Question 1195526: In a regular octagon, use its circumference to prove that the six angles between the adjacent diagonals at a vertex are all equal.
Find the value of Angle APB.
More generally, prove that the angles between adjacent diagonals at any vertex of an n-sided regular polygon are all equal and have the value of 180*/n.
Answer by ikleyn(52787)   (Show Source):
You can put this solution on YOUR website! .
(a) In a regular octagon, use its circumference to prove that the six angles
between the adjacent diagonals at a vertex are all equal. Find the value of Angle APB.
(b) More generally, prove that the angles between adjacent diagonals
at any vertex of an n-sided regular polygon are all equal and have the value of 180*/n.
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(a) Consider a regular octagon (8-gon) and draw the circumscribed circle about it.
The vertices of the regular octagon divide the circle in 8 equal/(congruent) arcs,
each such arc is 360°/8 = 45°.
Each angle between two adjacent diagonals of the regular octagon is an inscribed angle
in the circle leaning on one of such arcs.
Therefore, all angles between adjacent diagonals of the regular octagon
have the same angular measure, which is one half of the angular measure
of the corresponding arc, i.e. 45°/2 = 22.5°.
ANSWER. The angles between adjacent diagonals of a regular octagon are all congruent
and have angular measure of 22.5°.
(b) In the previous solution, replace regular octagon (8-gon) by regular n-gon for any
integer n >=3 to get the proof of the general statement by the same way.
Solved, proved and carefully/thoroughly explained.
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