Question

5. How many sides does a regular polygon have if each interior angle is
(a) 144°
(b) 160°
(c) 156°​

Answer 1

$\bf \: \underline {Formula \: to \: be \: used } :$ ↴

$\boxed{ \red{ \sf \: Each \: interior \: angles \: = \dfrac{(n - 2 \times 180\degree )}{n}}}$

Part : (A)

$\sf \longrightarrow \: \: Interior \: angle = 144 \degree$

$\sf \longrightarrow \: \: Sides = \: ?$

$\sf \: Each \: interior \: angles \: = \dfrac{(n - 2 \times 180\degree )}{n}$

$\sf \: Therefore, \dfrac{(n - 2 \times 180)}{n} = 144 \degree$

$\sf \implies \: ( n - 2 ) \times 180 = 144n$

$\sf \implies \: 180n - 360 = 144n$

$\sf \implies \: 180n - 144n = 360$

$\sf \implies \: 36n = 360$

$\sf \implies \: n = \dfrac{360}{36}$

$\sf \implies \: n = 10$

$\underline{ \boxed{ \sf\pink{Answer = 10 \: sides}}}$

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Part : (B)

$\sf \longrightarrow \: \: Interior \: angle = 160\degree$

$\sf \longrightarrow \: \: Sides = \: ?$

$\sf \: Each \: interior \: angles \: = \dfrac{(n - 2 \times 180\degree )}{n}$

$\sf \: Therefore, \dfrac{(n - 2 \times 180)}{n} = 160 \degree$

$\sf \implies \: ( n - 2 ) \times 180 = 160n$

$\sf \implies \: 180n - 360 = 160n$

$\sf \implies \: 180n - 160n = 360$

$\sf \implies \: 20n = 360$

$\sf \implies \: n = \dfrac{360}{20}$

$\sf \implies \: n = 18$

$\underline{ \boxed{ \sf\pink{Answer = 18 \: sides}}}$

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Part : (C)

$\sf \longrightarrow \: \: Interior \: angle = 156\degree$

$\sf \longrightarrow \: \: Sides = \: ?$

$\sf \: Each \: interior \: angles \: = \dfrac{(n - 2 \times 180\degree )}{n}$

$\sf \: Therefore, \dfrac{(n - 2 \times 180)}{n} = 156 \degree$

$\sf \implies \: ( n - 2 ) \times 180 = 156n$

$\sf \implies \: 180n - 360 = 156n$

$\sf \implies \: 180n - 156n = 360$

$\sf \implies \: 24n = 360$

$\sf \implies \: n = \dfrac{360}{24}$

$\sf \implies \: n = 15$

$\underline{ \boxed{ \sf\pink{Answer = 15 \: sides}}}$