angles between near diagonals in regular polygon

[shahar]

How I prove that angle between two adjacent polygon in regular polygon is equal to the angle of the polygon?
In Hebrew:
זווית בין אלכסונים סמוכים = זווית המצולע המשוכלל

 

[Dr.Peterson]

How I prove that angle between two adjacent polygon in regular polygon is equal to the angle of the polygon?
In Hebrew:
זווית בין אלכסונים סמוכים = זווית המצולע המשוכלל
View attachment 37099

According to Google, that says

Angle between adjacent diagonals = angle of the completed polygon​

(not "adjacent polygons").

What have you tried? Have you marked congruent angles in the figure, or marked some angle x and expressed others in terms of it?

Once you've shown how you started, we can help you continue.

 

[shahar]

I need hints.
I not see anyway how to solve it.

 

[Dr.Peterson]

I need hints.
I not see anyway how to solve it.

I just gave you a hint:

Label the diagram with some angles.​

There are several ways I might start; here is one:

​

Now label some other angles, using whatever theorems you know about angles. If you aren't sure, ask. But don't make us do all the thinking.

 

[shahar]

I see it.
x + x + 120 = 180 (sum of interior angles in isosceles triangle)
x = 60

In the small triangle in the picture:
we can see that it is isosceles triangle)
so his interior angles in isosceles triangle is 180
so it x + x + y = 180

I found x
so y = 120
I prove the situation in the picture that the angle y equal to angle 120).

I think I have void spaces in the proof.
Am I missing something?!

 

[Dr.Peterson]

What you've done is good; possibly one could want a little more explanation in a proof, but you did well.

What's missing is generality. As I understand it, you are to prove this in general, not just for a regular hexagon. So try replacing 120 with a variable, and repeat the process to make a proof for any regular polygon. You may need to change some parts of your work; I haven't tried writing a general proof.