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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.

Learn how to find m∠DGA when m∠BGC is given as 69° using the property of vertically opposite angles. This problem covers basic geometric concepts and angle properties, suitable for students in Grades 6-8.

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