NCERT Solutions for Class 8 Maths Chapter 3 - Understanding Quadrilaterals

 

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.

NCERT solutions for class 8 maths chapter 3 Understanding Quadrilaterals Excercise: 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.

 

Answer:

As shown in \triangleACD,

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

As shown in \triangleABC,

                               \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)].

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 is360\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.

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).

Answer:

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 Excercise: 3.1

Q1 (a) Given here are some figures.

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.


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.


Classify each of them on the basis of the following.

    (c) Polygon

 

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.


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.


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 :

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 :

Total number of diagonal a regular hexagon have are 9 .

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

A triangle

Answer:

(c) A triangle

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.

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.)

What can you say about the angle sum of a convex polygon with 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.)

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

    (b) 8

Answer:

 (b) 8

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

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

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

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.)

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

    (c) 10

Answer:

 (c) 10

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.)

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.

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.

Square has all angles of 90\degreeand 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.

All angles of the hexagon are 120\degree each.

Q6 Find the angle measure x in the following figures.

    

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.

    

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.

    

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.

    

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

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

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?

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

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\degree60\degree

 

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

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

 

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

 

NCERT solutions for class 8 maths chapter 3 Understanding Quadrilaterals Excercise: 3.2

Q1 (a) Find x in the following figures.

    

Answer:

Sum of all exterior angles of 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.

    

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 have all sides,interior angles and exterior angles equal.

Sum of exterior angles of a polygon = 360\degree

Let 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, 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 have all sides,interior angles and exterior angles equal.

Sum of exterior angles of a polygon = 360\degree

Let 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, 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°

Regular polygon has all exterior angles equal.

Sum of exterior angles of a polygon = 360\degree

Let 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°

Regular polygon has all exterior angles equal.

Sum of exterior angles of a polygon = 360\degree

Let 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,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\degree .

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

Answer:

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 = \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

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\degree

 

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 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 minimum number of sides i.e. 3.

Exterior angles a equilateral triangle has maximum measure.

 Sum of exterior angles of polygon = 360\degree

Let 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.

Solutions of NCERT 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).

    

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   

 

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?
                                           

Show that
\Delta ABC and
\Delta ADC are
congruent.
What do we
infer from
this?

                                    

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.

 

Answer:

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).

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:

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.

 

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:

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,


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:

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.

NCERT solutions for class 8 maths chapter 3 Understanding Quadrilaterals Excercise: 3.3

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

    

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.

        

\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.

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.

    

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 unknownsx, y, z.

    

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.

    

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.

    

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.

    

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.

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:

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.

    

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)

 

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)

    

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=3cm  and   y=13cm.
 

 

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)

Answer:

Given, \angle M+\angle L=100\degree+80\degree180\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}.

    

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 Pand \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?)

    

Answer:

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

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

             \angle P=180\degree-130\degree

             \angle P=50\degree

\angle R+\angle S=180\degree                    (angles on 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.

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 Excercise: 3.4

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

Answer:

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

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

Answer:

True.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 there opposite sides equal and parallel .Hence,they are parallelogram.

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

Answer:

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.

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 pair 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 square has four sides.

Q3 (ii) Explain how a square is a parallelogram

Answer:

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

Q3 (iii) Explain how a square is a rhombus

Answer:

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

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

Answer:

A square is a rectangle sinse 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).

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.

Solutions of NCERT 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

NCERT solutions for class 8 maths chapter 3 Understanding Quadrilaterals 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 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.

Solutions of NCERT 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 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.

NCERT solutions for class 8 maths: Chapter-wise

Chapter -1

NCERT solutions for class 8 maths chapter 1 Rational Numbers

Chapter -2 

Solutions of NCERT for class 8 maths chapter 2 Linear Equations in One Variable

Chapter-3

CBSE NCERT solutions for class 8 maths chapter 3 Understanding Quadrilaterals

Chapter-4

NCERT solutions for class 8 maths chapter 4 Practical Geometry

Chapter-5

Solutions of NCERT for class 8 maths chapter 5 Data Handling

Chapter-6

CBSE NCERT solutions for class 8 maths chapter 6 Squares and Square Roots

Chapter-7

NCERT solutions for class 8 maths chapter 7 Cubes and Cube Roots

Chapter-8

Solutions of NCERT for class 8 maths chapter 8 Comparing Quantities

Chapter-9

NCERT solutions for class 8 maths chapter 9 Algebraic Expressions and Identities

Chapter-10

CBSE NCERT solutions for class 8 maths chapter 10 Visualizing Solid Shapes

Chapter-11

NCERT solutions for class 8 maths chapter 11 Mensuration

Chapter-12

NCERT Solutions for class 8 maths chapter 12 Exponents and Powers

Chapter-13

CBSE NCERT solutions for class 8 maths chapter 13 Direct and Inverse Proportions

Chapter-14

NCERT solutions for class 8 maths chapter 14 Factorization

Chapter-15

Solutions of NCERT for class 8 maths chapter 15 Introduction to Graphs

Chapter-16

CBSE NCERT solutions for class 8 maths chapter 16 Playing with Numbers

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!

 

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