3.3    SUM OF THE MEASURES OF THE EXTERIOR ANGLES OF A POLYGON:

The sum of the exterior angles of any polygon will be 360°. Example 1:  If the sum of the measures of the interior angles of a polygon equals the sum of the measures of the exterior angles, how many sides does the polygon have?

Solution:  The sum of the measures of the interior angles of a polygon with n sides

=  ( n  –  2 )  x 1800

The sum of the exterior angles of any polygon  =   3600 ( n  –  2 )  x 1800

=   3600   Þ n  =  2  +  2  =  4 EXERCISE 3.2

1. Find x in the following figures. 125°   +   m   =   180°   ⇒   m   =   180°   –   125°   =   55° ( Linear pair )125°   +   n    =   180°   ⇒   n   =   180°   –   125°   =   55° ( Linear pair )

x   =   m   +   n ( exterior angle of a triangle is equal to the sum of 2 opposite interior 2 angles )

⇒   x   =   55°   +   55°   =   110°

b ) Two interior angles are right angles   =   90°

70°   +   m   =   180°   ⇒   m   =   180°   –   70°   =   110°  ( Linear pair )

60°   +   m   =   180°   ⇒   m   =   180°   –   60°   =   120° ( Linear pair ) the figure is having five sides and is a pentagon.

Thus, sum of the angles of pentagon   =   540° 90°   +   90°   +   110°   +   120°   +   y   =   540°

⇒   410°   +   y   =   540°   ⇒   y   =   540°   –   410°   =   130°

x   +   y   =   180° ( Linear pair )

⇒   x   +   130°   =   180°

⇒   x   =   180°   –   130°   =   50°

1. Find the measure of each exterior angle of a regular polygon of

( i ) 9 sides       ( ii ) 15 sides Solution:

Sum of angles a regular polygon having side n   =   ( n  –  2 )  ×  180°

( i ) Sum of angles a regular polygon having side 9   =   ( 9  –  2 )  ×  180°  =   7  ×  180°   =   1260°

Each interior angle  =  1260/9   =   140°

Each exterior angle   =   180°   –   140°   =   40°

Or,

Each exterior angle   =   sum of exterior angles/Number of angles   =   360/9   =   40°

( ii )      Sum of angles a regular polygon having side 15   =   ( 15  –  2 )  ×  180°

=   13  ×  180°   =   2340°

Each interior angle   =   2340/15   =   156°

Each exterior angle   =   180°   –   156°   =   24°

Or,

Each exterior angle   =   sum of exterior angles/Number of angles   =   360/15   =   24°

1. How many sides does a regular polygon have if the measure of an exterior angle is 24°?
2. Solution:

Each exterior angle   =   sum of exterior angles/Number of angles

24°  =   360/ Number of sides

⇒   Number of sides   =   360/24   =   15

Thus, the regular polygon has 15 sides.

1. How many sides does a regular polygon have if each of its interior angles is 165°.

Solution:  Interior angle   =   165°

Exterior angle   =   180°   –   165°   =   15°

Number of sides   =   sum of exterior angles/ exterior angles

⇒   Number of sides   =   360/15   =   24

Thus, the regular polygon has 24 sides.

1. 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 )        Exterior angle   =   22°

Number of sides   =   sum of exterior angles/ exterior angle

⇒   Number of sides   =   360/22   =   16.36

No, we can’t have a regular polygon with each exterior angle as 22° as it is not divisor of 360.

b )        Interior angle   =   22°

Exterior angle   =   180°   –   22°  =   158°

No, we can’t have a regular polygon with each exterior angle as 158° as it is not divisor of 360.

1. 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 )  Equilateral triangle is regular polygon with 3 sides has the least possible minimum interior angle because the regular with minimum sides can be constructed with 3 sides at least.. Since, sum of interior angles of a triangle   =   180°

Each interior angle   =   180/3   =   60°

b ) Equilateral triangle is regular polygon with 3 sides has the maximum exterior angle because the regular polygon with least number of sides has the maximum exterior angle possible. Maximum exterior possible   =   180   –   60°   =   120°