Question
Find the maximum number of acute angles which a convex quadrilateral, a pentagon and a hexagon can have. Observe the pattern and generalise the result for any polygon.
Find the maximum number of acute angles which a convex quadrilateral, a pentagon and a hexagon can have. Observe the pattern and generalise the result for any polygon.
Answer:
:
Let us assume that a convex polygon has four or more acute angles. If an interior angle is acute, then the corresponding exterior angle will be obtuse ( greater than 90∘). So, the sum of the exterior angles of such a polygon will begreater than 4×90∘=360∘.
However, this is impossible, since the sum of the exterior angles of a polygon must always be equal to 360∘. Hence, a polygon can have, at most, 3 obtuse exterior angles.
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Let us assume that a convex polygon has four or more acute angles. If an interior angle is acute, then the corresponding exterior angle will be obtuse ( greater than 90∘). So, the sum of the exterior angles of such a polygon will begreater than 4×90∘=360∘.
However, this is impossible, since the sum of the exterior angles of a polygon must always be equal to 360∘. Hence, a polygon can have, at most, 3 obtuse exterior angles.
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