Question

use the polygon sum to solve for x.

Explanation:

Step1: Calculate the sum of interior angles

The formula for the sum of interior angles of a pentagon ($n = 5$) is $(n - 2)\times180^{\circ}$.
$$(5 - 2)\times180^{\circ}= 540^{\circ}$$

Step2: Set up the equation

Let the unknown angle be $x$. Then $x+75^{\circ}+105^{\circ}+75^{\circ}+115^{\circ}+45^{\circ}+105^{\circ}=540^{\circ}$ (Note: assume the figure is a pentagon - if it's a hexagon, adjust $n$ accordingly. But from the sum of given angles and options, pentagon is more likely as $75 + 105+75+115 + 45+105+x=540$).
Simplify the left - hand side: $(75 + 105+75+115 + 45+105)+x=520 + x$

Step3: Solve for $x$

Set $520+x = 540$
Subtract 520 from both sides: $x=540 - 520$

Answer:

$x = 20^{\circ}$