Find the number of sides in a regular polygon, if its each interior angle is:

(i) 160°

(ii) 135°

(iii) $1\dfrac{1}{5}$ of a right angle.

(i) 160°

According to the properties of a polygon, if there are n sides, then each of its interior angles is $\dfrac{(2n - 4) \times 90°}{n}$.

Given that each interior angle is 160°,

$\dfrac{(2n - 4) \times 90°}{n}$ = 160°

By cross multiplying, we get

⇒ (2n - 4) x 90° = 160°n

⇒ 180°n - 360° = 160°n

⇒ 180°n - 160°n = 360°

⇒ 20°n = 360°

⇒ n = $\dfrac{360°}{20°}$

⇒ n = 18

Hence, the number of sides is 18.

(ii) 135°

According to the properties of a polygon, if there are n sides, then each of its interior angles is $\dfrac{(2n - 4) \times 90°}{n}$.

Given that each interior angle is 135°,

⇒ $\dfrac{(2n - 4) \times 90°}{n}$ = 135°

By cross multiplying, we get

⇒ (2n - 4) x 90° = 135°n

⇒ 180°n - 360° = 135°n

⇒ 180°n - 135°n = 360°

⇒ 45°n = 360°

⇒ n = $\dfrac{360°}{45°}$

⇒ n = 8

Hence, the number of sides is 8.

(iii) $1\dfrac{1}{5}$ of a right angle.

= $\dfrac{6}{5}$ x 90°

= $\dfrac{540°}{5}$

= 108°

According to the properties of a polygon, if there are n sides, then each of its interior angles is $\dfrac{(2n - 4) 90°}{n}$.

Given that each interior angle is 108°,

⇒ $\dfrac{(2n - 4) \times 90°}{n}$ = 108°

By cross multiplying, we get

⇒ (2n - 4) x 90° = 108°n

⇒ 180°n - 360° = 108°n

⇒ 180°n - 108°n = 360°

⇒ 72°n = 360°

⇒ n = $\dfrac{360°}{72°}$

⇒ n = 5

Hence, the number of sides is 5.