To find the measure of angle $\angle DGB$, we need to use the information given in the diagram and apply the properties of angles formed by intersecting lines.

Step-by-Step Solution:

1. Given Information:

   • $\angle AGF = 63^\circ$
   • $\angle CGE = 86^\circ$

2. Understanding the Diagram:

   • Angles $\angle AGF$ and $\angle DGB$ are vertically opposite angles.
   • Vertically opposite angles are always equal.

3. Finding $\angle DGB$: Since $\angle AGF = 63^\circ$, and $\angle DGB$ is its vertically opposite angle, we have:

$\angle DGB = 63^\circ$

4. Reason for the Step: The reason for the equality is the "Vertical Angles Theorem," which states that vertically opposite angles are congruent.

Conclusion:

$\text{m} \angle DGB = 63^\circ \quad (\text{Vertical Angles Theorem})$

Would you like more details or have any questions about this solution? Here are some related questions for further practice:

1. What is the sum of angles around point G?
2. If $\angle CGF = x$, how can we find $x$?
3. What is the relationship between $\angle CGE$ and $\angle BGD$?
4. Can we determine $\angle AGC$ using the given information?
5. How do we know that angles $\angle AGF$ and $\angle DGB$ are vertical angles?

Tip: When two lines intersect, always check for vertical angles, as they are congruent!