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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!

In the diagram, m∠AGF = 63° and m∠CGE = 86°. Find m∠DGB.

Learn how to find the measure of angle DGB when given angles AGF and CGE. The solution applies the Vertical Angles Theorem, a fundamental geometry concept. Suitable for students in Grades 7-9.

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