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
- 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°
- 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°
- How many sides does a regular polygon have if the measure of an exterior angle is 24°?
- 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.
- 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.
- 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.
- 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°
ADDITIONAL QUESTIONS:
1 ). In a quadrilateral ABCD, the angles A, B, C and D are in the ratio 1: 2: 3: 4. Find the measure of each angle of the quadrilateral.
2 ). The measures of two adjacent angles of a parallelogram are in the ratio 4: 5. Find the measure of each of the angles of the parallelogram.
