NCERT Solutions For Class 8 Maths Chapter 3

NCERT Solutions Class 8 Maths Understanding Quadrilaterals

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals provides complete set of solved questions for students to helps them in their assignments. These solutions help students to learn all the formulas used for finding the measurement of the particular given angles.

NCERT Solutions for Class 8 Maths Chapter 3 are available in PDF format for free in our website BYJU’S and are prepared by the subject professionals under the guidance of CBSE class 8 syllabus. Those students who find difficulties in solving Quadrilaterals problems can download NCERT Solutions for Class 8 Maths Chapter 3 pdf files and use for their preparations.

NCERT Solutions Class 8 Maths Chapter 3 Exercises

Exercise 3.1

Q1) Given below are some shapes.

Identify the below diagrams based on the following category

(a) Concave polygon             (b) Simple curve        (c)Convex polygon

(d)Simple closed curve          (e) Polygon

1

Ans.)

(a) Concave polygon: 1

(b) Simple curve: 1, 2, 5, 6 and 7

(c) Convex polygon: 2

(d) Simple closed curve: 1, 2, 5, 6 and 7

(e) Polygon: 1 and 2

 

Q2) Identify the number of diagonals does each figure have?

(a)Regular hexagon               (b)Triangle                 (c)Convex quadrilateral

Ans.)

(a)Regular hexagon

2

It has 9 diagonals

(b) Triangle

3

It has no diagonals.

(c) Convex quadrilateral

4

It has 2 diagonals.

 

Q3) In a convex quadrilateral, determine the sum of measures of angles? If the quadrilateral is not convex, will the property be the same?

Ans.)

5

Let the sides of the quadrilateral be ABCD. We can see that the quadrilateral is formed by the combination of 2 triangles that is ADC and ABC .

Since ,we are aware that the total sum of the interior angles of triangle will be 180

Thus, the sum of the interior angles of both the triangles are  180

+180 = 360

6

Let us consider another quadrilateral ABCD which is not convex and join BC which divides it into two triangles BCD and ABC .

In ΔABC,

∠1 + ∠2 + ∠3 = 180°   (angle sum property of triangle)

In ΔBCD,

∠4 + ∠5 + ∠6 = 180°   (angle sum property of triangle)

Therefore,  ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 180° + 180°

⇒ ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 360°

⇒ ∠A + ∠B + ∠C + ∠D = 360°
Thus, this property is valid only if the quadrilateral is not convex.

 

Q4) Check the table (Each diagram is separated into triangles and the sum of the angles is taken out from that)

7

Find out the angle sum of convex polygon with following number of sides?

(a)n     (b)11   (c)9      (d)7

Ans.

(a)n

Given n = n

Therefore, angle sum = (n2)×180

(b)11

Given, n=11

Therefore, angle sum = (112)×180=1620

(c)9

Given, n=9

Therefore, angle sum = (92)×180=1260

(d)7

Given, n=7

Therefore, angle sum = (72)×180=900

 

Q5) What do mean by a regular polygon?

Tell the name of the regular polygon which has

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

Ans.) A polygon which has sides of equal length and the angles whose measures are equal is called a regular polygon.

(i) Regular polygon which has 6 sides is called regular hexagon.

(ii) Regular polygon which has 3 sides is called equilateral triangle.

(iii) Regular polygon which has 4 sides is square.

Q6) Calculate the measure of the angle x in the figures given below:

8

Ans.)

(a) The diagram has 4 sides. Hence, Quadrilateral.

Total sum of the angles of quadrilateral = 360

50+130+120+x=360 300+x=360 x=360300=60

 

(b) The diagram is having four sides. It’s a quadrilateral.

And , one side is 90

Sum of the interior angles of the quadrilateral = 360

90+70+60+x=360 220+x=360 x=360220=140

 

(c) There are 5 sides in the figure. It’s a pentagon.

Sum of the interior angles of pentagon = 540

Angles which are at the bottom are a linear pair

Hence, 18070=110

18060=120 30+110+120+x+x=540 260+2x=540 2x=540260=280 x=280/2=140

 

(d) The diagram given has five equal sides. Therefore a regular pentagon .Thus, all interior angles are equal.

5x=540 x=540/5 x=108

 

Q7) (a) Find x + y + z + w

10

(b)Find x + y + z

28

 

Ans.

(a) Sum of all interior angles of quadrilateral = 360

Single side of quadrilateral = 360(60+80+120)=360260

x+120=180180120=60 y+80=180y=18080=100 z+60=180z=18060=120 w+100=180w=180100=80 x+y+z+w=60+100+120+80=360

 

(b) Sum of interior angles of triangle = 180

Single side of triangle = 180(90+30)=60

x+90=180x=18090=90 y+60=180y=18060=120 z+30=180z=18030=150 x+y+z=90+120+150=360

 

EXERCISE – 3.2

 QUESTIONS-:

1Find the value of ‘p’ from the following figures

12

Ans.-(a)

13

Here, 125° + a = 180° => 180° – 125° = 55° (Linear pair)

125° + b = 180° =>180° – 125° = 55° (Linear pair)

p = a + b (exterior angle of a triangle is equal to the sum of two opposite interior two angles)

=> p = 55° + 55° = 110°

 

 

(b)

14

Two interior angles are right angles = 90°

70° + a = 180° => a = 180° – 70° = 110°    (Linear pair)

60° + b = 180° =>b = 180° – 60° = 120°    (Linear pair)
The given figure consisting of five sides and it is a pentagon.
Hence, the sum of the angles of a pentagon = 540°
90° + 90° + 110° + 120° + q = 540°
=>410° + y = 540° => q = 540° – 410° = 130°
p + q = 180°    (Linear pair)
=> p + 130° = 180°
=> p = 180° – 130° = 50°

 

2. Obtain the value for each of the exterior angle of a regular polygon with

(i) 10 sides           (ii) 25 sides

 

Ans.- Sum of angles a regular polygon having side a = (a-2)×180°
(i) Sum of angles a regular polygon having side 10 = (10-2)×180°
= 8×180° = 1440°
Each interior angle = 1440°/10 = 144°
Each exterior angle = 180° – 144° = 36°
Or,
Each exterior angle = Sum of exterior angles/Number of sides = 360°/10 = 36°

(i) Sum of angles a regular polygon having side 25 = (25-2)×180°
= 23×180° = 4140°
Each interior angle = 4140°/25 = 165.6°
Each exterior angle = 180° – 165.6° = 14.4°
Or,
Each exterior angle = Sum of exterior angles/Number of sides = 360°/25 = 14.4°

 

3. Calculate the number of sides a regular polygon will have if the value of an exterior angle is 36°?

Ans.-As we know that, Each exterior angle = Sum of exterior angles/Number of sides
So, 36° = 360°/Number of sides
=>Number of sides = 360°/36° = 10
Hence therefore, the regular polygon have 15 sides.

 

4. A regular polygon having each of its interior angles 144°. Calculate the number of sides it will have.

Ans.-Given, Interior angle = 144°
Exterior angle = 180° – 144° = 36°
Number of sides = Sum of exterior angles/exterior angle
=>Number of sides = 360°/36° = 10
Thus, the regular polygon have 10 sides.

 

5. (a) Can a regular polygon have each exterior angle with a measure of 37°?

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

 

Ans.-(a) Exterior angle = 22°
Number of sides = Sum of exterior angles/exterior angle
=>Number of sides = 360°/37° = 9.72
No, we cannot have a regular polygon with each exterior angle as 37° as it is not divisor of 360.

(b) Interior angle = 37°
Exterior angle = 180° – 37°= 143°
No, we can’t have a regular polygon with each exterior angle as 143° as it is not divisor of 360.

 

6. (a) Find the minimum possible interior angle for a regular polygon? Why?
(b) Find the maximum possible exterior angle for a regular polygon?

 

Ans.-(a)An equilateral triangle is a regular polygon having 3 sides with the least possible minimum interior angle as the regular polygon with minimum sides can be constructed with 3 sides at least..
Since, sum of the interior angles of a triangle = 180°
Each interior angle = 180°/3 = 60°

(b) Equilateral triangle is regular polygon with 3 sides has the maximum exterior angle because the regular polygon consisting least number of sides have the maximum exterior angle possible.
Maximum exterior possible = 180 – 60° = 120°

 

Exercise 3.3

1. Given a parallelogram  ABCD. Complete each statement along with the definition or property used.
(i) AD = ……           (ii) DCB = ……
(iii) OC = ……         (iv)  DAB +  CDA = ……

Answer

(i) AD = BC   (Opposite sides of a parallelogram are equal)
(ii) DCB = DAB (Opposite angles of a parallelogram are equal)
(iii) OC = OA    (Diagonals of a parallelogram are equal)
(iv)  m DAB + m CDA = 180

 

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

1516181917

Answer

(a)

15

y = 100 (opposite angles of a parallelogram)

x + 100= 180   (Adjacent angles of a parallelogram)

⇒ x = 180100= 80

x = z = 80 (opposite angles of a parallelogram)
Thus, x = 80, y = 100and y = 100

(b)

16

50+ x = 180⇒ x = 18050 = 130 (Adjacent angles of a parallelogram)
x = y = 130 (opposite angles of a parallelogram)
x = z = 130 (corresponding angle)

(c)

18

x = 90 (vertical opposite angles)

x + y + 30 = 180 (angle sum property of a triangle)

90 + y + 30 = 180
⇒ y = 180120 = 60
also, y = z = 60                                                                                                                                                                                                             (alternate angles)

(d)

19

z = 80 (corresponding angle)

z = y = 80 (alternate angles)

x + y = 180 (adjacent angles)

⇒ x + 80 = 180 ⇒ x 18080 = 100

 

(e)

17

y = 112  [Opposite angles are equal in a parallelogram

40+y+x=180 40+112+x=180 152+x=180 x=180152=28

and z = x = 28          [Alternate angles]

 

3. Can a quadrilateral ABCD be a parallelogram if

(i) D + B = 180?         (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?
(iii) A = 70and C = 65?

 Answer

(i)Yes, a quadrilateral ABCD be a parallelogram if D + B = 180but it should also fulfilled some conditions which are:

  • The sum of the adjacent angles should be 180
  • Opposite angles must be equal.

 

(ii) No, opposite sides should be of same length. Here, AD ≠ BC

 

(iii) No, opposite angles should be of same measures. AC

 

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

20

Answer

ABCD is a figure of quadrilateral that is not a parallelogram but has exactly two opposite angles that is B = D of equal measure. It is not a parallelogram because AC

 

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.

 Answer

Let the measures of two adjacent angles A and B be 3x and 2x respectively in parallelogram ABCD.

A + B = 180

⇒ 3x + 2x = 180

⇒ 5x = 180

⇒ x = 36

We know that opposite sides of a parallelogram are equal.

A = C = 3x = 3 × 36 = 108

B = D = 2x = 2 × 36 = 72

 

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

 Answer

Let ABCD be a parallelogram.

Sum of adjacent angles of a parallelogram =  180

A + B = 180

⇒ 2A = 180

A = 90

also, 90+ B  = 180

B = 18090= 90

A = C = 90

B = D = 90

 

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

Answer

21

y = 40 (alternate interior angle)

P = 70 (alternate interior angle)

P = H = 70 (opposite angles of a parallelogram)

z = H40 = 7040 = 30

H + x = 180

70 + x = 180

⇒ x = 18070 = 110

 

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

 22

Answer

(i) SG = NU and SN = GU   (opposite sides of a parallelogram are equal)

3x = 18

⇒ x = 18/3 = 6

3y – 1 = 26 and,

⇒ 3y = 26 + 1

⇒ y = 27/3 = 9

x = 6 and y = 9

 

(ii) 20 = y + 7 and 16 = x + y   (diagonals of a parallelogram bisect each other)

y + 7 = 20

⇒ y = 20 – 7 = 13 and,

x + y  = 16

⇒ x + 13 = 16

⇒ x = 16 – 13 = 3

x = 3 and y = 13

 

9. In the below figure both RISK and CLUE are parallelograms. Find the value of x.

23

Answer

K + R = 180 (adjacent angles of a parallelogram are supplementary)

⇒ 120 + R = 180

R = 180120 = 60°

also, R = SIL (corresponding angles)

SIL = 60

also, ECR = L = 70(corresponding angles)

x + 60+ 70= 180 (angle sum of a triangle)

⇒ x + 130 = 180

⇒ x = 180130 = 50

 

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

 24

Answer

 

When a transversal line intersects two lines in such a way that the sum of the adjacent angles on the same side of transversal is 180

then the lines are parallel to each other.

Here, M + L  =100

+ 80

= 180

 

Thus, MN || LK

As the quadrilateral KLMN has one pair of parallel line therefore it is a trapezium.

MN and LK are parallel lines.

 

11. Find mC in Fig 3.33 if  AB || DC ?

 25

Answer

 

mC  + mB  = 180

(angles on the same side of transversal)

⇒mC  + 60

= 180

 

⇒mC 180

60

= 120

 

 

12. Find the measure of p andS if  SP || RQ ? in Fig 3.34. (If you find mR , is there more than one method to find mp ?)

 26

Answer

 

p  +Q  = 180

(angles on the same side of transversal)

p  +130= 180

 

p  =180

130= 50°

also, R  + S  = 180

(angles on the same side of transversal)

90+ S  = 180

 

S  =180

90= 90

Thus, p  =50and S  = 90

 

Yes, there are more than one method to find mp .

PQRS is a quadrilateral. Sum of measures of all angles is 360

Since, we know the measurement of Q ,R  and S .

Q  = 130, R  = 90and S  = 90

p  + 130+ 90+ 90= 360

p  +310= 360

p  =360 310 = 50

 

 

Exercise 3.4

1. State whether the following questions given below are True or False.

(a) All rectangles are squares.
(b) All rhombuses are parallelograms.
(c) All squares are rhombuses and also rectangles.
(d) All squares are not parallelograms.

(e) All kites are rhombuses.

(f) All rhombuses are kites.

(g) All parallelograms are trapeziums.

(h) All squares are trapeziums.

Answer

(a) False. All square are rectangles but all rectangles are not square.

(b) True

(c) True

(d) False. All squares are parallelograms as opposite sides are parallel and opposite angles are equal.

(e) False. A kite doesn’t have all sides of same length.

(f) True

(g) True

(h) True

 

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

(b) four right angles

Answer

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

(b) Square and rectangle have four right angles.

 

3. Explain how a square is.
(i) a quadrilateral

(ii) a parallelogram

(iii) a rhombus

(iv) a rectangle

Answer

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

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

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

(iv) Square is a rectangle because its interior angle is 90°

 

4. Name the quadrilaterals whose diagonals are equal to each other.
(i) bisect each other

(ii) are perpendicular bisectors of each other

(iii) are equal

Answer

(i) Parallelogram, Rhombus, Square and Rectangle

(ii) Rhombus and Square

(iii) Rectangle and Square

 

5. Explain why a rectangle is a convex quadrilateral.

Answer

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

 

6. PQR is a right-angled triangle and W is the midpoint of the side opposite to the right angle. Explain why W is equidistant from P, Q and R. (The dotted lines are drawn additionally to help you).

 27

Answer

PS and SR are drawn so that PS || QR and PQ || SR

PS = QR and PQ = SR

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

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

Hence, PW = WR = QW = WS

Thus, W is equidistant from P, Q and R.