Polygon with 180 degrees? (polygon has n>5 sides, int. angles form arith. seq., smallest angle 84*, common diff. 4*)

[apple2357]

Can a polygon have 180 degrees? This question which recently appeared in an examination doesn't feel right.
If you do the maths the 25th term in the arithmetic series is 180.
Have i missed something? Can anyone picture the polygon?

 

[The Highlander]

Can a polygon have 180 degrees? This question which recently appeared in an examination doesn't feel right.
If you do the maths the 25th term in the arithmetic series is 180.
Have i missed something? Can anyone picture the polygon?

I'm not about to draw ("picture") the polygon but I get n to lie somewhere between 43 and 47 (We're not supposed to post answers for 4 days so I've just quoted a range ).

I think I see you point (about can an interior angle of a polygon be 180°, if that's what you're asking?) and, as you say, the 25th interior angle in the given sequence would, indeed, be 180°.

I suppose it might be possible to have a polygon where all the sides were of equal length except one that was twice the length of the others.
(If you took n sticks of equal length and laid them together as per the question then two of them would form a straight line.)

Could that still be regarded as a regular polygon?
Would it actually have (n - 1) sides then?

There is also the possibility that the author of the question simply didn't bother to think that through and just came up with 84° as a 'nice, round' starting point (if an odd number like 83 or 85 had been chosen the problem wouldn't arise). S/he may have just wanted to create an exercise where the student was required to use the formula for the sum of the interior angles of a polygon coupled with the formula for the sum of the first n terms of an arithmetic sequence and the ability to solve the resulting quadratic.

We often see questions that are defective in a similar respect but an answer can still be obtained.

Hope that helps.

 

[apple2357]

Many thanks. Thats really helpful and yes i was thinking can.a polygon have an angle of 180 degrees.

 

[blamocur]

Many thanks. Thats really helpful and yes i was thinking can.a polygon have an angle of 180 degrees.

There is a solution where internal angles are less than 180 degrees.

 

[apple2357]

There is a solution where internal angles are less than 180 degrees.

say more? but n>5 in the question?

 

[The Highlander]

There is a solution where internal angles are less than 180 degrees.

Would that not be n = 4?
(Which is excluded by the question; deliberately I expect. )

 

[blamocur]

say more? but n>5 in the question?

Ooops --- I was thinking [imath]n=4[/imath]. Sorry.

 

[Dr.Peterson]

Can a polygon have 180 degrees? This question which recently appeared in an examination doesn't feel right.
If you do the maths the 25th term in the arithmetic series is 180.
Have i missed something? Can anyone picture the polygon?

I wouldn't write out the sequence at all; this sequence has some number n of terms, and clearly will not reach your 25th term.

I'd write an equation that says the sum of the sequence of n terms equals that sum of the interior angles of an n-gon. That leads to a quadratic equation with two solutions, one of which is not greater than 5.

This is not a defective question.

(But the wording of your question confused me, because usually when students talk about a polygon "having" some number of degrees, they are talking about the sum of all the interior angles, not about one angle being 180 degrees or whatever.)

 

[Steven G]

Can a polygon have 180 degrees?

Of course it can. Just consider a triangle.
Of course, you asked the wrong question.

 

[The Highlander]

I am somewhat loth to say this (given their authors) but I think that, in both of the previous two posts, there have been some misunderstandings/misreadings (or lack of reading?) of the prior posts.

Of course it can. Just consider a triangle.
Of course, you asked the wrong question.

The OP wasn't asking whether the sum of the interior angles of a polygon could be 180° (as in your example of a triangle). He was questioning, rather, whether one of a polygon's interior angles could be 180°!

If so, then, clearly, the two sides connected by such an interior angle would effectively become a single side (ie: form a straight line)!

I did say (in Post #2:-.

I think I see your point (about can an interior angle of a polygon be 180°, if that's what you're asking?) and, as you say, the 25th interior angle in the given sequence would, indeed, be 180°.

and (slightly more worryingly)...

I wouldn't write out the sequence at all; this sequence has some number n of terms, and clearly will not reach your 25th term.

I don't believe anyone suggested that one should "write out the sequence" (it was a simple matter to calculate that the 25th term in the sequence would be 180°) and, clearly, the required sequence did reach beyond that (25th) term!

I'd write an equation that says the sum of the sequence of n terms equals that sum of the interior angles of an n-gon. That leads to a quadratic equation with two solutions, one of which is not greater than 5.

I believe that is exactly what the OP, @blamocur and myself all did (getting 4 & 45 for our answers) but the question specifically states that n must be greater than 5!

So, although one result is not greater than 5 (and is therefore, excluded), the other result is significantly greater than 25 (so 180° would be included as one of the interior angles)

This is not a defective question.

Given the resulting anomaly (of an interior angle = 180°), the question is defective, (I would argue) in a similar respect to that one about the 24/25 glasses fitting on a tray. I believe what's happened is what I also wrote at Post #2, ie:-

"...the author of the question simply didn't bother to think it through (about 180° being included as one of the interior angles) and just came up with 84° as a 'nice, round' starting point (if an odd number like 83 or 85 had been chosen the problem wouldn't have arisen).​

​

S/he may have just wanted to create an exercise where the student was required to use the formula for the sum of the interior angles of a polygon coupled with the formula for the sum of the first n terms of an arithmetic sequence and the ability to solve the resulting quadratic."​

 

[apple2357]

I am somewhat loth to say this (given their authors) but I think that, in both of the previous two posts, there have been some misunderstandings/misreadings (or lack of reading?) of the prior posts.


The OP wasn't asking whether the sum of the interior angles of a polygon could be 180° (as in your example of a triangle). He was questioning, rather, whether one of a polygon's interior angles could be 180°!

If so, then, clearly, the two sides connected by such an interior angle would effectively become a single side (ie: form a straight line)!

I did say (in Post #2:-.


and (slightly more worryingly)...

I don't believe anyone suggested that one should "write out the sequence" (it was a simple matter to calculate that the 25th term in the sequence would be 180°) and, clearly, the required sequence did reach beyond that (25th) term!


I believe that is exactly what the OP, @blamocur and myself all did (getting 4 & 45 for our answers) but the question specifically states that n must be greater than 5!

So, although one result is not greater than 5 (and is therefore, excluded), the other result is significantly greater than 25 (so 180° would be included as one of the interior angles)


Given the resulting anomaly (of an interior angle = 180°), the question is defective, (I would argue) in a similar respect to that one about the 24/25 glasses fitting on a tray. I believe what's happened is what I also wrote at Post #2, ie:-

"...the author of the question simply didn't bother to think it through (about 180° being included as one of the interior angles) and just came up with 84° as a 'nice, round' starting point (if an odd number like 83 or 85 had been chosen the problem wouldn't have arisen).​

​

S/he may have just wanted to create an exercise where the student was required to use the formula for the sum of the interior angles of a polygon coupled with the formula for the sum of the first n terms of an arithmetic sequence and the ability to solve the resulting quadratic."​

Agree completely. I solved the problem using algebra ( forming a quadratic equation) got n=45 and then wondered if one of the interior angles hit 180? I again using algebra found when n=25 it did. This then suggested an unusual polygon (!) to say the least. I suppose one of the sides could be 'imaginary'?!

 

[blamocur]

Agree completely. I solved the problem using algebra ( forming a quadratic equation) got n=45 and then wondered if one of the interior angles hit 180? I again using algebra found when n=25 it did. This then suggested an unusual polygon (!) to say the least. I suppose one of the sides could be 'imaginary'?!

Drawing unit sides with the computed angles produces a non-closed polygon -- see attached.

 

[The Highlander]

Drawing unit sides with the computed angles produces a non-closed polygon -- see attached.

Good Job!

I wondered about that too. I tried to just imagine how the sides would 'progress' but it did seem like they would start to spiral outwards and I couldn't see how they might eventually form a closed figure.

Is there even such a thing as a non-closed polygon? All the definitions I've ever seen say a polygon is a closed figure. I guess this is just another respect in which this is a defective question.

Of course, its defects don't make it any less valid as an exam question. It's clear that the candidates are expected just to arrive at an algebraic solution as they will, no doubt, have been taught that they can equate the formulae and solve the resulting quadratic.

​

 

[apple2357]

Thanks for that image!

 

[blamocur]

Drawing unit sides with the computed angles produces a non-closed polygon -- see attached.

Did not like the non-closed version, but the closed one does not look that much better:

 

[Dr.Peterson]

I don't believe anyone suggested that one should "write out the sequence" (it was a simple matter to calculate that the 25th term in the sequence would be 180°) and, clearly, the required sequence did reach beyond that (25th) term!

Yes, I missed that fact that the actual (supposed) solution is beyond the 25th term. I was supposing that the OP had not actually found that solution, because there was no mention of that. And I probably didn't think carefully about what everyone had said, because I had just come back from a trip and was reading too fast. Or because I was focused on the confusion I mentioned about the meaning of "have 180°", which you also pointed out less directly than I did.

You are entirely right. The difficulty in the problem is precisely that, after finding the "solution", one needs to think about the entire sequence in order to see that such a polygon is, at the least, questionable.

I can imagine some contexts in which one would accept a 180° angle in a polygon, though I didn't find any such explicit statements, though many just fail to mention it; definitions do vary (e.g. some allow self-intersection). What interests me more is how a polygon can have so many reflex angles (making it very concave, if not necessarily self-intersecting).

On reading the problem yet again, though, I see that the angles forming the sequence do not have to be arranged in order (64, 68, 72, ...) consecutively around the polygon, but might largely alternate between large and small, which might help.

I left this unsent for some time to ponder whether the polygon can actually exist; I see that others have now discovered this issue, but appear to be assuming the sequence has to be in order. I don't intend to dig any deeper, as it is obvious that the author of the problem can't have intended anyone to actually visualize the "polygon".

 

[Steven G]

The OP wasn't asking whether the sum of the interior angles of a polygon could be 180° (as in your example of a triangle). He was questioning, rather, whether one of a polygon's interior angles could be 180°!

Sorry, but I disagree with you here. Of course it is all silliness, but the OP's question was Can a polygon have 180 degrees?
I was just pointing out to the OP letting him/her know that they need to be more precise when speaking math.

 

[The Highlander]

Sorry, but I disagree with you here. Of course it is all silliness, but the OP's question was Can a polygon have 180 degrees?
I was just pointing out to the OP letting him/her know that they need to be more precise when speaking math.

OK, I take your point, Steven.
(Not entirely sure whether the OP will agree with you, though. )

 

[apple2357]

OK, I take your point, Steven.
(Not entirely sure whether the OP will agree with you, though. )

Yes its a fair point and apologies for being a bit slack. But i hope most people reading the problem would know what i meant and would forgive me. I posted it in haste. We all make mistakes with language and sometimes laying into people ( who might be more sensitive than i am) puts them off the subject.

 

[Steven G]

Yes its a fair point and apologies for being a bit slack. But i hope most people reading the problem would know what i meant and would forgive me. I posted it in haste. We all make mistakes with language and sometimes laying into people ( who might be more sensitive than i am) puts them off the subject.

I'm just trying to make you a tougher math student. If I did the reverse, then I am truly sorry.