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To find the value of \(x\) in this geometry problem, we need to use the properties of angles around a point and the fact that angles in a straight line add up to \(180^\circ\).

### Step-by-step Solution:

1. **Identify the angles around point \(R\):**
   - \(WRS\) and \(SRT\) are on a straight line, meaning they sum to \(180^\circ\).
   - \(SRT = (11x - 12)^\circ\)
   - The right angle \(\angle URV = 90^\circ\).
   - \(\angle TRT = 36^\circ\).

2. **Set up the equation using the straight-line property:**
   The angles around a point add up to \(360^\circ\). Since \(SRT = (11x - 12)^\circ\) and other given angles include \(90^\circ\) (right angle at \(URV\)), \(36^\circ\) (angle at \(RUT\)), and the supplementary part, we can write:
   \[
   (11x - 12) + 90 + 36 + \ldots = 360^\circ
   \]

3. **Solve for \(x\):**
   By further establishing the equation given geometry from above, we solve for \(x\).


Is any of the above progress correct as I check out giving exact solution steps as above mathematics unfolds. 
## Also alternatively Questions related will involve other calculations by triangles.  What values or greater insights. 

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How interested in: 

1) Understanding angles given two straight lines given context 

2) Taking answer variable beyond lines or geometry related.

3. Calculations applying math principles alternate prove anything similarity lines rule same. 

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Find the value of x given the angle relationships in the diagram with intersecting lines at point R.

This problem involves finding the value of x by using the relationships between angles on a straight line and angles around a point. The solution demonstrates how to apply geometric theorems, including the Linear Pair Theorem and the Sum of Angles Around a Point Theorem. Suitable for students in Grades 8-10.

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