The image provided shows a geometry problem asking for the angle $\angle BOC$ in a diagram where multiple lines intersect at point $O$.

The diagram presents three angles labeled as:

• $2x^\circ$ for $\angle AOF$,
• $6x^\circ$ for $\angle EOD$,
• $x^\circ$ for $\angle COD$.

To solve for $\angle BOC$, we can utilize the fact that these angles are adjacent to each other, and the total sum of angles around point $O$ must be $360^\circ$.

Step-by-step solution:

1. We have three adjacent angles:

   • $\angle AOF = 2x^\circ$,
   • $\angle EOD = 6x^\circ$,
   • $\angle COD = x^\circ$.

2. Since these angles form a straight line (line $AC$), their sum must be equal to $180^\circ$ (linear pair of angles): $2x + 6x + x = 180^\circ$ $9x = 180^\circ$ $x = 20^\circ$

3. Now that we have $x = 20^\circ$, we can calculate $\angle BOC$, which is vertically opposite to $\angle EOD$. Therefore: $\angle BOC = \angle EOD = 6x = 6 \times 20^\circ = 120^\circ$

Conclusion:

The measure of $\angle BOC$ is $120^\circ$.

Would you like further clarification or details on this solution? Here are some related questions to explore:

1. How are vertically opposite angles related to each other?
2. What are linear pairs of angles, and how are they used in this problem?
3. What are some other ways to solve for unknown angles in intersecting lines?
4. How can we prove that the sum of angles around a point is $360^\circ$?
5. How do transversal lines affect angle relationships?

Tip: Always remember that vertically opposite angles are equal when two lines intersect. This can simplify many geometry problems!