Given: D is the midpoint of AC, AE congruent to CF, AB perpendicular to DE and BC perpendicular to DF. Prove: angle A is congruent to angle C.

We are given that:
- \( D \) is the midpoint of \( AC \),
- \( AE \cong CF \),
- \( AB \perp DE \),
- \( BC \perp DF \).

We need to prove that \( \angle A \cong \angle C \).

### Step-by-step proof:

1. **Identify Midpoint Properties:**
   Since \( D \) is the midpoint of \( AC \), we have:
   \[
   AD = DC
   \]
   This means that \( \triangle ADE \) and \( \triangle CDF \) share a symmetry across line \( DE \) and \( DF \).

2. **Perpendicularity Implies Right Angles:**
   - \( AB \perp DE \) implies that \( \angle ADE = 90^\circ \).
   - \( BC \perp DF \) implies that \( \angle CDF = 90^\circ \).

3. **Congruence of Segments \( AE \) and \( CF \):**
   We are given that \( AE \cong CF \). This helps in proving that the triangles involving these sides might be congruent.

4. **Prove Triangle Congruence:**
   Consider triangles \( \triangle ADE \) and \( \triangle CDF \):
   - \( AD = DC \) (since \( D \) is the midpoint of \( AC \)),
   - \( \angle ADE = \angle CDF = 90^\circ \) (from the perpendicularity),
   - \( AE = CF \) (given).

   Thus, by the **SAS (Side-Angle-Side)** postulate, we can conclude that:
   \[
   \triangle ADE \cong \triangle CDF
   \]

5. **Conclusion:**
   Since \( \triangle ADE \cong \triangle CDF \), corresponding angles are congruent, meaning:
   \[
   \angle A \cong \angle C
   \]
   This completes the proof.

### Would you like more details or have any questions?

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Here are 5 related questions:
1. What is the significance of midpoint properties in triangle congruence?
2. How does the SAS postulate work in triangle congruence?
3. What is the impact of perpendicular lines in geometric proofs?
4. Can we apply a different method (such as the HL Theorem) to prove the same statement?
5. What are some common mistakes when proving congruent angles?

**Tip:** Always mark congruent triangles clearly in geometric diagrams to visualize angle and side relationships more easily.

In this geometric proof, we use the SAS postulate and properties of midpoints and perpendicular lines to prove that angle A is congruent to angle C. The problem involves key concepts like triangle congruence and perpendicularity, making it suitable for students in Grades 9-11.

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