To solve for the value of $x$ in this problem, we need to use the fact that the interior angles of any polygon add up to a specific total based on the number of sides.

Looking at the two polygons (one being quadrilateral $LMNOPQ$ and the other being quadrilateral $RSTUVW$), we have the following:

1. Sum of interior angles of any polygon:
   The sum of the interior angles of any polygon is given by the formula: $\text{Sum of interior angles} = (n - 2) \times 180^\circ$ where $n$ is the number of sides of the polygon.

2. Quadrilateral angles:
   For both quadrilaterals shown in the problem, the sum of their interior angles will be $360^\circ$.

3. Using the angles given for $\triangle RSTUVW$:
   The angle $\angle T$ is given as $(5x - 20)^\circ$, and $\angle T$ is part of the interior angles. The sum of the angles of the quadrilateral should be 360°, and we can set up an equation to solve for $x$.

Let’s calculate this and find the value of $x$. I will now work through the calculations to identify which value of $x$ matches the correct angles sum.

sympy import symbols, Eq, solve

Define variable

x = symbols('x')

Set up the equation based on the sum of interior angles

The sum of the angles in quadrilateral RSTUVW is 360°

Angles given: 125°, 125°, 140°, and (5x - 20)° for angle T

angle_sum = 125 + 125 + 140 + (5 * x - 20)

Set up equation

equation = Eq(angle_sum, 360)

Solve for x

solution = solve(equation, x) solutionIt seems that the solution for $x$ is $-2$, which is not a valid answer for the angle measure, as angle measures cannot be negative.

Let's revisit the setup. The total sum of angles in the quadrilateral should be 360°. Given this information and the equation setup, it seems there may have been an error in interpreting the angles. Let me correct this and reattempt the calculations to determine the correct value of $x$.Let's correctly reconsider the given angles and the equation setup:

The interior angles of the quadrilateral $RSTUVW$ should sum up to 360°. The angles are:

1. $125^\circ$
2. $125^\circ$
3. $140^\circ$
4. The angle $\angle T$ is given as $(5x - 20)^\circ$

Thus, the equation becomes: $125 + 125 + 140 + (5x - 20) = 360$ Now let's solve this properly for $x$. Recalculate the correct equation setup for the sum of angles angle_sum_corrected = 125 + 125 + 140 + (5 * x - 20)

Set up the correct equation

equation_corrected = Eq(angle_sum_corrected, 360)

Solve for x

solution_corrected = solve(equation_corrected, x) solution_correctedIt appears that the value of $x = -2$ still results from the equation, which doesn't fit the context of the problem. Let's carefully recheck everything.

Considering that the setup for the angles in a quadrilateral is valid, it may be possible that the angles in the figure or some other aspect of the problem setup should be reconsidered, but this angle equation approach produces an impossible negative result.

Given this, I suggest reexamining the problem visually.