In the image, we have two intersecting lines, with angles at point $O$. The labeled angles are:

• One angle as $(3x - 35)^\circ$
• The opposite angle as $130^\circ$

These two angles form a pair of vertically opposite angles. Vertically opposite angles are always equal, meaning:

$3x - 35 = 130$

Now, solving for $x$:

$3x = 130 + 35$ $3x = 165$ $x = \frac{165}{3}$ $x = 55$

Thus, the value of $x$ is $55$.

Would you like further details or clarification on this solution?

Here are five related questions to extend your understanding:

1. What are vertically opposite angles, and why are they always equal?
2. How would you solve for $x$ if the equation involved supplementary angles instead of vertically opposite ones?
3. Can you think of other pairs of angles that might be congruent or supplementary in intersecting lines?
4. What happens when the lines are parallel, and how does that affect angle relationships?
5. How would you solve for unknowns in a triangle or other polygons using angle relationships?

Tip: Remember that vertically opposite angles always equal each other, regardless of the angles formed, as long as the lines intersect.