Understanding Quadrilaterals - Solutions 1

CBSE Class –VIII Mathematics
NCERT Solutions
CHAPTER - 3
Understanding Quadrilaterals (Ex. 3.1)
1. Given here are some figures:
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
Ans. (a) Simple curve
(b) Simple closed curve
  
(c) Polygons
  
(d) Convex polygons
  
(e) Concave polygon
  
2. How many diagonals does each of the following have?
(a) A convex quadrilateral  
(b) A regular hexagon
(c) A triangle
Ans. (a) A convex quadrilateral has two diagonals.
Here, AC and BD are two diagonals.
(b) A regular hexagon has 9 diagonals.
Here, diagonals are AD, AE, BD, BE, FC, FB, AC, EC and FD.
(c) A triangle has no diagonal.
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)
Ans. Let ABCD is a convex quadrilateral, then we draw a diagonal AC which divides the quadrilateral in two triangles.
A + B + C + D 
= 1 + 6 + 5 + 4 + 3 + 2
= (1 + 2 + 3) + (4 + 5 + 6)
   
=     
[By Angle sum property of triangle]
= 
Hence, the sum of measures of the triangles of a convex quadrilateral is 
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 ABD, 1 + 2 + 3 =  ……….(i)
In BDC, 4 + 5 + 6 =  ……….(i)
Adding eq. (i) and (ii),
1 + 2 + 3 + 4 + 5 + 6 = 
 1 + 2 + (3 + 4) + 5 + 6
= 
 A + B + C + D = 
Hence proved.
4. Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)
Figure
Side
3
4
5
6
Angle sum
What can you say about the angle sum of a convex polygon with number of sides?
Ans. (a) When  = 7, then
Angle sum of a polygon = 
(b) When  = 8, then
Angle sum of a polygon = 
(c) When  = 10, then
Angle sum of a polygon = 
(d) When  =   then
Angle sum of a polygon = 
5. What is a regular polygon? State the name of a regular polygon of:
(a) 3 sides 
(b) 4 sides  
(c) 6 sides
Ans. 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.
6. Find the angle measures  in the following figures:
Ans. (a) Using angle sum property of a quadrilateral,
 
 
 
 (b) Using angle sum property of a quadrilateral,
 
 
 
(a) First base interior angle 
= 
Second base interior angle
= 
There are 5 sides,  = 5
 Angle sum of a polygon = 
 =  = 
 
   
 
    
 
(b) Angle sum of a polygon = 
 =  = 
 
  
 
Hence each interior angle is 
7. (a) Find     
(b) Find 
Ans. (a) Since sum of linear pair angles is 
 
 
And 
 
Also   
[Exterior angle property]
 
(b) Using angle sum property of a quadrilateral,
 
 
 
 
Since sum of linear pair angles is 
   ……….(i)
   ……….(ii)
   ……….(iii)
   ……….(iv)
Adding eq. (i), (ii), (iii) and (iv),