To solve for each angle, let's analyze the geometry in the figure step by step:

1. Identify known angles:

   • $\angle ABC = 63^\circ$
   • $\angle BCD = 84^\circ$
   • $\angle GEF = 90^\circ$ (right angle)
   • $\angle AFG = 39^\circ$

2. Use angle sum properties:

   • For any triangle, the sum of interior angles is $180^\circ$.
   • For straight lines, the sum of angles on a straight line is $180^\circ$.

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Step 1: Solve $\triangle BCD$

In $\triangle BCD$:

• $\angle BCD = 84^\circ$,
• $\angle ABC = 63^\circ$,
• Remaining $\angle CBD = 180^\circ - 84^\circ - 63^\circ = 33^\circ$.

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Step 2: Solve $\triangle AFG$

In $\triangle AFG$:

• $\angle GEF = 90^\circ$,
• $\angle AFG = 39^\circ$,
• Remaining $\angle FAG = 180^\circ - 90^\circ - 39^\circ = 51^\circ$.

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Step 3: Solve $\triangle FEG$

In $\triangle FEG$, since $\angle GEF = 90^\circ$:

• $\angle EFG = 180^\circ - 90^\circ - 39^\circ = 51^\circ$.

Thus, all angles in $\triangle FEG$ are resolved.

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Step 4: Summary of Angles

Here are the resolved angles:

• $\angle ABC = 63^\circ$,
• $\angle BCD = 84^\circ$,
• $\angle CBD = 33^\circ$,
• $\angle GEF = 90^\circ$,
• $\angle AFG = 39^\circ$,
• $\angle FAG = 51^\circ$,
• $\angle EFG = 51^\circ$.

Let me know if you'd like further clarifications or diagrams to explain this!

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Related Questions

1. What is the relationship between $\triangle AFG$ and $\triangle FEG$?
2. How does the right angle at $G$ influence calculations in this figure?
3. Can we determine the type of triangles (e.g., scalene, isosceles) in the figure?
4. How can trigonometry be used to find the lengths of the sides?
5. What are the supplementary angles in the diagram?

Tip:

Always start solving angles in geometric problems by focusing on triangles and straight-line rules!