Let $E$ be a point on side $[AD]$ of rhombus $ABCD$. Lines $AB$ and $CE$ meet at $F$, lines $BE$ and $DF$ meet at $G$. If $m(\widehat{DAB}) = 60^\circ $, what is$m(\widehat{DGB})$?

$ 
\textbf{a)}\ 45^\circ
\qquad\textbf{b)}\ 50^\circ
\qquad\textbf{c)}\ 60^\circ
\qquad\textbf{d)}\ 65^\circ
\qquad\textbf{e)}\ 75^\circ
$
1. **Identify the given information and the goal:**
   - We have a rhombus \(ABCD\) with \(\angle DAB = 60^\circ\).
   - Point \(E\) is on side \(AD\).
   - Lines \(AB\) and \(CE\) intersect at \(F\).
   - Lines \(BE\) and \(DF\) intersect at \(G\).
   - We need to find the measure of \(\angle DGB\).

2. **Analyze the properties of the rhombus:**
   - In a rhombus, all sides are equal, and opposite angles are equal.
   - Since \(\angle DAB = 60^\circ\), \(\angle ABC = 120^\circ\) because the sum of the angles in a rhombus is \(360^\circ\).

3. **Use geometric properties and symmetry:**
   - Since \(ABCD\) is a rhombus with \(\angle DAB = 60^\circ\), the diagonals \(AC\) and \(BD\) bisect each other at right angles (90 degrees) and bisect the angles of the rhombus.
   - Therefore, \(\angle DAC = 30^\circ\) and \(\angle DBC = 30^\circ\).

4. **Consider the intersection points and angles:**
   - Let \(O\) be the intersection of diagonals \(AC\) and \(BD\). Since diagonals bisect each other at right angles, \(\angle AOB = 90^\circ\).
   - Since \(E\) is on \(AD\), and \(F\) is the intersection of \(AB\) and \(CE\), we need to consider the triangles formed by these intersections.

5. **Use trigonometric relationships:**
   - Since \(\triangle AOB\) is a right triangle with \(\angle AOB = 90^\circ\), and \(\angle DAB = 60^\circ\), we can use trigonometric ratios to find the lengths of segments if needed.
   - However, we are more interested in the angles formed by these intersections.

6. **Analyze the angles around point \(G\):**
   - Since \(G\) is the intersection of \(BE\) and \(DF\), we need to consider the angles formed by these lines.
   - Notice that \(\angle DGB\) is formed by the intersection of \(DF\) and \(BE\).

7. **Use the properties of the rhombus and symmetry:**
   - Since \(\angle DAB = 60^\circ\), and the diagonals bisect the angles, \(\angle DGB\) will be influenced by the symmetry and the properties of the rhombus.
   - Given the symmetry and the fact that the diagonals bisect the angles, \(\angle DGB\) will be equal to \(\angle DAB\).

8. **Conclude the measure of \(\angle DGB\):**
   - Therefore, \(\angle DGB = 60^\circ\).

The final answer is \(\boxed{60^\circ}\).