Question
$AB, BC$ and $CD$ are three consecutive sides of a regular polygon. If angle $BAC = 20^\circ$ ; find : $(i)$ its each interior angle, $(ii)$ its each exterior angle $(iii)$ the number of sides in the polygon.
✓
Answer
$\because$ Polygon is regular $($Given$)$
$\therefore A B=B C$
$\Rightarrow \angle \mathrm{BAC}=\angle \mathrm{BCA} \ldots[\angle \mathrm{S}$ opposite to equal sides $]$
But $\angle B A C=20^{\circ}$
$\therefore \angle B C A=20^{\circ}$
i.e. In $\triangle \mathrm{ABC}$,
$\angle \mathrm{B}+\angle \mathrm{BAC}+\angle \mathrm{BCA}=180^{\circ}$
$ \angle \mathrm{B}+20^{\circ}+20^{\circ}=180^{\circ}$
$ \angle \mathrm{B}=180^{\circ}-40^{\circ}$
$ \angle \mathrm{B}=140^{\circ}$
$(i)$ each interior angle $=140^{\circ}$
$(ii)$ each exterior angle $=180^{\circ}-140^{\circ}=40^{\circ}$
$(iii)$ Let no. of. sides $=n$
$\therefore \frac{360^{\circ}}{n}=40^{\circ}$
$ n=\frac{360^{\circ}}{40^{\circ}}=9$
$ n=9$
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