MHB Understanding the Relationship between Angles & Diagonals in a Polygon

[highmath]

There a relationships between angles to diagonals in a polygon?

 

[Joppy]

Welcome to MHB!

Do you have any particular polygons in mind?

 

[highmath]

Can we start with pentagon?
What can I say on the pentagon itself, the angles of it, and diagonal?
I do a presentation and the topic of it is as above: "relationships between angles and diagonal". I want to show the topic and investigate it.
I don't get a mark on the presentation. It is only for adult course in the center... (community center)
So what do you say?
Thanks for any help...

 

[MarkFL]

A convex polygon with \(n\) sides has \(n\) vertices, and a diagonal can be drawn from each vertex to all but 2 of the other vertices. Iterating over all vertices, and observing the diagonals will be drawn twice, we may hypothesize that the number of diagonals \(D_n\) is given by:

$$D_n=\frac{n(n-3)}{2}$$ where \(3\le n\)

Observing the base case \(D_3=0\) is true, for a triangle has no diagonals, we may use as our inductive step, the addition of another vertex. From this new vertex, diagonals may be drawn to all but \(n-2\) of the other vertices and a new diagonal may now be drawn between the two existing vertices on either side of the new vertex, for a total of \(n-1\) new diagonals. Hence:

$$D_{n+1}=\frac{n(n-3)}{2}+n-1=\frac{n(n-3)+2(n-1)}{2}=\frac{n^2-n-2}{2}=\frac{(n+1)(n-2)}{2}=\frac{(n+1)((n+1)-3)}{2}$$

We have derived \(D_{n+1}\) from \(D_n\), thereby completing the proof by induction.

 

[Wilmer]

There a relationships between angles to diagonals in a polygon?

...and at how many other sites did you post this?