Question

The sum of all but one of the interior angles of a convex polygon equals $2570^{\circ }$. Then

• A. number of sides in the polygon is 16.
• B. number of sides in the polygon is 17 .
• C. remaining angle of the polygon is $130^{\circ }$
• D. remaining angle of the polygon is $144^{\circ }$

Answer 1

Answer: (B), (C)

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 $x^{\circ }$
$(n-2)180^{\circ }-x^{\circ }=2570^{\circ }$
$x=(n-2)180^{\circ }-2570^{\circ }$
$x=180^{\circ }n-360^{\circ }-2570^{\circ }=180^{\circ }n-2930^{\circ }$
$\therefore 0<180^{\circ }n-2930^{\circ }<180^{\circ }\text{ (as }x\text{ is the interior angle of the polygon) }$
$2930<180n<3110$
$16.2<n<17.3$
$\Rightarrow n=17$
$\text{ hence }x=15\cdot 180-2570$
$=10[270-257]=130^{\circ }$