The image shows two intersecting lines that form four angles. Two of the angles are marked, one with $105^\circ$ and the other with $x$. Since these are vertically opposite angles, they are equal.

Step-by-step Solution:

1. Identify the Angles:

   • The angles marked $105^\circ$ and $x$ are vertically opposite.
   • Vertically opposite angles are equal. Therefore, $x = 105^\circ$.

2. Find the Other Pair of Vertically Opposite Angles:

   • The sum of angles on a straight line is $180^\circ$.
   • Thus, the angle adjacent to $105^\circ$ is: $180^\circ - 105^\circ = 75^\circ.$
   • The angle vertically opposite to $75^\circ$ is also $75^\circ$.

Conclusion:

• $x = 105^\circ$
• The other two angles are $75^\circ$ each.

Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. What are complementary and supplementary angles?
2. How can you determine if two angles are vertically opposite?
3. What is the sum of all interior angles in a quadrilateral?
4. Can two adjacent angles be equal? If yes, what are they called?
5. What are the properties of intersecting lines in geometry?

Tip: Vertically opposite angles are always equal when two lines intersect!