Diagram illustrating parallel lines and transversal with an angle of 39 degrees.

Problem Analysis:

The given image shows two parallel lines, \( AB \) and \( EF \), with a transversal \( CD \) intersecting them at points \( P \) and \( Q \), respectively. The angle \( \angle APD \) is given as \( 39^\circ \). We are tasked with finding the measure of another angle in the diagram, likely involving the properties of parallel lines and transversals.

Key Concepts:

1. Corresponding Angles: When a transversal intersects two parallel lines, corresponding angles are equal.
2. Alternate Interior Angles: When a transversal intersects two parallel lines, alternate interior angles are equal.
3. Co-Interior Angles (Consecutive Interior Angles): When a transversal intersects two parallel lines, co-interior angles are supplementary (sum to \( 180^\circ \)).
4. Vertically Opposite Angles: When two lines intersect, vertically opposite angles are equal.

Step-by-Step Solution:

#### Step 1: Identify the given information
- \( AB \parallel EF \)
- \( CD \) is a transversal.
- \( \angle APD = 39^\circ \)

#### Step 2: Determine the relationship between \( \angle APD \) and other angles
Since \( AB \parallel EF \) and \( CD \) is a transversal, we can use the properties of angles formed by a transversal intersecting parallel lines.

- \( \angle APD \) and \( \angle PQF \) are corresponding angles because they are on the same side of the transversal and above the parallel lines.
- Corresponding angles are equal when the lines are parallel.

Thus,
\[
\angle PQF = \angle APD = 39^\circ
\]

#### Step 3: Find the measure of \( \angle CQF \)
The angle \( \angle CQF \) is a vertically opposite angle to \( \angle PQF \). Vertically opposite angles are always equal.

Thus,
\[
\angle CQF = \angle PQF = 39^\circ
\]

Final Answer:

The measure of \( \angle CQF \) is \( \boxed{39^\circ} \).