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Exercise - 3.2

Question-1 :-  Find x in the following figures. quadrilateral

Solution :-
(a) By linear pair angle, 
    125° + m = 180°
           m = 180° - 125°
           m = 55° 
    By linear pair angle, 
    125° + n = 180°
           n = 180° - 125°
           n = 55°
    Exterior angle x = sum of opposite interior angles
                   x = 55° + 55°
                   x = 110°
(b) Sum of angles of a pentagon = (n-2) x 180° = (5-2) x 180° = 3 x 180° = 540°
    By using of linear pair angles,
    a + 90° = 180°
          a = 180° - 90° 
          a = 90°
    
    b + 60° = 180°
          b = 180° - 60°
          b = 120°
        
    c + 90° = 180°
          c = 180° - 90°
          c = 90°
        
    d + 70° = 180°
          d = 180° - 70°
          d = 110°
     
    e + x = 180° 
        e = 180° - x 

    Now, a + b + c + d + e = 540°
    90° + 120° + 90° + 110° + 180° - x = 540°
    590° - x = 540°
           x = 590° - 540°
           x = 50°
    

Question-2 :-  Find the measure of each exterior angle of a regular polygon of
(i) 9 sides   (ii) 15 sides

Solution :-
(i) Sum of angles of a regular polygon = (n-2) x 180° = (9 - 2) x 180° = 7 x 180° = 1260° 
    Each interior angle = sum of interior angles ÷ no. of sides = 1260° ÷ 9 = 140°
    So, each exterior angle = 180° - 140° = 40°

(ii)Sum of exterior angles of a regular polygon = 360°
    Each interior angle = sum of interior angles ÷ no. of sides = 360° ÷ 15 = 24°
    

Question-3 :-  How many sides does a regular polygon have if the measure of an exterior angle is 24°?

Solution :-
  Let no. of sides be n.
  Sum of exterior angles of a regular polygon = 360°
  No. of sides = sum of exterior angles ÷ each inetrior angle = 360° ÷ 24° = 15
  Hence, the regular polygon has 15 sides.
    

Question-4 :-  How many sides does a regular polygon have if each of its interior angles is 165°?

Solution :-
  Let no. of sides be n.
  Exterior angle = 180° - 165° = 25°
  Sum of exterior angles of a regular polygon = 360°
  No. of sides = sum of exterior angles ÷ each inetrior angle = 360° ÷ 15° = 24
  Hence, the regular polygon has 24 sides.
    

Question-5 :-  (a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?
(b) Can it be an interior angle of a regular polygon? Why?

Solution :-
(a) No, because 22° is not divisior of 360°.
(b) No, because each exterior angle is 180° - 22° = 158°, which is not divisior of 360°.
    

Question-6 :-  (a) What is the minimum interior angle possible for a regular polygon? Why?
(b) What is the maximum exterior angle possible for a regular polygon?

Solution :-
(a) The Equilateral Trianglebeing a regular polygon of 3 sides has the last measure of an interior angles of 60°.
    Sum of all angles of triangle = 180°
    x + x + x = 180°
           3x = 180°
            x = 180°/3
            x = 60°

(b) By first (a), we can observe that the greatest interior angle = 180° - 60° = 120°. 
    
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