document.write( "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.
\n" ); document.write( "Find the value of Angle APB.
\n" ); document.write( "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. \n" ); document.write( "

Algebra.Com's Answer #828043 by ikleyn(52794) $\"\"$ $\"About$

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\n" ); document.write( "(a) In a regular octagon, use its circumference to prove that the six angles
\n" ); document.write( "between the adjacent diagonals at a vertex are all equal. Find the value of Angle APB.
\n" ); document.write( "(b) More generally, prove that the angles between adjacent diagonals
\n" ); document.write( "at any vertex of an n-sided regular polygon are all equal and have the value of 180*/n.
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document.write( "(a)  Consider a regular octagon (8-gon) and draw the circumscribed circle about it.\r\n" );
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document.write( "     The vertices of the regular octagon divide the circle in 8 equal/(congruent) arcs,\r\n" );
document.write( "     each such arc is  360°/8 = 45°.\r\n" );
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document.write( "     Each angle between two adjacent diagonals of the regular octagon is an inscribed angle \r\n" );
document.write( "     in the circle leaning on one of such arcs.\r\n" );
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document.write( "     Therefore, all angles between adjacent diagonals of the regular octagon\r\n" );
document.write( "     have the same angular measure, which is one half of the angular measure\r\n" );
document.write( "     of the corresponding arc, i.e.  45°/2 = 22.5°.\r\n" );
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document.write( "     ANSWER.  The angles between adjacent diagonals of a regular octagon are all congruent \r\n" );
document.write( "               and have angular measure of 22.5°.\r\n" );
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document.write( "(b)  In the previous solution, replace regular octagon (8-gon) by regular n-gon for any \r\n" );
document.write( "     integer  n >=3  to get the proof of the general statement by the same way.\r\n" );
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