Let's analyze the problem.

We are tasked to find the measure of $\angle x$ in a diagram where $\overline{BC} \parallel \overline{DE}$. The given angles are:

• $\angle DBA = 42^\circ$,
• $\angle DAE = 109^\circ$,
• $\angle EAF = 29^\circ$.

Using the properties of parallel lines and transversals:

1. $\angle DBA$ and $\angle BAF$ are supplementary because they are on a straight line.
   $\angle BAF = 180^\circ - 42^\circ = 138^\circ.$

2. $\angle DAE = 109^\circ$ is part of the triangle $\triangle AEF$. Using the triangle sum property: $\angle x + \angle DAE + \angle EAF = 180^\circ.$ Substituting the known values: $\angle x + 109^\circ + 29^\circ = 180^\circ.$

3. Solve for $\angle x$: $\angle x = 180^\circ - 109^\circ - 29^\circ = 42^\circ.$

Thus, $\angle x = 42^\circ$.

Would you like a detailed breakdown of the reasoning?

Related Questions:

1. Why are $\angle DBA$ and $\angle BAF$ supplementary?
2. How do parallel lines influence angle relationships in this diagram?
3. What is the triangle sum property, and why does it apply here?
4. How would the solution change if any angle measurement were altered?
5. How do we determine corresponding or alternate angles with parallel lines?

Tip: Remember, in diagrams with parallel lines, corresponding and alternate angles are equal!