R is a convex polygon. Does R have at least 8 sides? (1)

Question

R is a convex polygon. Does R have at least 8 sides?

(1) Exactly 3 of the interior angles of R are greater than 80 degrees.

(2) None of the interior angles of R are less than 60 degrees.

Please explain your answer.
Thanks

alpha_plus_gamma · Answer

A.

1) consider an 8 sides polygon, the sum of angles= 6*180 = 1080
Only 3 angles can be greater than 80 and the polygon is convex polygon so each angle must be < 180
let the measure of 3 angles each be 170
170* 3 = 510 
sum of remaining angles = 1080 - 510 = 570

each of remaining angles= 570/3 = 190..which cant be true...

thus the polygon cant have atleast 8 sides.

suff

2) This can be true for any polygon, e.g. rectangle, hexagon or 9 sided polygon as well.

Thus insuff

rahulgoyal1986 · Answer

A very very good question. 
Answer A.

The sum total of an 8 sided convex polygon = 1080
If, three of the interior angles have a degree measure of more than 80, the rest of the 5 angles will have a degree measure of less than 80.

In a convex polygon none of the angle can be greater than 180.

considering the worst case scenario:

The 3 angles greater than 80, can at max have a degree measure of 179.
And the rest of the five angles, can at max have a degree measure of 79.

179*3 + 79*5 < 1080

The second is too broad of an option to give any kind of an answer.

x2suresh · Answer

SUM OF THE INTERNAL ANGLES FOR Convex POLYGON with n sides = (n-2)*180 = 6*180 1)Exactly 3 of the interior angles of R are greater than 80 degrees. important for onvex polygon .. all angles must be <180 degrees. these three angles must be between netween 80 and 180 assume that 3 angles are ~ 180 (max possible angles. 3*180 + A1+A2+...An= 6*180 for atleast 8 sides n must be >=5 A1+A2+... An ~ 3*180 A1.. An must be <=80 assume that all are 80 n= (3*180/80)=54/8=6.7 ~7 So n must be atleast 7 .. means.. no of angles >10 Suffcieint. 2)None of the interior angles of R are less than 60 degrees. Clearly this is infufficient... possible.. n>8 and n<8 rectangle. -- all are 90 >60 octogon -- possible.. all angles must be equal to 120. Insufficient. A is the answer

tarek99 · Answer

Thanks everyone for your great input! The OA is A, but I have a question regarding your approach. You said that when 3 angles are over 80 degrees, then the remaining 5 angles must be less than 80 degrees? Would you please explain why the remaining 5 angles must be less than 80 degrees?

tarek99 · Answer

question guys. Does a convex polygon mean that each angle is less than 180 degrees or does it mean that each angle is less than or equal to 180 degrees?
thanks

jallenmorris · Answer

An angle of 180 degrees isn't really much of an angle in geometry. Technically speaking, it's possible, but with drawing a polygon, show me an enclosed shape with 10 angles of 180 degrees each

If you trust Wikipedia: https://en.wikipedia.org/wiki/Convex_polygon It says: Convex = Angle < 180 degrees Concave = Angle > 180 degrees Angle = 180 degrees = straight line. EDIT: I think the best definition should be: Convex = All Angles <= 180 degrees Concave = All Angles >= 180 degrees A circle is not a polygon but it "could" satisfy the definition of Covnex = Angle < 180 degrees. If you thought about each pixel on your monitor being a point on the circle as it is displayed, and measured the angles between them, there would technically not be an angle that measures 180 degrees. But the midpoint of a side of a square does technically measure 180 degrees. This is a point that doesn't really matter in the end for GMAT. Does a circle even have "angles"?

x2suresh · Answer

Convex polygon means.. all angles must be < 180 If you try to draw any polygon with all angles equal to 180 degrees you will get a straight line :lol:

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rahulgoyal1986 · Answer

I meant, as per the question, if three out of eight angles of an octagon are above 80 degrees, the rest 5 will be less than 80. Had the rest of the angles been greater than 80 thn the question would have said more than 3 angles of the polygon are more thn 80.

Hope above helps. Feel free to quote again

tarek99 · Answer

so, if 3 angles are suppose to be greater than 80 degrees, shouldn't that mean that the remaining 5 angles must be less than or equal to 80? I was just wondering why you rather specifically went for the less than 80 degrees.

rahulgoyal1986 · Answer

Well Well Well. Mr. Tarek u're so very right.

Thanks for pointing it out. Cheers!!

R is a convex polygon. Does R have at least 8 sides? (1)

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