NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

Class 8: NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

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Understanding Quadrilaterals

Exercise 3.1

Question 1:
 Given here are some figures:

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8
NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8
NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8
NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

Classify each of them on the basis of the following:

(a) Simple curve
(b) Simple closed curve
 (c) Polygon
 (d) Convex polygon
 (e) Concave polygon

Answer 1:

(a) Simple curve

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8
NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

(b) Simple closed curve

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

(c) Polygons

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

(d) Convex polygons

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

(e) Concave polygon

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

Question 2:
 How many diagonals does each of the following have?
 (a) A convex quadrilateral
 (b) A regular hexagon
 (c) A triangle

Answer 2:
(a) A convex quadrilateral has two diagonals.
Here, AC and BD are two diagonals.

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

(b) A regular hexagon has 9 diagonals.
Here, diagonals are AD, AE, BD, BE, FC, FB, AC, EC and FD.

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

(c) A triangle has no diagonal.

Question 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)
 Answer 3:

Let ABCD is a convex quadrilateral, then we draw a diagonal AC which divides the quadrilateral in two triangles.

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

= 180+180o                              [By Angle sum property of triangle]
= 360o

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8
Hence, the sum of measures of the triangles of a convex quadrilateral is 360o.
Yes, if quadrilateral is not convex then, this property will also be applied.

Let ABCD is a non-convex quadrilateral and join BD, which also divides the quadrilateral in two triangles.
Using angle sum property of triangle,
In ΔABD, NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 1 + NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 2 + NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 3 = 180o ……….(i)
In ΔBDC, NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 4 + NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 5 + NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 6 = 180o ……….(i)

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

Adding eq. (i) and (ii),

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

Hence proved.

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

FigureNCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

Side

3

4

5

6

Angle sum

1 x 1800 = (3- 2) x 1800

2x1800 = (4-2)xl800

3x1800 = (5-2)xl800

4x1800 = (6-2)xl800

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

Answer 4:
(a) When n = 7, then
Angle sum of a polygon = (n - 2)x 180o = (7 - 2)x180o = 5x180o = 900o
(b) When n = 8, then
Angle sum of a polygon = (n - 2)x 180o = (8 - 2)x180o = 6x180o =1080o
(c) When n = 10, then
Angle sum of a polygon = (n - 2)x 180o = (10 - 2)x180o = 8x180o =1440o
(d) When n = n, then
Angle sum of a polygon = (n - 2)x 180o

Question 5:
 What is a regular polygon? State the name of a regular polygon of:
 (a) 3 sides
 (b) 4 sides
 (c) 6 sides

Answer 5:
A regular polygon: A polygon having all sides of equal length and the interior angles of equal size is known as regular polygon.
(i) 3 sides - Polygon having three sides is called a triangle.
(ii) 4 sides - Polygon having four sides is called a quadrilateral.
(iii) 6 sides - Polygon having six sides is called a hexagon.

Question 6:
 Find the angle measures x in the following figures:

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

 

Answer 6:
(a) Using angle sum property of a quadrilateral,

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

50o+130o+120o+ x = 360o
NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 300o+ x = 360o
NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 x = 360- 300o
NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 x = 60o

(b) Using angle sum property of a quadrilateral,

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8
90o+60o+70o+ x = 360o
NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 220o+ x = 360o
NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 x = 360o-220o
NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 x =140o

(c) First base interior angle = 180o- 70o =110o
Second base interior angle = 180o- 60o = 120o
There are 5 sides, n = 5

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8
NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 Angle sum of a polygon = (n - 2)x180o
= (5-2)x 180= 3x180= 540o

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

(d) Angle sum of a polygon = (n - 2)x180o

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

Hence each interior angle is 108o

Question 7:
 (a) Find x + y + z

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

(b) Find x+y+z+w

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

Answer 7:
(a) Since sum of linear pair angles is 180o.

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

[Exterior angle property]

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 x+y+x=90o+120o+150o=360o

(b) Using angle sum property of a quadrilateral,

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8

Since sum of linear pair angles is 180.
NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 w+100 +180o ……….(i)
x+120o +180o ……….(ii)
y +80o +180o ……….(iii)
z +60o +180o ……….(iv)
Adding eq. (i), (ii), (iii) and (iv),
NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 x + y + z + w+ 100o +120o + 80o + 60o +180o +180o +180o +180o
NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 x + y + z + w+ 360o = 720o
NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 x + y + z + w = 720o - 360o
NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 x + y + z + w = 360o

 

The document NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8 Notes - Class 8 is a part of Class 8 category.
All you need of Class 8 at this link: Class 8

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