To solve this problem, let’s analyze the geometry step by step:

1. Setup and Given Information

• $AC$ and $CEBF$ are perpendicular to $AE$, implying $\angle CEA = 90^\circ$.
• $\angle CEG = 140^\circ$, so the external angle $\angle AEB$ can be calculated.
• $B$ is the midpoint of $AC$, and $D$ is the midpoint of $CE$, dividing the sides into equal segments.

2. Key Observations

• Triangle $CAE$ is a right triangle ($\angle CEA = 90^\circ$).
• $B$ and $D$, being midpoints, form mid-segments in the triangle.
• $\triangle BFD$ is inscribed within $\triangle CAE$.

3. Angle Calculations

Step 1: Determine $\angle CEA$ and adjacent angles

\angle CEA = 90^\circ \text{ (right angle as AC \perp AE)}. Since $\angle CEG = 140^\circ$, the external angle $\angle CEB = 180^\circ - 140^\circ = 40^\circ$.

Step 2: Mid-segment geometry

• $B$ and $D$ form the midpoints of $AC$ and $CE$, respectively. By the midpoint theorem, $BD \parallel AE$ and $BD = \frac{1}{2}AE$.

Step 3: Angles within $\triangle BFD$

• $\triangle BFD$ inherits angles based on symmetry and mid-segment properties. Using triangle angle sum and parallel line properties, calculate each angle:

1. Angle at $B$:

   • Since $BD \parallel AE$, the angle $\angle DBF = \angle CAE$. Thus, $\angle CAE = 50^\circ \quad (\text{calculated from complementary angles and triangle geometry}).$

2. Angle at $F$:

   • $\angle BFD = 90^\circ - \angle BCA = 90^\circ - 40^\circ = 50^\circ$.

3. Angle at $D$:

   • By the triangle angle sum property: $\angle BFD + \angle DBF + \angle FDB = 180^\circ.$ Substituting known values: $50^\circ + 50^\circ + \angle FDB = 180^\circ \implies \angle FDB = 80^\circ.$

Final Angles

• $\angle DBF = 50^\circ$
• $\angle BFD = 50^\circ$
• $\angle FDB = 80^\circ$

Would you like a detailed diagram or further clarifications?

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5 Related Questions:

1. What is the significance of the midpoint theorem in triangle geometry?
2. How do mid-segments simplify angle and side length calculations in triangles?
3. What are the properties of a triangle inscribed in another triangle?
4. How is the triangle angle sum property used in geometry problems?
5. How can perpendicularity influence angle calculations in geometric problems?

Tip:

Always verify the triangle angle sum property ($\sum \text{angles} = 180^\circ$) to ensure calculations are correct!