Question : In a regular polygon if one of its internal angles is greater than the external angle by $132^\circ$, then the number of sides of the polygon is:
Option 1: $14$
Option 2: $12$
Option 3: $15$
Option 4: $16$
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Correct Answer: $15$
Solution :
In a regular polygon, the relationship between an internal angle $(I)$ and an external angle $(E)$,
$⇒I = 180^\circ - E$ ...(1)
Given that the internal angle is greater than the external angle by $132^\circ$.
$⇒I = E + 132^\circ$ ...(2)
Solving the above equations, we get,
$⇒180^\circ - E = E + 132^\circ$
$\therefore E = \frac{180^\circ - 132^\circ}{2} = 24^\circ$
The number of sides $(n)$ in a polygon,
$⇒n = \frac{360^\circ}{E} = \frac{360^\circ}{24^\circ} = 15$
Hence, the correct answer is $15$.
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