A person walked completely around the edge of a park beginning at the

Responding to a pm:

You need to find the sum of exterior angles of a polygon. We know that the sum of exterior angles of a convex polygon is always 360. But the question stem does not give us that the polygon is convex. What if the park looks like this:


Certainly feasible! But this is a concave polygon and sum of all angles turned here will not be 360.
So what we need from the statements is whether the park is a convex polygon or not.

(1) One of the turns is 80 degrees.
No information about other angles.

(2) The number of sides of the park is 4, all of the sides are straight, and each interior angle is less than 180 degrees.
"Each interior angle is < 180" means it is a convex polygon. Sufficient.


Answer (B)
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This question relates to a straightforward geometric rule so we'll go for a Precise approach:
The sum of the internal angles of a polygon with n sides is 180(n -2), so to answer we need to know how many sides the park has.

(1) does not give us this information but (2) does.

(B) is our answer.

Please comment on my flow of thoughts,

Statement 1:
Not Sufficient - because do not know the shape of it

Statement 2:
Because the interior angle + exterior angle = 180
The shape given is quadrilateral.
Therefore, the sum of the exterior angle must be 360.

Close, but not quite.
The sum of angles the person walked is not the sum of exterior angles (which complete the interior angles to 180 as you write), but rather 360 - each internal angle (as the person makes a 'full turn' around each corner).
So the calculation you would need to to do is (360 - internal angle 1) + (360 - internal angle 2) + .... for all 4 angles.
Since the internal angles of a quadrilateral sum to 360, this gives 360*4 - 360 = 1080.
You obviously don't have to do the actual math though...

But the measure of interior angel is less than 180. So my confusion is if angle is 160/150 then multiple answer comes.

Posted from my mobile device

Hey shaonkarim,

You are correct that any one angle can be any value smaller than 180.
But the SUM of the internal angles in a quadrilateral is always 360.
Since the person walks around the ENTIRE park, then they walk around all corners. Thus, the value of any specific angle does not matter.

Does this help?

Thank you very much! You're helpful ☺

Statement B tells is that its a four sides convex polygon.

and we know that sum of all exterior angles of a convex polygon taken one per vertex in clockwise or counter clockwise adds up to 360

Hence Sufficient
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Hey DavidTutorexamPAL
Can you please elaborate this explanation a bit more? How are you able to say that the sum of angles = 360 - each internal angle?
As per the diagram, I thought that the person is making turns in such a way that we need to count the sum of exterior angles and as that is always 360 degrees, I marked D.

Can you please help me explain the same?
Thanks

I think a drawing is the best answer at this point.
See attached, note that the blue angles sum to 360 because, given stmt (2), the park is a quadrilateral and that the required number is the sum of the degrees of all the red angles.
Stmt (1) gives no information on the shape of the park, only on one specific angle (so is insufficient).
If it is still unclear please ask.

DavidTutorexamPAL
Hey David, Thanks for your reply.

A drawing indeed explains things well, so here is my exact questions for you in a drawing format. I mean to ask why are the angles not the sum of all green angles? Why are we taking the red angles to count the total sum of the degrees by which the man has turned? Please see image attached.

Thanks

Hey aalekhoza,

Going back to the original question's drawing, I think you may be right -- the question could be referring to the 'green angles' only.
Even if it is, however, you still need to know that the park is a polygon* to know that the degree-sum is 360 (and if you don't count turns in the opposite direction as having 'negative' degrees, then you need to know that it is a convex polygon). Note that the original question does not state that the park is a polygon, though the drawing makes us want to think that it is.
The shape of the park is not given in the question stem or in stmt (1), but is given in (2), which tells us that it is a (convex) quadrilateral. Therefore (2) is sufficient and (B) is the answer.

*Technically speaking, if the exterior of the park is a closed curve which doesn't cross itself, and if you measure degrees with respect to a specific orientation (i.e. if turning 'left' is 'positive' then turning' 'right' is 'negative'), then their sum is always 360. Look up 'winding number' on Wikpedia if you like, but keep in mind that this is well above anything GMAT-related.

DavidTutorexamPAL
Thanks for your explanation David. I got the catch now. After you explanation, I re-read the statements for convex vs concave and understood why statement 2 is correct as it explicitly states the "bold part" >> "The number of sides of the park is 4, all of the sides are straight, and each interior angle is less than 180 degrees." and with that we know that it's a regular convex polygon and thus 180(n-2) will work here, otherwise the correct answer would have been E.

Thanks.

DavidTutorexamPAL, could you please explain your below statement-
" (and if you don't count turns in the opposite direction as having 'negative' degrees, then you need to know that it is a convex polygon)"

I want to understand why the qs needed to specify that it is convex quadrilateral for us to arrive at the answer

Hey Bhavyam,

The simplest answer is that only convex polygons are part of the GMAT material, so the question writers had to make sure to tell you that the shape of the park is a convex polygon and not just any polygon.

The more detailed answer goes into college-level and grad-level math so isn't really relevant here.
In brief, the sum of a polygon's exterior angles is ALWAYS 360, no matter whether it is concave or convex. The idea behind this is that you are doing '1 full turn' around a point, but formalizing and proving this idea is involved. The one caveat is what I wrote above -- that you need to define the angle of a 'left' turn or 'right' turn in a specific way for the claim to hold true.

This question is asking you to decide if statement number (1) or (2) gives you sufficient information to solve the problem. In this case, we can see that number 1 states "one of the turns is 80 degrees."

Because this doesn't tell us how many sides the park has, we cannot solve for the sum of the degrees of all the turns that the walker made if we only know the angle of one turn. This is due to the fact that we aren't told how many turns there were, and there is no way to find out! This is insufficient information.

Looking at statement (2), it says "The number of sides of the park is 4, all of the sides are straight, and each interior angle is less than 180 degrees." Since we now know the park has 4 straight sides with each angle less than 180 degrees, we know it must be a quadrilateral shape. We should also know that the basic geometrical rule for a quadrilateral is that all of its 4 angles add up to 360 degrees.

So, with this information, we easily have the information to tell "What is the sum of the degrees of all the turns that the person made?" We know they must have made 4 turns, due to the 4 straight sides of the park, giving us 4 angles. So, the answer would be 360 degrees, if we did have to solve this mathematically. In this case, though, all we have to do is know that (2) does give sufficient information to solve the question.

Do we need to find the internal angles or exterior angles. I think it is Sum of 180 - (All Interier angles)
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Bunuel
Hi Bunuel,

Could you provide a detailed solution to this problem? I couldn't find any helpful solution.

Thanks
Anusha

VeritasKarishma

I have a doubt in the highlighted portion

If the park is a convex polygon, say a regular pentagon, wouldn't the turns be taken around vertex and so the degree of turn should be interior angle, and in case it is not convex as shown in the image attached by you, then for the obtuse angle we'll take exterior angle's measure. That's why we need to know if it is convex + the no. of sides

Kindly clarify!