Question
Is it possible to have a polygon whose each interior angle is $105^\circ ?$
✓
Answer
Given each interior angle $=105^{\circ}$
So, each exterior angle $=180^{\circ}-105^{\circ}=75^{\circ}$
Thus, the number of sides of the polygon
$ =\frac{360^{\circ}}{\text { Each exterior angle }}$
$=\frac{360^{\circ}}{75^{\circ}}$
$=4 \frac{4}{5}$
which is not a natural number
Therefore, no polygon is possible whose each interior angle is $105^{\circ}$.
So, each exterior angle $=180^{\circ}-105^{\circ}=75^{\circ}$
Thus, the number of sides of the polygon
$ =\frac{360^{\circ}}{\text { Each exterior angle }}$
$=\frac{360^{\circ}}{75^{\circ}}$
$=4 \frac{4}{5}$
which is not a natural number
Therefore, no polygon is possible whose each interior angle is $105^{\circ}$.
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