**NCERT solutions for class 8 maths chapter 3 Understanding Quadrilaterals- ** 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. Solutions of NCERT for class 8 maths chapter 3 Understanding Quadrilaterals is covering the chapter to help you with the answers and conceptual clarity. It’s really an important segment under geometry; without the conceptual knowledge, your efficiency to crack the geometry problem will downfall and thus we request you to give an attentive approach while going through this chapter. It carries around 35% of geometry. The most important types of rectangles are- rectangles, square, rhombus, parallelogram, and trapezium, etc. CBSE NCERT solutions for class 8 maths chapter 3 Understanding Quadrilaterals is covering the problems' solutions from all the variety of 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 consist of a total of 31 questions. NCERT solutions for class 8 maths chapter 3 Understanding quadrilaterals is covering every question in a comprehensive manner.

As shown in ACD,

As shown in ABC,

( Since, , ,,

)

Hence proved, the sum of the measures of and

amounts to .

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.

As we know the sum of all four angles of a quadrilateral is.

are four angles of quadrilateral ABCD.

Hence, the sum of these angles is =

Angles ∠1, ∠2, ∠3, ∠4 meet at a point and sum of these angles is = .

As we know

Hence, sum of .

Draw a line matching points B and D.Line BD divides ABCD in two triangles . and .

Sum of angles of =

Sum of angles of =

Here,

Adding equation 1 and equation 2,

{ and

i.e. = . }

The sum of the interior angles in a quadilateral, = .

**Q1 (a) **Given here are some figures.

Classify each of them on the basis of the following.

(a) Simple curve

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

Some simple curves are

**Q1 (b) **Given here are some figures.

Classify each of them on the basis of the following.

(b) Simple closed curve

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

Some simple closed curve are

**Q1 (c) **Given here are some figures.

Classify each of them on the basis of the following.

(c) Polygon

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

Some polygons are shown in figures

**Q1 (d) **Given here are some figures.

Classify each of them on the basis of the following.

(d) Convex polygon

(d) Convex polygon :Convex polygon are polygons having all interior angles less than .

Convex polygon = 2.

**Q1 (e) **Given here are some figures.

Classify each of them on the basis of the following.

(e) Concave polygon

(e) Concave polygon : Concave polygon have one or more interior angles greater than .

Concave polygon = 1.

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

A convex quadrilateral

(a) A convex quadrilateral :

There are 2 diagonals in a convex quadrilateral.

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

A regular hexagon

(b) A regular hexagon :

Total number of diagonal a regular hexagon have are 9 .

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

A triangle

(c) A triangle

A triangle does not have any diagonal.

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

Here,as we can see this quadrilateral has two triangles and the sum of angles of these triangles is = sum of all angles of a quadrilateral.

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

(a) 7

(a) 7

the sum of angles of a convex polygon

the sum of angle of a convex polygon with 7 sides

the sum of angles of a convex polygon with 7 sides

the sum of angles of a convex polygon with 7 sides

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

(b) 8

(b) 8

sum of the angles of a convex polygon

sum of the angles of a convex polygon with 8 sides

sum of the angles of a convex polygon with 8 sides

sum of the angles of a convex polygon with 8 sides

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

(c) 10

(c) 10

angle sum of a convex polygon

angle sum of a convex polygon with 10 sides

angle sum of a convex polygon with 10 sides

angle sum of a convex polygon with 10 sides

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

(d)

(d)

All the angles of a convex polygon added up to

the sum of the angles of a convex polygon having n sides

**Q5 (i) **What is a regular polygon?

State the name of a regular polygon of 3 sides

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.

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

Each angle =

**Q5 (ii) **What is a regular polygon?

State the name of a regular polygon of 4 sides

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

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

Square has all angles of and all sides are equal.

**Q5 **What is a regular polygon?

State the name of a regular polygon of 6 sides

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.

All angles of the hexagon are each.

**Q6 **Find the angle measure in the following figures.

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

Sum of angles of a quadrilateral

**Q6 **Find the angle measure in the following figures.

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

Sum of angles of a quadrilateral

**Q6 **Find the angle measure in the following figures.

This pentagon has all sides and all angles equal.

Sum of all angles of pentagon is

**Q6 **Find the angle measure in the following figures.

(linear pair) (linear pair)

Sum of all angles of pentagon =

**Q7 (a) **Find

** **

(a) Find

Sum of all angles of triangle is

the unmarked angle of triangle be A.

=

=

**Q7 (b) **Find

** **

**Answer:**

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

Sum of all angles of quadrilateral =

Let the unmarked angle be A.

=

=

Take a regular hexagon Fig 3.10.

**Q1** What is the sum of the measures of its exterior angles

the sum of all its exterior angles will be equal to .

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

Hence,

**Q2 **Take a regular hexagon Fig 3.10.

It is a hexagon with all sides equal.

All interior angles are also equal.

(linear pairs)

Hence,

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

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

It is a hexagon with all sides equal.

All interior angles are also equal.

Sum of all angles of hexagon is .

(linear pairs)

Each exterior angle = = =

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

It is a hexagon with all sides equal.

All interior angles are also equal.

Sum of all angles of hexagon is .

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

(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.

.

All exterior angles be B.

**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.

.

All exterior angles be B.

**Q1 (a) **Find in the following figures.

Sum of all exterior angles of polygon is

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

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

Sum of exterior angles of a polygon =

Let interior angle be A.

Sum of exterior angles of 9 sided polygon =

Exterior angles of 9 sided polygon

Hence, measure of each exterior angle of a regular polygon of 9 sides is

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

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

Sum of exterior angles of a polygon =

Let interior angle be A.

Sum of exterior angles of 15 sided polygon =

Exterior angles of 15 sided polygon

Hence, measure of each exterior angle of a regular polygon of 15 sides is

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

The measure of an exterior angle is 24°

Regular polygon has all exterior angles equal.

Sum of exterior angles of a polygon =

Let number of sides be X.

Sum of exterior angles of a polygon =

Exterior angles of 15 sided polygon

Hence, 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°?

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 =

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°?

The measure of an exterior angle is 22°

Regular polygon has all exterior angles equal.

Sum of exterior angles of a polygon =

Let number of sides be X.

Sum of exterior angles of a polygon =

Exterior angles of 15 sided polygon

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

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

The measure of an interior angle is 22°

Regular polygon has all interior angles equal.

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

Sum of interior angles of a polygon =

Sum of interior angles of a polygon = =

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

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

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

Sum of interior angles of 3 sided polygon =

Interior angles of regular polygon are equal = A .

Hence,the minimum interior angle possible for a regular polygon is .

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

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

Exterior angles a equilateral triangle has maximum measure.

Sum of exterior angles of polygon =

Let exterior angle be A.

Hence,the maximum exterior angle possible for a regular polygon is .

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:**

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-

and

Show that |

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

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.

Kite has symmetry along AC diagonal. and are congruent and equal triangles.

Diagonals AC and BD are of different lengths.

Diagonals bisect each other.

The two diagonals AC and BD bisect .

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

Here is same as except for the name. Similarly the other corresponding

sides are equal too.

Place over . Do they coincide? What can you now say about the lengths

and ?

Similarly examine the lengths and . What do you find?

You may also arrive at this result by measuring and .

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

Here is same as .

Also, is same as

Place over . They coincide with each other.

The lengths and are equal and parallel lines.

The lengths and are equal and parallel lines.

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

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.

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

As shown in the above figure opposites angles are equal and are equal to .

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,

ABCD (Fig 3.29). Let its diagonals and meet at O.

Find the mid point of by a fold, placing C on A. Is the

mid-point same as O?

Does this show that diagonal bisects the diagonal at the point O? Discuss it

with your friends. Repeat the activity to find where the mid point of could lie.

ABCD . Let its diagonals and meet at O.

The mid point of is at point O.Hence, the

mid-point is same as O.

This show that diagonal bisects the diagonal at the point O.

The mid point of is point O.

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

......

In a parallelogram, opposite sides are equal in lengths.

Hence,

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

......

In a parallelogram ,opposite angles are equal.

(ii)

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

......

In a parallelogram, both diagonals bisect each other.

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

.....

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

(iv)

**Q2 (i) **Consider the following parallelograms. Find the values of the unknowns.

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

Opposite angles are equal.

Hence, z =

and y = 100

**Q2 (ii) **Consider the following parallelograms. Find the values of the unknowns .

( Two adjacent angles are supplementary to each other)

x=y= (opposite angles are equal)

z=x= ( corresponding angles are equal)

**Q2 (iii) **Consider the following parallelograms. Find the values of the unknowns .

x= (vertically opposite angles)

(sum of angles of a triangle is )

y=z= (alternate interior angles)

**Q2 (iv) **Consider the following parallelograms. Find the values of the unknowns .

(adjacent angles are supplementary)

y = ( opposite angles are equal)

z= (corresponding angles are equal)

**Q2 (v) **Consider the following parallelograms. Find the values of the unknowns .

y = (opposite angles are equal)

(adjacent angles are supplementary)

x = z = (alternate angles are equal)

**Q3 (i) **Can a quadrilateral ABCD be a parallelogram if

**Answer:**

(i)

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

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

and ?

Opposite sides of a parallelogram are equal in length.

Since, and are opposite sides and have different lengths.

No, it is not a parallelogram.

**Q3 (iii) **Can a quadrilateral ABCD be a parallelogram if and ?

Opposite angles of a parallelogram are equal

Since here and are different.

So, it is not a parallelogram.

The above-shown figure shows two opposite angles equal..

But, it's not a parallelogram because the other two angles are different i.e. .

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

Sum of adjacent angles is .

Hence, angles are and .

Let there be a parallelogram ABCD then, and . (Opposite angles are equal).

Given: Two adjacent angles of a parallelogram have equal measure = .

( adjacent angles of a parallelogram are supplementary)

and ( Opposite angles of a parallelogram are equal)

Hence,

The adjacent figure HOPE is a parallelogram.

(linear pairs)

(opposite angles of a parallelogram are equal)

( adjacent angles are supplementary )

y= (Alternate interior angles are equal)

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

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

Hence, cm and cm.

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

Diagonals of a parallelogram intersect each other.

Hence,cm and cm.

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

** ( adjacent angles are supplemantary)**

(oppsite angles are equal)

(sum of angles of a triangle is )

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

Given, = .

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

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

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

**Q12 **Find the measure of and if in Fig 3.34.

(If you find , is there more than one method to find ?)

Given,

(angles on same side of transversal)

(angles on same side of transversal)

Yes, to find there is more than one method.

PQRS is quadrilateral so the sum of all angles is 360

and we know

so put values of and we get a measurement of

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

PO = OR ( Square is a type of parallelogram)

By SSS congruency rule, and are congruent triangles.

&

Hence,.

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

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

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

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

True.Opposite sides of the rhombus are parallel and equal.

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

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 .

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

False.

All squares have there opposite sides equal and parallel .Hence,they are parallelogram.

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

False,

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

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

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

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

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

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

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

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

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

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

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

**Q3 (i) **Explain how a square is a quadrilateral

A square is a quadrilateral because square has four sides.

**Q3 (ii) **Explain how a square is a parallelogram

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

**Q3 (iii) **Explain how a square is a rhombus

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

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

A square is a rectangle sinse it has all interior angles of .

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

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

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

**Q4 (iii) **Name the quadrilaterals whose diagonals are equal

The quadrilaterals whose diagonals are equal are squares and rectangles.

**Q5 **Explain why a rectangle is a convex quadrilateral.

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

Draw line AD and DC such that and .

and

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

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

Hence, AO = BO = CO = DO

O is equidistant from A,B,C,D.

(1).All the properties of a parallelogram.

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

(3) Diagonals are equal

rhombus with equal angles? Explore this idea.

Properties of 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 rhombus are :

(1) All the properties of a parallelogram.

(2) Diagonals are perpendicular to each other.

A square satisfies all the properties of rhombus so we can define it as

rhombus with equal angles.

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

Trapezium has two sides parallel and 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.

- 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!*

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