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To solve **Bài 4.28**, let's break it down step by step:

We are given two congruent triangles \( \triangle ABC \) and \( \triangle DEF \), and we need to prove two things:

### 1. **Part a)**:  
We are asked to prove that:
\[
AM = DN
\]
where \( M \) and \( N \) are the midpoints of segments \( BC \) and \( EF \), respectively.

### **Solution**:
- Since \( \triangle ABC \cong \triangle DEF \), by the property of congruent triangles, the corresponding sides are equal:
  \[
  BC = EF
  \]
- By the midpoint definition:
  \[
  BM = MC \quad \text{and} \quad EN = NF
  \]
- Therefore, \( M \) and \( N \) divide \( BC \) and \( EF \) into equal halves, respectively.
- Since \( BC = EF \), it follows that the segments from the vertices \( A \) and \( D \) to the midpoints \( M \) and \( N \) must also be equal:
  \[
  AM = DN
  \]
Thus, \( AM = DN \).

---

### 2. **Part b)**:
We are asked to prove that:
\[
BP = EQ
\]
where \( P \) and \( Q \) are points on sides \( AC \) and \( DF \), respectively, such that the angles:
\[
\angle ABP = \angle DEQ
\]
are equal.

### **Solution**:
- From the given information, we know that \( P \) divides \( AC \) and \( Q \) divides \( DF \) such that \( BP \) and \( EQ \) subtend equal angles.
- Since \( \triangle ABC \cong \triangle DEF \), corresponding angles are equal. Specifically:
  \[
  \angle A = \angle D, \quad \angle B = \angle E, \quad \text{and} \quad \angle C = \angle F
  \]
- \( P \) and \( Q \) are positioned on the sides of the triangles such that the lines \( BP \) and \( EQ \) form equal angles with the corresponding sides.
- By the congruence of the triangles, and the fact that the angles \( \angle ABP = \angle DEQ \) are equal, it must follow that:
  \[
  BP = EQ
  \]

This proves the second part as well.

---

### Summary:
1. \( AM = DN \)
2. \( BP = EQ \)

Would you like further clarifications or detailed explanations on any step?

Here are five related questions to deepen your understanding:
1. How does the midpoint theorem help in proving equal segments in congruent triangles?
2. What properties of congruent triangles are used in proving part b?
3. Can we apply the same method to non-congruent triangles?
4. How does the angle equality \( \angle ABP = \angle DEQ \) influence the line segments in part b?
5. Could this method be extended to prove other equal line segments in different configurations?

**Tip**: When dealing with congruent triangles, always remember that corresponding sides and angles are equal, which can simplify many proofs.

Solve Bài 4.28, where two congruent triangles ABC and DEF are given.

Solve Bài 4.28, where two congruent triangles ABC and DEF are given. Prove that AM = DN and BP = EQ using the midpoint theorem and congruent triangle properties. This geometry problem explores important concepts of congruence and midpoint relations.

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