We are given that $m \angle BGC = 69^\circ$ and we need to find $m \angle DGA$.

Step-by-Step Solution:

1. Identify vertical angles:

   • Angle $\angle BGC$ is vertically opposite to $\angle DGA$, because they are formed by the intersection of two straight lines (the horizontal line $AB$ and the diagonal line $DF$).
   • Vertically opposite angles are always equal. Thus, $m \angle DGA = m \angle BGC$.

2. Apply the given information:

   • Since $m \angle BGC = 69^\circ$, we conclude that $m \angle DGA = 69^\circ$.

Final Answer:

$m \angle DGA = 69^\circ$

Would you like more details or have any questions?

Here are five related questions to consider:

1. How do you identify corresponding angles in a similar diagram?
2. What are the properties of vertically opposite angles?
3. How can you calculate the sum of angles around a point?
4. How would this problem change if another angle were given instead of $\angle BGC$?
5. Can vertically opposite angles be unequal under any conditions?

Tip: Always remember, vertically opposite angles formed by intersecting lines are congruent, which is a helpful property in many geometric problems.