Answer:
a) The smallest possible number of sides of the polygon is a regular polygon with 5 sides
b) The largest possible number of sides of the polygon is a regular polygon with 9 sides
Step-by-step explanation:
a) The smallest exterior angle of the polygon = 40°
The largest exterior angle of the polygon = 80°
Therefore, we have the interior angles are;
The smallest interior angle of the polygon = 180° - 40° = 140°
The largest interior angle of the polygon = 180° - 80° = 100°
Therefore, we have
The interior angles of a regular polygon are given by the following formula;
![Interior \ angle =\dfrac{(n-2) \times 180 ^{\circ}}{n}](https://tex.z-dn.net/?f=Interior%20%5C%20angle%20%3D%5Cdfrac%7B%28n-2%29%20%5Ctimes%20180%20%5E%7B%5Ccirc%7D%7D%7Bn%7D)
For an interior angle of 100°, we have;
![100^{\circ} =\dfrac{(n-2) \times 180 ^{\circ}}{n}](https://tex.z-dn.net/?f=100%5E%7B%5Ccirc%7D%20%3D%5Cdfrac%7B%28n-2%29%20%5Ctimes%20180%20%5E%7B%5Ccirc%7D%7D%7Bn%7D)
100/180 × n = n - 2
2 = n - 100/180 × n = 4/9·n
2 = 4/9·n
n = 18/4 = 4.5
Therefore, given that a square has an interior angle of 90° we have the smallest possible number of sides polygon = 5 sides
b) For 140°, we have;
![140^{\circ} =\dfrac{(n-2) \times 180 ^{\circ}}{n}](https://tex.z-dn.net/?f=140%5E%7B%5Ccirc%7D%20%3D%5Cdfrac%7B%28n-2%29%20%5Ctimes%20180%20%5E%7B%5Ccirc%7D%7D%7Bn%7D)
140/180 × n = n - 2
2 = n - 140/180 × n = 2/9·n
2 = 2/9·n
n = 18/2 = 9
n = 9 sides
Therefore, the largest possible number of sides = 9 Sides.