ML Aggarwal: Understanding Quadrilaterals - 1 - Class 8 PDF Download

 Page 1


 
1. Some figures are given below. 
 
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:- 
 
The given figure are classified as, 
(a) Figure (i), Figure (ii), Figure (iii), Figure (v) and Figure (vi) are Simple 
curves. 
Simple curve is a curve that does not cross itself. 
(b) Figure (iii), Figure (v) and Figure (vi) are Simple closed curves. 
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1. Some figures are given below. 
 
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:- 
 
The given figure are classified as, 
(a) Figure (i), Figure (ii), Figure (iii), Figure (v) and Figure (vi) are Simple 
curves. 
Simple curve is a curve that does not cross itself. 
(b) Figure (iii), Figure (v) and Figure (vi) are Simple closed curves. 
In simple closed curves the shapes are closed by line-segments or by a 
curved line. 
(c) Figure (iii) and Figure (vi) are Polygons. 
A Polygon is any 2-dimensional shape formed with straight lines. 
(d) Figure (iii) is a Convex polygon. 
In a convex polygon, every diagonal of the figure passes only through 
interior points of the polygon. 
(e) Figure (vi) is a Concave polygon. 
In a concave polygon, at least one diagonal of the figure contains points 
that are exterior to the polygon. 
 
2. How many diagonals does each of the following have? 
(a) A convex quadrilateral 
(b) A regular hexagon 
Solution: 
 
(a) A convex quadrilateral has two diagonals. 
 
(b) A regular hexagon has 9 diagonals as shown. 
Page 3


 
1. Some figures are given below. 
 
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:- 
 
The given figure are classified as, 
(a) Figure (i), Figure (ii), Figure (iii), Figure (v) and Figure (vi) are Simple 
curves. 
Simple curve is a curve that does not cross itself. 
(b) Figure (iii), Figure (v) and Figure (vi) are Simple closed curves. 
In simple closed curves the shapes are closed by line-segments or by a 
curved line. 
(c) Figure (iii) and Figure (vi) are Polygons. 
A Polygon is any 2-dimensional shape formed with straight lines. 
(d) Figure (iii) is a Convex polygon. 
In a convex polygon, every diagonal of the figure passes only through 
interior points of the polygon. 
(e) Figure (vi) is a Concave polygon. 
In a concave polygon, at least one diagonal of the figure contains points 
that are exterior to the polygon. 
 
2. How many diagonals does each of the following have? 
(a) A convex quadrilateral 
(b) A regular hexagon 
Solution: 
 
(a) A convex quadrilateral has two diagonals. 
 
(b) A regular hexagon has 9 diagonals as shown. 
 
3. Find the sum of measures of all interior angles of a polygon with 
the number of sides: 
(i) 8 
(ii) 12 
Solution: 
From the question it is given that, 
(i) 8 
We know that, 
Sum of measures of all interior angles of 8 sided polygons = (2n – 4) × 
90
o
 
Where, n = 8 
= ((2 × 8) – 4) × 90
o
 
= (16 – 4) × 90
o
 
= 12 × 90
o
 
= 1080
o
 
(ii) 12 
We know that, 
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1. Some figures are given below. 
 
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:- 
 
The given figure are classified as, 
(a) Figure (i), Figure (ii), Figure (iii), Figure (v) and Figure (vi) are Simple 
curves. 
Simple curve is a curve that does not cross itself. 
(b) Figure (iii), Figure (v) and Figure (vi) are Simple closed curves. 
In simple closed curves the shapes are closed by line-segments or by a 
curved line. 
(c) Figure (iii) and Figure (vi) are Polygons. 
A Polygon is any 2-dimensional shape formed with straight lines. 
(d) Figure (iii) is a Convex polygon. 
In a convex polygon, every diagonal of the figure passes only through 
interior points of the polygon. 
(e) Figure (vi) is a Concave polygon. 
In a concave polygon, at least one diagonal of the figure contains points 
that are exterior to the polygon. 
 
2. How many diagonals does each of the following have? 
(a) A convex quadrilateral 
(b) A regular hexagon 
Solution: 
 
(a) A convex quadrilateral has two diagonals. 
 
(b) A regular hexagon has 9 diagonals as shown. 
 
3. Find the sum of measures of all interior angles of a polygon with 
the number of sides: 
(i) 8 
(ii) 12 
Solution: 
From the question it is given that, 
(i) 8 
We know that, 
Sum of measures of all interior angles of 8 sided polygons = (2n – 4) × 
90
o
 
Where, n = 8 
= ((2 × 8) – 4) × 90
o
 
= (16 – 4) × 90
o
 
= 12 × 90
o
 
= 1080
o
 
(ii) 12 
We know that, 
Sum of measures of all interior angles of 12 sided polygons = (2n – 4) × 
90
o
 
Where, n = 12 
= ((2 × 12) – 4) × 90
o
 
= (24 – 4) × 90
o
 
= 20 × 90
o
 
= 1800
o
 
 
4. Find the number of sides of a regular polygon whose each exterior 
angles has a measure of 
(i) 24
o
 
(ii) 60
o
 
(iii) 72
o
 
Solution:- 
(i) The number of sides of a regular polygon whose each exterior angles 
has a measure of 24
o
 
Let us assume the number of sides of the regular polygon be n, 
Then, n = 

°
°
 
n = 15 
Therefore, the number of sides of a regular polygon is 15. 
(ii) The number of sides of a regular polygon whose each exterior angles 
has a measure of 60
o
 
Let us assume the number of sides of the regular polygon be n, 
Then, n = 

°
°
 
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1. Some figures are given below. 
 
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:- 
 
The given figure are classified as, 
(a) Figure (i), Figure (ii), Figure (iii), Figure (v) and Figure (vi) are Simple 
curves. 
Simple curve is a curve that does not cross itself. 
(b) Figure (iii), Figure (v) and Figure (vi) are Simple closed curves. 
In simple closed curves the shapes are closed by line-segments or by a 
curved line. 
(c) Figure (iii) and Figure (vi) are Polygons. 
A Polygon is any 2-dimensional shape formed with straight lines. 
(d) Figure (iii) is a Convex polygon. 
In a convex polygon, every diagonal of the figure passes only through 
interior points of the polygon. 
(e) Figure (vi) is a Concave polygon. 
In a concave polygon, at least one diagonal of the figure contains points 
that are exterior to the polygon. 
 
2. How many diagonals does each of the following have? 
(a) A convex quadrilateral 
(b) A regular hexagon 
Solution: 
 
(a) A convex quadrilateral has two diagonals. 
 
(b) A regular hexagon has 9 diagonals as shown. 
 
3. Find the sum of measures of all interior angles of a polygon with 
the number of sides: 
(i) 8 
(ii) 12 
Solution: 
From the question it is given that, 
(i) 8 
We know that, 
Sum of measures of all interior angles of 8 sided polygons = (2n – 4) × 
90
o
 
Where, n = 8 
= ((2 × 8) – 4) × 90
o
 
= (16 – 4) × 90
o
 
= 12 × 90
o
 
= 1080
o
 
(ii) 12 
We know that, 
Sum of measures of all interior angles of 12 sided polygons = (2n – 4) × 
90
o
 
Where, n = 12 
= ((2 × 12) – 4) × 90
o
 
= (24 – 4) × 90
o
 
= 20 × 90
o
 
= 1800
o
 
 
4. Find the number of sides of a regular polygon whose each exterior 
angles has a measure of 
(i) 24
o
 
(ii) 60
o
 
(iii) 72
o
 
Solution:- 
(i) The number of sides of a regular polygon whose each exterior angles 
has a measure of 24
o
 
Let us assume the number of sides of the regular polygon be n, 
Then, n = 

°
°
 
n = 15 
Therefore, the number of sides of a regular polygon is 15. 
(ii) The number of sides of a regular polygon whose each exterior angles 
has a measure of 60
o
 
Let us assume the number of sides of the regular polygon be n, 
Then, n = 

°
°
 
n = 6 
 Therefore, the number of sides of a regular polygon is 6. 
(iii) The number of sides of a regular polygon whose each exterior angles 
has a measure of 72
o
 
Let us assume the number of sides of the regular polygon be n, 
Then, n = 

°
°
 
n = 5 
Therefore, the number of sides of a regular polygon is 5. 
 
5. Find the number of sides of a regular polygon if each of its interior 
angles is 
(i) 90
o
 
(ii) 108
o
 
(iii) 165
o
 
Solution:- 
(i) The number of sides of a regular polygon whose each interior angles 
has a measure of 90
o
 
Let us assume the number of sides of the regular polygon be n, 
Then, we know that 90
o
 = 
	
 – 


 × 90
o
 
°
°
 = 
	
 – 


 
  1 = 
	
 – 


 
2n – 4 = n 
By transposing we get,