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Let A1 A2 A3 ... A15 be a regular polygon. A rotation centered at $A_4$ with an angle of $\alpha$ takes $A_3$ to $A_5$. Given that $\alpha < 180^\circ$, find $\alpha,$ in degrees.

 Apr 30, 2024
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This is the diagram. You can try to rotate A_3 around A_4 in your mind. Notice the angle in question is just an interior angle of the regular polygon.

 

We use the formula for an interior angle of a regular n-gon: \(\alpha = \dfrac{(n-2)\times 180^\circ}n\). Here, n is the number of sides, which is 15 in this case.

We get \(\alpha = \dfrac{(15 - 2) \times 180^\circ}{15} = 156^\circ\).

 Apr 30, 2024

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