(a) a convex, 11-sided polygon can have at most how many obtuse interior angles? (b) a convex, 11-sided polygon can have at most how many acute interior angles? provide an explanation!

A)  All 11 interior angles may be obtuse.
B)  At most 3 interior angles may be acute.

Explanation:
A)  To find the sum of the angles of a polygon, we use (n-2)(180), where n is the number of sides.  We would then have (11-2)(180)=9(180)=1620.  If the polygon is regular, we divide this by the number of sides; 1620/11 = 147.23.  Thus all 11 angles would be obtuse in a regular 11-sided figure.

B)  All interior angles of a polygon have an exterior angle that is supplementary to it.  If the interior angles are acute, that means the exterior angles are obtuse.  Likewise, if the interior angles are obtuse, the exterior angles would have to be acute.

The sum of the exterior angles of any polygon is 360.  This means there can be no more than 3 obtuse exterior angles; since obtuse means greater than 90, if there were 4 obtuse angles, 4 angles greater than 90, the sum would be more than 360.  If there can be no more than 3 obtuse exterior angles, then there can be no more than 3 acute interior angles.

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