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Question Number 51942 by mr W last updated on 01/Jan/19

Commented by mr W last updated on 01/Jan/19

$F i n d t h e s u m o f a l l c o r n e r a n g l e s$ $o f a n - c o r n e r s t a r .$ $α_{1} + α_{2} + α_{3} + . . . + α_{n} = ?$ $(n = o d d i n t e g e r, n ⩾ 5)$
Commented by tanmay.chaudhury50@gmail.com last updated on 01/Jan/19

Commented by mr W last updated on 01/Jan/19

$t h a n k y o u s i r!$
Commented by tanmay.chaudhury50@gmail.com last updated on 01/Jan/19

$n s i d e d s t a r f o r m n s i d e d p o l y g o n .$ $T h e s u m o f i n t e r n a l a n g l e s o f n s i d e d p o l y g o n$ $x_{1} + x_{2} + x_{3} + . . . + x_{n} = n π - 2 π$ $w e h a v e t o f i n d t h e v a l u e o f$ $α_{1} + α_{2} + α_{3} + . . . α_{n}$ $n o w c o n s i d e r a n y o n e t r i a n g l e$ $α_{1} + a_{1} + a_{2} = π$ $a_{1} + x_{1} = π a_{1} = π - x_{1}$ $a_{2} + x_{2} = π a_{2} = π - x_{2}$ $s o α_{1} + π - x_{1} + π - x_{2} = π$ $α_{1} = x_{1} + x_{2} - π$ $α_{2} = x_{2} + x_{3} - π$ $. . . . .$ $. . . . .$ $α_{1} + α_{2} + α_{3} + . . . . + α_{n} = 2 (x_{1} + x_{2} + x_{3} + . . + x_{n}) - n π$ $s o$ $α_{1} + α_{2} + . . . + α_{n} = 2 [n π - 2 π] - n π$ $= n π - 4 π$ $p l s c h e c k i h a v e d e r i v e d t h e f o r m u l a . . .$
Commented by mr W last updated on 01/Jan/19

$Σ α = π!$ $c o n s i d e r t h e c a s e t h a t t h e c o r n e r s a r e$ $o n a c i r c l e, t h e n w e h a v e$ $s u m o f a l l c e n t r a l a n g l e s = Σ (2 α) = 2 π$ $\Rightarrow Σ α = π$
	Answered by mr W last updated on 01/Jan/19

	Commented by mr W last updated on 01/Jan/19

	$(α_{1} + β_{1}) + (α_{2} + γ_{2}) + α_{n - 2} = π (t r i a n g l e)$ $Σ α + Σ β + Σ α + Σ γ + Σ α = n π$ $2 Σ α + (Σ α + Σ β + Σ γ) = n π$ $s i n c e Σ α + Σ β + Σ γ = (n - 2) π (n - s i d e p o l y g o n)$ $\Rightarrow 2 Σ α + (n - 2) π = n π$ $\Rightarrow Σ α = π$ $t h i s i s i n d e p e n d e n t f r o m n .$
	Commented by mr W last updated on 02/Jan/19

	$y e s s i r . b u t h e r e i t {d o e s n}^{'} t m a t t e r h o w$ $w e c a l l i t .$
	Commented by mr W last updated on 01/Jan/19

	$s u c h s t a r s c a n o n l y h a v e o d d n u m b e r$ $o f c o r n e r s . t h e y a r e n o ‘ ‘ {n o r m a l}^{″} p o l y g o n s, b u t$ $c o n s i s t o f d i a g o n a l s o f a ‘ ‘ {n o r m a l}^{″} p o l y g o n w i t h$ $o d d n u m b e r o f s i d e s .$
	Commented by mr W last updated on 01/Jan/19

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