pretzel wrote:
IMO is E.
A convex polygon with 4 sides can still have angles greater than 80 and less than 60. Experts advice on this?
No, the correct answer is A.
R is a convex polygon. Does R have at least 8 sides?The Sum of Interior Angles of a polygon is \(180(n-2)\) degrees, where \(n\) is the number of sides (so is the number of angles). So, the greater the number of sides the greater is the sum of the angles.
For 8 sided polygon the sum of the angles is \(180(n-2)=180*6=1080\) degrees. The question basically asks whether the sum of the angles of the polygon is more than or equal to 1080 degrees.
(1) Exactly 3 of the interior angles of R are greater than 80 degrees. This implies that each of the remaining angles must be less than or equal to 80 degrees.
Assume that the polygon IS 8-sided. In this case, the sum of those 3 angles would be \(3*80<(sum \ of \ the \ given \ 3 \ angles)<3*180\) --> \(240<(sum \ of \ the \ given \ 3 \ angles)<540\).
Thus the sum of the reaming 5 angles must be \(1080-540<(sum \ of \ the \ remaining \ 5 \ angles)<1080-240\) --> \(540<(sum \ of \ the \ remaining \ 5 \ angles)<840\). But the sum of the reaming 5 angles must be less than or equal to 5*80=400, and not greater than 540 degrees. Therefore the assumption that the polygon could be 8-sided was wrong.
If the polygon cannot be 8-sided, then it cannot be more sided too. So, R must have less than 8 sides. Sufficient.
(2) None of the interior angles of R are less than 60 degrees. Not sufficient: consider equilateral triangle (all angles 60 degrees) and regular octagon (all angles 1080/8=135 degrees).
Answer: A.
Hope it's clear.