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I am wondering how to show that a polygon P, in the plane such that at most two angles of P exceed $180$ , then P can be guarded by two guards.

I know that by Chavatal we will need $n \le 8$ and I also know that the sum of interior angles of a polygon is $180(n-2)$ but then I don't know how to prove the statement.

I do not know very much about polygon. I know the bound involving floor function of a third of the vertices.

So maybe we can use that we can only have 8 walls or less? If at most two angles are more then 180, then n-2 angles less then 180. I think a region can be guarded if we can draw a straight line from the guard to the point , where the straight line does not leave the polygon

Can anyone let me know any way to do this? What approach should I try? I really have no idea. I have been sitting trying to do anything but I just make zero progress. I really need some advice

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    Try to prove at first that a polygon with at most one angle exceeding $180^\circ$ can be guarded by one guard. – 2017-02-04
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    @Smylic I will try but I really don't have any idea how to prove things related to polygon and such. When I draw it out I see why , but I don't know how to prove – 2017-02-04
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    You have to start with a mathematical definition of what it means for a set of guards to "guard" a polygon. You may then need to work up through simple polygons such as triangles (or get facts from your textbook or lecture notes prior to the presentation of this question) in order to solve this more complicated question. I don't know if anyone is willing to lay out all those steps for you; can you meet us halfway by editing the questions to show the definitions and as much work as you can do, so we know what next step you need? – 2017-02-04
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    Why don't you use the classical keyword "art gallery problems/theorems" ?It's important for further searches. – 2017-02-04
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    I updated it. and I really know very little. I want to learn it though, it isn't that I am lazy but I have never seen it before. Is their standard text for this? – 2017-02-05
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    I tried to update with anything I know and a title relating to art gallery problem – 2017-02-05
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    What happens if you place guards on the reflex vertices? – 2017-02-06
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    I guess we would want to put the guards at these two angle that exceed 180, but I don't know how to prove every point is still cover – 2017-02-06

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As already suggested, let's prove at first that a polygon with at most one angle exceeding $180^\circ$ can be guarded by one guard. Suppose that $P:=A_1...A_k...A_n$ is a polygon. If $\widehat{A_i}<180^\circ$ for all $1\leq i\leq n$, then $P$ is convex and the result is trivial. On the other hand, if only $\widehat{A_k}>180^\circ$, only one guard $G$ at $A_k$ can guard $P$, because he guards each of the triangles $A_{p}A_{k}A_{p+1}$, where $p\in\{1,...,k-2,k+1,...,n\}$.

Now, let $Q$ be a polygon in the plane such that $n\leq2$ angles exceed $180^\circ$. If $n=0$ ou $n=1$, we get the result. If $n=2$ and $\widehat{A}$ and $\widehat{B}$ are the mentioned angles, divide $Q$ into two polygons so that $A$ and $B$ are in distinct polygons and use the previous conclusion.