Question 1201837
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Answer: <font color=red size=4>100 degrees</font>


Explanation:


A regular polygon has equal angles and equal sides. 


Let's draw out a regular nonagon with 9 sides.
{{{
drawing(400,400,-4.94,4.94,-4.76,5,
line(0,4,-2.57,3.06),line(-2.57,3.06,-3.94,0.69),line(-3.94,0.69,-3.46,-2),line(-3.46,-2,-1.37,-3.76),line(-1.37,-3.76,1.37,-3.76),line(1.37,-3.76,3.46,-2),line(3.46,-2,3.94,0.69),line(3.94,0.69,2.57,3.06),line(2.57,3.06,0,4),line(0,4,0,4)
)
}}}


Highlight 3 consecutive sides. 
{{{
drawing(400,400,-4.94,4.94,-4.76,5,
line(0,4,-2.57,3.06),line(-2.57,3.06,-3.94,0.69),line(-3.94,0.69,-3.46,-2),
red(
line(-3.46,-2,-1.37,-3.76),
line(-1.37,-3.76,1.37,-3.76),
line(1.37,-3.76,3.46,-2)
),
line(3.46,-2,3.94,0.69),line(3.94,0.69,2.57,3.06),line(2.57,3.06,0,4),line(0,4,0,4)
)
}}}
These are adjacent neighboring sides.
I picked the 3 sides at the bottom marked in red.


Pick two sides, of those 3, that aren't neighbors. Extend them until they meet at point T.
{{{
drawing(400,400,-4.94,4.94,-5.26,4.5,
line(0,4,-2.57,3.06),line(-2.57,3.06,-3.94,0.69),line(-3.94,0.69,-3.46,-2),
red(
line(-3.46,-2,0,-4.91),
line(-1.37,-3.76,1.37,-3.76),
line(0,-4.91,3.46,-2),
locate(0,-4.91,T)
),
line(3.46,-2,3.94,0.69),line(3.94,0.69,2.57,3.06),line(2.57,3.06,0,4),line(0,4,0,4)
)

}}}


We'll need to know the measure of each interior angle for a regular nonagon.
S = sum of interior angles of a polygon of n sides
S = 180(n-2)
S = 180(9-2)
S = 180*(9-2) 
S = 1260
Then we divide that into nine equal pieces
1260/9 = 140
The measure of each interior angle of a regular nonagon is 140 degrees.


{{{
drawing(400,400,-4.94,4.94,-5.26,4.5,
line(0,4,-2.57,3.06),line(-2.57,3.06,-3.94,0.69),line(-3.94,0.69,-3.46,-2),
line(-3.46,-2,0,-4.91),
line(-1.37,-3.76,1.37,-3.76),
line(0,-4.91,3.46,-2),
locate(0,-4.91,T),
locate(0.8,-3,140^o),
locate(-1.5,-3,140^o),
locate(0.5,-3.8,x),
locate(-0.8,-3.8,x),
line(3.46,-2,3.94,0.69),line(3.94,0.69,2.57,3.06),line(2.57,3.06,0,4),line(0,4,0,4)
)

}}}
Each adjacent pair of angles marked 140 and x are supplementary.
They add to 180 degrees to form a straight line.
x+140 = 180
x = 180-140
x = 40



{{{
drawing(400,400,-4.94,4.94,-5.26,4.5,
line(0,4,-2.57,3.06),line(-2.57,3.06,-3.94,0.69),line(-3.94,0.69,-3.46,-2),
line(-3.46,-2,0,-4.91),
line(-1.37,-3.76,1.37,-3.76),
line(0,-4.91,3.46,-2),
locate(0,-4.91,T),
locate(0.8,-3,140^o),
locate(-1.5,-3,140^o),
locate(0.3,-3.8,40^o),
locate(-0.7,-3.8,40^o),
line(3.46,-2,3.94,0.69),line(3.94,0.69,2.57,3.06),line(2.57,3.06,0,4),line(0,4,0,4)
)

}}}


Focus on the triangle at the very bottom. The three interior angles add to 180.
x+x+T = 180
40+40+T = 180
80+T = 180
T = 180-80
T = <font color=red>100</font> is the final answer.
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