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To solve for \( x \) in the given diagram, we need to use the concept of vertically opposite angles, which states that opposite angles formed by two intersecting lines are equal.

From the image:
- Angle \( \angle MSO \) is given as \( (15x + 11)^\circ \).
- Angle \( \angle RTN \) is given as \( (5x + 69)^\circ \).

Since \( \angle MSO \) and \( \angle RTN \) are vertically opposite angles, we can set them equal to each other:

\[
15x + 11 = 5x + 69
\]

Now, solve for \( x \):

1. Subtract \( 5x \) from both sides:
   \[
   15x - 5x + 11 = 69
   \]
   \[
   10x + 11 = 69
   \]

2. Subtract 11 from both sides:
   \[
   10x = 69 - 11
   \]
   \[
   10x = 58
   \]

3. Divide by 10:
   \[
   x = \frac{58}{10} = 5.8
   \]

So, the value of \( x \) is \( 5.8 \).

Would you like any further explanation or details?

Here are 5 questions related to this concept:
1. What are the properties of vertically opposite angles?
2. How do we identify corresponding angles in parallel lines cut by a transversal?
3. Can you solve for angles when given a system of intersecting lines?
4. What is the sum of angles formed by intersecting lines at a point?
5. How does the exterior angle theorem relate to this problem?

**Tip**: Always check if angles are complementary, supplementary, or vertically opposite when working with line intersections.

Solve for x in the following diagram using the vertically opposite angles theorem, where angle MSO is (15x + 11)° and angle RTN is (5x + 69)°.

Learn how to solve for x in a geometry problem involving vertically opposite angles. Given the angles (15x + 11)° and (5x + 69)°, this step-by-step solution covers key algebraic manipulations. Suitable for students in Grades 8-10.

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