Question

what is the sum of the meausures of the interior angles of a polygon which has 20 diagonals​

Answer 1

Ответ: $1080^o$.

Объяснение:

If the number of diagonals is 20, then the number of corners is:

$N=\dfrac{n(n-3)}{2} \\\\N \: - \: \text {number of diagonals};\\n \: - \: \text {number of angles}.$

$N=20\\\\20=\dfrac{n(n-3)}{2} \\\\n(n-3)=2 \cdot 20\\\\n^{2} -3n=40\\\\n^{2} -3n-40=0\\\\D=(-3)^{2} -4 \cdot (-40)=9+160=169=13^{2} \\\\n =\dfrac{3+13}{2} =8$

Formula to find the sum of interior angles of a n-sided polygon:

$S_{n} =(n-2) \cdot 180^o\\\\S_{n} =(8-2) \cdot 180^o=6 \cdot 180^o=1080^o$