Question

is it possible to have a regular polygon each of whose angles measure 145 degrees?

Answer 1

Dear student
$$\begin{gathered} \mathrm{Sum}\mathrm{of}\mathrm{each}\mathrm{interior}\mathrm{angle}\mathrm{of}\mathrm{a}\mathrm{regular}\mathrm{polygon}=\frac{\left(\mathrm{n}-2\right)\times 180°}{\mathrm{n}} \\ 145°=\frac{\left(\mathrm{n}-2\right)\times 180°}{\mathrm{n}} \\ \Rightarrow 145°\mathrm{n}=180°\mathrm{n}-360° \\ \Rightarrow 360°=180°\mathrm{n}-145°\mathrm{n} \\ \Rightarrow 360°=35°\mathrm{n} \\ \Rightarrow \mathrm{n}=\frac{72}{7}\left(\mathrm{which}\mathrm{is}\mathrm{not}\mathrm{possible}\mathrm{because}\mathrm{sides}\mathrm{cannot}\mathrm{be}\mathrm{in}\mathrm{fraction}\right) \\ \mathrm{So}\mathrm{it}\mathrm{is}\mathrm{not}\mathrm{possible}\mathrm{to}\mathrm{have}\mathrm{a}\mathrm{regular}\mathrm{polygon}\mathrm{each}\mathrm{of}\mathrm{whose}\mathrm{angles}\mathrm{measure}145° \end{gathered}$$

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