NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev

Mathematics (Maths) Class 8

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Class 8 : NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev

The document NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev is a part of the Class 8 Course Mathematics (Maths) Class 8.
All you need of Class 8 at this link: Class 8

EXERCISE 3.1 
Question 1. Given here are some figures.
1. NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev

2. NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
3.NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
4.NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
5. NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
6.NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
7.NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
8.NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
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 
Solution:
(a) Simple curves are: (1), (2), (5), (6) and (7).
(b) Simple closed curves are: (1), (2), (5), (6) and (7).
(c) Polygons are: (1), (2) and (4).
(d) Convex polygon is: (2).
(e) Convex polygons are (1) and (4).

Question 2. How many diagonals does each of the following have? 
(a) A convex quadrilateral 
(b) A regular hexagon 
(c) A triangle
Solution:
Note: Number of diagonals in a polygon of n-sides =  NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
(a) In quadrilateral, number of sides (n) = 4

∴ Number of diagonalsNCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev

 NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
(b) In a regular hexagon, number of sides (n) = 6 
∴  Number of diagonals
  NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
(c) In a triangle, number of sides (n) = 3 
∴ Number of diagonals   
NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev

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!)
Solution: The sum of measures of angles of a convex quadrilateral = 360° Yes, this property holds, even if the quadrilateral is not convex.

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


Figure
NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRevNCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRevNCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRevNCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
Side
3456
Angle sum
180°
2 x 180°
= (4 – 2) x 180°
3 x 180°
= (5 – 2) x 180°
4 x 180°
= (6 – 2) x 180°


What can you say about the angle sum of a convex polygon with number of sides? (a) 7 (b) 8 (c) 10 (d) n
Solution: From the above table, we conclude that sum of the interior angles of polygon with n-sides = (n – 2) x 180°
(a) When n = 7
Substituting n = 7 in the above formula, we have
Sum of interior angles of a polygon of 7 sides (i.e. when n = 7)
= (n – 2) x 180° = (7 – 2) x 180°
= 5 x 180° = 900°

(b) When n = 8
Substituting n = 8 in the above formula, we have
Sum of interior angles of a polygon having 8 sides
= (n – 2) x 180° = (8 – 2) x 180°
= 6 x 180° = 1080°

(c) When n = 10
Substituting n = 10 in the above formula, we have
Sum of interior angles of a polygon having 10 sides
= (n – 2) x 180°
= (10 – 2) x 180°
= 8 x 180° = 1440°

(d) When n = n 

The sum of interior angles of a polygon having n-sides = (n – 2) x 180°

 

Question 5. What is a regular polygon?
State the name of a regular polygon of
(i) 3 sides 
(ii) 4 sides 
(iii) 6 sides

Solution: A polygon is said to be a regular polygon if
(a) The measures of its interior angles are equal, and
(b) The lengths of its sides are equal.
The name of a regular polygon having
(i) 3 sides is ‘equilateral-triangle’
(ii) 4 sides is ‘square’
(iii) 6 sides is ‘regular hexagon’

Question 6. Find the angle measure x in the following figures.

NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
Solution: (a) ∵ The sum of interior angles of a quadrilateral = 360°
∴ x + 120° + 130° + 50° = 360°
or  x + 300° = 360°
or  x = 360° – 300° = 60° 

(b) ∵ The sum of interior angles of a quadrilateral = 360°
∴  x + 60° + 70° + 90° = 360°
or  x + 220° = 360°
or x = 360° – 220° = 140° 

(c) Interior angles are: 30°, x°, x°, (180° – 70°)
and   (180° – 60°),i.e.  30°, x°, x°, 110° and 120° 

The given figure is a pentagon.
∵ Sum of interior angles of a pentagon = 540°
∴ 30° + x + x + 110° + 120° = 540°
or  

2x + 260° = 540°
or

NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
(d) It is a regular pentagon.
Sum of all interior angles of a regular pentagon = 540°.
∴  Its each angle is equal.
∴  x + x + x + x + x = 540°
or  
5x = 540°
or 
 x = 540° ÷ 5 = 108°
Question 7.
NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev
(a) Find x + y + z. 
(b) Find x + y + z + w.
Solution:
(a) ∵ x + 90° = 180° [Linear pair]
∴  x = 180° – 90° = 90°
y = 30° + 90° = 120° [∵ Sum of interior opposite angles = exterior angle]

z = 180° – 30° = 150°
Now, x + y + z = 90° + 120° + 150° = 360°
(b) ∵ The sum of interior angles of a quadrilateral = 360°
∴   ∠1 + 120° + 80° + 60° = 360°
or  1 + 260° = 360°
or  1 = 360° – 260° = 100°
NCERT Solutions(Part- 1)- Understanding Quadrilaterals Class 8 Notes | EduRev

Now, x + 120° = 180° [Linear pair]
∴  x = 180° – 120° = 60° 
y + 80° = 180°   [Linear pair]
∴  y = 180° – 80° = 100° 
z + 60 = 180° [Linear pair]
∴  z = 180° – 60° = 120° 
w + 100 = 180°"  [Linear pair]
∴ w = 180° – 100° = 80°

 Thus,
x + y + z + w = 60° + 100° + 120° + 80° = 360°

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