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Slovenia — National Math Olympiad, 2012

Geometry Difficulty 3.9 AMC 10/12 Find the answer Slovenia

At most how many interior angles of an $n$ -gon can be greater than $180^\circ$ ?

Pick one

Solution

The answer is $n - 3$ . It can be seen from the picture that only $n - 3$ interior angles of an $n$ -gon can be greater than $180^\circ$ . The sum of all the interior angles of an $n$ -gon is equal to $(n - 2) \cdot 180^\circ$ . If at least $n - 2$ interior angles were greater than $180^\circ$ , the sum of all the interior angles would be greater than $(n - 2) \cdot 180^\circ$ , which would be a contradiction.

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