figures having at least or more than three sides. They are made of line segments only. Polygons are classified according to the number of sides they have. Some of the most common polygons and their properties are given in the table below.

Understanding Quadrilaterals Class 8

According to geometry, a quadrilateral is a covered, two-dimensional shape that has four straight sides. The polygon has four vertices or corners. Quadrilaterals will typically imply approved forms with four sides like rectangle, square, Trapezoid, kite, or uneven and uncharacterized. From the polygon formula, we can also derive the Sum of interior angles, i.e. (n - 2) × 180, where n stands for the polygon's number of sides. However, squares, rectangles, etc., are particular types of quadrilaterals with some of their sides and equal angles.

Different Types of Quadrilaterals

There are five types of quadrilaterals based on their shape:

1. Rectangle

2. Square

3. Parallelogram

4. Rhombus

5. Trapezium

Rectangle

A rectangle is a kind of quadrilateral having four right angles. Hence, every angle in a rectangle is equal (360°/4 = 90°). Moreover, the opposite planes of a rectangle are parallel and similar. Diagonals bisect each other. Letting the length of the rectangle L and breadth B then,

1. Area of a Rectangle = Length(L) × Breadth(B).

2. Perimeter = 2 × (L + B).

Properties:

• Every angle of a rectangle are 90°.

• Opposite sides are equal and Parallel.

• Diagonals of a rectangle bisect each other.

Square

Square is another quadrilateral having four equal sides and angles. It's also a normal quadrilateral as both its sides and angles are equal. Accurately like a rectangle, a square has four angles of 90 degrees each. We can also call it a rectangle whose two adjacent sides are equal. Letting the side of a square 'a' then,

1. Area = a × a = a².

2. Perimeter = 2 × (a + a) = 4a.

Properties:

1. All the angles are 90°.

2. Each and every side is parallel and also equal to each other.

3. Diagonals bisect each other perpendicularly.

Parallelogram

A parallelogram is a simple quadrilateral whose opposite sides are parallel, as we can understand by the name itself. Thus, it consists of two pairs of parallel sides. Besides, the opposite angles in a parallelogram are alike, and its diagonals divide each other.

Properties:

• Opposite angles are equal.

• Opposite sides are equal and parallel.

• Diagonals bisect each other.

• The summation of any two adjacent angles is 180 degrees.

Rhombus

A rhombus is also a quadrilateral whose all four sides are identical in length and opposite sides parallel. However, the angles are not similar to 90°. A rhombus with right angles would match a square. We often call rhombus a diamond' as it looks similar to the diamond suit in playing cards. Letting the side of a rhombus is 'a' then, the perimeter = 4a.

Considering the length of two diagonals of the rhombus are d1 and d2, then the rhombus area = ½ × d1 × d2.

Properties:

• All planes are equal, and opposite planes are parallel.

• Opposite angles are equal.

• The diagonals bisect each other at 90°.

Trapezium

A trapezium is also a quadrilateral having one parallel side pair. The parallel sides are known as 'bases,' and the rest are known as 'legs' or lateral sides. Letting the height of a trapezium 'h' then:

1. Perimeter = Sum of lengths of all the sides = AB + BC + CD + DA.

2. Area = ½ × (Sum of lengths of parallel sides) × h = ½ × (AB + CD) × h.

Properties:

A trapezium is another type of quadrilateral in which it follows a single property where only one pair of opposite sides of trapezium should be parallel to each other.

Kite: A Special Quadrilateral

Kite is a quadrilateral that has the following properties.

1. It has two pairs of consecutive sides that are equal in size.

2. Diagonals intersect each other at 90°, therefore the diagonals of a kite are perpendicular to each other.

3. When diagonals intersect each other, only one of them will be bisected.

NCERT Solutions for Class 8 Maths - Chapterwise Solutions

• Chapter 1 - Rational Numbers

• Chapter 2 - Linear Equations in One Variable

• 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 - Visualising Solid Shapes

• Chapter 11 - Mensuration

• Chapter 12 - Exponents and Powers

• Chapter 13 - Direct and Inverse Proportions

• Chapter 14 - Factorisation

• Chapter 15 - Introduction to Graphs

• Chapter 16 - Playing with Numbers

We Cover all the Given Exercises of Chapter 3 Understanding Quadrilaterals:-

Benefits of NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Our subject specialists worked hard to make the solutions to NCERT Answers for Class 8 Mathematics Chapter 3 Understanding Quadrilaterals easier to comprehend for students. We have answered all of the questions from the chapter in the NCERT textbook. Students will be able to recognise the problems and solve them correctly in the test if they refer to these solutions.

Quick Revision 

Quadrilaterals are majorly of 6 types - Squares, Rectangles, Parallelograms, Trapeziums, Rhombuses, and Kites. It is important that students learn the formulas of area and perimeter for these quadrilaterals. It is also imperative that they revise the same so that they can use these to solve sums from this chapter quickly and efficiently.

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.

Conclusion

In conclusion, NCERT Solutions for Class 8 Maths Chapter 3 - Understanding Quadrilaterals provide 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 begins by introducing the concept of a quadrilateral and its different types, such as parallelograms, rectangles, squares, rhombuses, and trapeziums. Each type is explained in terms of its defining properties, including sides, angles, diagonals, and symmetry.