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

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R is a convex polygon. Does R have at least 8 sides? [#permalink]  13 Aug 2006, 14:44
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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.
[Reveal] Spoiler: OA

Last edited by Bunuel on 22 Apr 2014, 01:48, edited 1 time in total.
Edited the question and added the OA.
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What is the sum of the interior angles for an octagon. I know it is 540 for a pentagon..
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Sum of interior angles of any "regular" shape = (n-2)*180

n = number of sides
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What do we know about interior angles of convex polygons?
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Re: DS: Polygon [#permalink]  14 Aug 2006, 09:57
kevincan wrote:
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.

The sum of the interion angles of the polygon = 180(n-2) where n is the number of sides in the polygon.

From (1) 180(n-2) >240
n-2 > 4/3
n>10/3
therefore n should be atleast 4 but we cannot say it is atleast 8

From(2) 180(n-2) >=60n
n-2 >= n/3
2n/3 >= 2
n >=3
we cannot say n is atleast 8

Combining both, we can still say it is atleast 4 but not 8.

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Thanks for the formula. I will have to remember that...

sum of interior angles = (n-2)*180
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In a convex polygon, the maximum degree measure of an interior angle is....?
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kevincan wrote:
In a convex polygon, the maximum degree measure of an interior angle is....?

180.... thats the definitition of a convex polygn; all angles less than 180
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Should be E.

St1: Could be 8 sided or any other.
8 sided = {80,80,80 and 168 each for other 5 angles}
4 sided = {90,90,110,70}: INSUFF

St2: Clearly INSUFF. True for all regular polygons of sides 4 or more.

Combined:
Same as statement 1.: INSUFF
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Re: DS: Polygon [#permalink]  15 Aug 2006, 06:06
kevincan wrote:
kevincan wrote:
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.

Yes, I got it. I always do silly mistakes in your questions.

Statement1 is SUFF. Lets assume this is a 8 sided polygon. Then 5 angles are 80 or less. Then other three angles are 1080 - 400 = 680. Because this is a convex polygon, sum of remaining three angles should not be greater than 540. So this is not an 8 sided polygon.
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R is a convex polygon. Does R have at least 8 sides? (1) [#permalink]  23 Sep 2006, 08:56
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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.
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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.

What is convex polygon
A convex polygon is a simple polygon whose interior is a convex set. The following properties of a simple polygon are all equivalent to convexity:

* Every internal angle is at most 180 degrees.
* Every line segment between two vertices of the polygon does not go
exterior to the polygon

To have minimun 8 sides a polygon should have sum of angles 180(n-2)
180(8-2) = 1080

(1). Exactly 3 of the interior angles of R are greater than 80 degrees.
Lets take the greatest value for these angles- 180 and smallest for other 5 -80.
180*3+80*5= 940 < 1080 ..SUFFI

(2) None of the interior angles of R are less than 60 degrees.
Angles can have any value ..NOT SUFFI

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Re: R is a convex polygon. Does R have at least 8 sides? (1) [#permalink]  21 Apr 2014, 18:05
IMO is E.

A convex polygon with 4 sides can still have angles greater than 80 and less than 60. Experts advice on this?
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Re: R is a convex polygon. Does R have at least 8 sides? (1) [#permalink]  22 Apr 2014, 02:45
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Expert's post
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).

Hope it's clear.
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Re: R is a convex polygon. Does R have at least 8 sides? (1) [#permalink]  01 May 2015, 02:48
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Re: R is a convex polygon. Does R have at least 8 sides? (1)   [#permalink] 01 May 2015, 02:48
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