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Q.
The sum of all but one of the interior angles of a convex polygon equals $2570^{\circ}$. Then
Sequences and Series
Solution:
Sum of all the interior angles of a polygon of side $n =( n -2) \cdot 180^{\circ}$ sum of the interior angles if one angle is $x ^{\circ}$
$( n -2) 180^{\circ}- x ^{\circ}=2570^{\circ} $
$x =( n -2) 180^{\circ}-2570^{\circ} $
$x =180^{\circ} n -360^{\circ}-2570^{\circ}=180^{\circ} n -2930^{\circ} $
$\therefore 0<180^{\circ} n -2930^{\circ}<180^{\circ} \text { (as } x \text { is the interior angle of the polygon) }$
$2930<180 n <3110$
$16.2< n <17.3 $
$\Rightarrow n =17$
$\text { hence } x =15 \cdot 180-2570$
$=10[270-257]=130^{\circ} $