To solve the problem, we need to find the measure of $x^\circ$.

Step-by-Step Explanation:

1. Identify the angle relationships: The lines are intersecting, and $121^\circ$ is vertically opposite to the angle on the left, meaning that angle is also $121^\circ$.

2. Recognize the linear pair relationship: The angle $x^\circ$ and $121^\circ$ lie on a straight line, which means they form a linear pair. Linear pairs of angles always sum up to $180^\circ$.

3. Use the linear pair formula: $x + 121 = 180$

4. Solve for $x$: $x = 180 - 121$ $x = 59^\circ$

Final Answer:

$x = 59^\circ$

Would you like a detailed explanation or visualization of the geometry involved?

Related Questions:

1. How do vertically opposite angles work?
2. What is the rule for angles on a straight line?
3. Can this setup involve supplementary angles?
4. How do parallel lines and a transversal affect angle relationships?
5. What are other examples of angles forming linear pairs?

Tip: Always check for supplementary and complementary angle relationships in geometric problems!