To solve for $x$, let's analyze the angles provided in the diagram:

1. The given angles are:

   • $(5x + 29)^\circ$ at point $A$,
   • $(9x - 7)^\circ$ at point $B$.

2. Based on the figure, the two angles are vertically opposite angles formed by two intersecting lines. For vertically opposite angles, the rule is: $\text{Vertically Opposite Angles are Equal.}$

3. Therefore, we can set the two expressions equal: $5x + 29 = 9x - 7.$

4. Solve for $x$:

   • Subtract $5x$ from both sides: $29 = 4x - 7.$
   • Add $7$ to both sides: $36 = 4x.$
   • Divide by $4$: $x = 9.$

Final Answer:

$x = 9.$

Let me know if you need a detailed explanation of the steps or have further questions!

5 Related Questions:

1. What is the definition of vertically opposite angles, and how are they used in geometry?
2. How would the solution change if the lines were not intersecting?
3. What are other types of angle relationships in intersecting lines?
4. Can vertically opposite angles ever be supplementary? Why or why not?
5. How can we verify our solution geometrically once $x$ is found?

Tip:

When solving problems involving angle relationships, always identify key geometric rules like vertical angles, corresponding angles, or alternate angles. These simplify the solution process!