Ncert Solutions For Class 8 Maths Ex 3.1

Ncert Solutions For Class 8 Maths Chapter 3 Ex 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