To solve for the measure of $\angle BEF$, let's analyze the diagram and apply angle relationships:

Step-by-step solution:

1. Identify the known angles:

   • $\angle ABC = 96^\circ$ (as given).
   • $\angle ABE = (x + y)^\circ$ (as labeled).
   • $\angle EBF = y^\circ$ (as labeled).
   • $\angle BEF = 2x^\circ$ (as labeled).

2. Using the straight-line property: Since $\angle ABC$ and $\angle ABE$ are adjacent angles that form a straight line, their sum must be $180^\circ$: $\angle ABC + \angle ABE = 180^\circ$ Substituting the known values: $96^\circ + (x + y)^\circ = 180^\circ$ Simplifying: x + y = 84^\circ \tag{1}

3. Using the vertically opposite angles: The angles $\angle EBF$ and $\angle ABC$ are vertically opposite angles, so they must be equal: $\angle EBF = \angle ABC = 96^\circ$ Therefore, $y = 96^\circ$.

4. Substitute $y = 96^\circ$ into Equation (1): $x + 96^\circ = 84^\circ$ Solving for $x$: $x = 84^\circ - 96^\circ$ $x = -12^\circ$

   Since having a negative angle is not physically meaningful in this context, there may have been a misunderstanding in the setup or the diagram itself. Based on this analysis, it seems there might be additional information or clarification needed for a valid solution.

Please clarify the values or provide additional details if available.

Would you like to ask any specific follow-up questions or request further clarification? Here's a set of related questions:

1. How do vertically opposite angles help in determining unknown angles in geometry?
2. Could there be any alternative angle relationships in this problem that we've missed?
3. What are some common reasons negative angle results might appear in solutions like these?
4. Can the straight-line property be applied to multiple angles simultaneously?
5. How do supplementary angles differ from complementary angles in problems like these?

Tip: Always verify angle constraints based on geometric properties before finalizing your answer, especially when dealing with linear pairs or vertical angles.