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

1. Observe Given Angles:

   • $\angle ABC = 114^\circ$
   • $\angle ABE = 66^\circ$
   • $\angle BED = 42^\circ$

2. Determine $\angle DBF$:

   • Since $\angle ABE$ and $\angle DBF$ are supplementary angles along line $BF$ (they form a linear pair), we can use the fact that the sum of supplementary angles is $180^\circ$.

   $\angle DBF = 180^\circ - \angle ABE$ $\angle DBF = 180^\circ - 66^\circ$ $\angle DBF = 114^\circ$

Therefore, the measure of $\angle DBF$ is $114^\circ$.

Would you like more detailed explanations or have any other questions?

Here are some related questions you might find useful:

1. How can I find other angles in a similar parallelogram?
2. How can I determine if two angles are supplementary or complementary?
3. What is the significance of linear pairs in geometry?
4. How do parallel lines affect angle measurements in polygons?
5. How would the calculation change if $\angle ABE$ was different?

Tip: Remember, in geometry, knowing relationships like supplementary, complementary, and linear pairs can often simplify finding unknown angles.