To find the measure of angle $\angle BEC$ in the diagram, let's analyze the given information step-by-step.

Step-by-Step Solution

1. Identify given angles and relationships:

   • We have $\angle BCD = 93^\circ$.
   • $\angle ABE = 2x^\circ$.
   • $\angle BEF = x^\circ$.
   • $\angle ABE$ and $\angle BEF$ are supplementary, as they form a straight line on $AB$.

2. Calculate $x$ using the straight line relationship: Since $\angle ABE$ and $\angle BEF$ are supplementary, they add up to $180^\circ$: $2x + x = 180^\circ$ $3x = 180^\circ$ $x = 60^\circ$

3. Substitute $x = 60^\circ$ into the angles:

   • $\angle ABE = 2x = 2 \times 60^\circ = 120^\circ$.
   • $\angle BEF = x = 60^\circ$.

4. Determine $\angle BEC$: $\angle BEC$ is the angle adjacent to $\angle BCD$, and together they form a straight line. Therefore, they add up to $180^\circ$: $\angle BEC + 93^\circ = 180^\circ$ $\angle BEC = 180^\circ - 93^\circ = 87^\circ$

Conclusion

The measure of $\angle BEC$ is $87^\circ$, but none of the answer choices provided (29°, 58°, 122°, 140°) include this value.

It appears there may have been a misalignment in the answer options with the actual solution, as the correct answer based on the given information and calculations is 87 degrees.

Would you like more details on any step or have any questions?

Here are five related questions that can expand on this information:

1. How do we determine when angles are supplementary?
2. Why do $\angle ABE$ and $\angle BEF$ add up to $180^\circ$?
3. What methods can we use to find unknown angles in geometric figures?
4. How does the concept of supplementary angles apply in different types of polygons?
5. How can we verify if the diagram and calculations align correctly in a problem like this?

Tip: Always check if the answer options match your calculated solution, as discrepancies may suggest either an error in options or a need to re-evaluate the approach.