Question

Question 183
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 1

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 ${90}^{\circ }$). So, the sum of the exterior angles of such a polygon will be greater than $4\times {90}^{\circ }={360}^{\circ }$.
However, this is impossible, since the sum of the exterior angles of a polygon must always be equal to ${360}^{\circ }$. Hence, a polygon can have, at most, 3 obtuse exterior angles.