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To solve question 4.28 from the provided image, we can approach it as follows:

**Given:**
- Triangles \( ABC \) and \( DEF \) are congruent triangles, as stated.
- \( M \) and \( N \) are the midpoints of segments \( BC \) and \( EF \), respectively.

### **Part a: Prove that \( AM = DN \)**

**Step-by-step explanation:**
1. Since \( \triangle ABC \cong \triangle DEF \), corresponding sides and angles of these triangles are equal. Therefore, we have:
   - \( AB = DE \)
   - \( BC = EF \)
   - \( AC = DF \)
2. Because \( M \) and \( N \) are midpoints of \( BC \) and \( EF \) respectively, it follows that:
   \[
   BM = \frac{BC}{2} \quad \text{and} \quad EN = \frac{EF}{2}
   \]
3. From the congruency \( BC = EF \), we know that:
   \[
   BM = EN
   \]
4. Also, since \( AB = DE \) and \( \triangle ABC \cong \triangle DEF \), the line segments \( AM \) and \( DN \) are equal because they correspond to each other in the congruent triangles:
   \[
   AM = DN
   \]
Thus, **\( AM = DN \)** is proven.

### **Part b: Prove that \( BP = EQ \)**

**Step-by-step explanation:**
1. The points \( P \) and \( Q \) are taken on the sides \( AC \) and \( DF \) such that \( BP \) and \( EQ \) are angle bisectors of \( \angle ABC \) and \( \angle DEF \), respectively.
2. Since \( \triangle ABC \cong \triangle DEF \), the angles \( \angle ABC \) and \( \angle DEF \) are congruent:
   \[
   \angle ABC = \angle DEF
   \]
3. The angle bisectors of congruent angles will divide the angles in the same proportion. This implies that the lengths of the bisectors \( BP \) and \( EQ \), which are drawn from the corresponding points \( B \) and \( E \), will be equal due to the equality of the sides and the symmetry of the triangles:
   \[
   BP = EQ
   \]
Thus, **\( BP = EQ \)** is proven.

---

### Do you have any questions or would you like further details on any step? Here are some additional related questions:

1. How can you prove two triangles are congruent using the SSS or SAS criteria?
2. Why do midpoints play a significant role in triangle congruency problems?
3. What properties of congruent triangles help prove that angle bisectors are equal?
4. How does the Angle Bisector Theorem apply to this type of problem?
5. Can you find the length of the bisectors \( BP \) and \( EQ \) if you are given side lengths?

**Tip:** In triangle congruency problems, identifying corresponding sides and angles is crucial, especially when midpoints and bisectors are involved, as they often create symmetry.

Cho tam giác ABC bằng tam giác DEF (Hình 4.28). Gọi M và N lần lượt là trung điểm các đoạn thẳng BC và EF. Chứng minh rằng AM = DN. Trên hai cạnh AC và DF lấy hai điểm P và Q sao cho BP, EQ lần lượt là phân giác của các góc ABC và DEF. Chứng minh rằng BP = EQ.

This problem involves proving equal line segments in congruent triangles using midpoints and angle bisectors. Learn the steps to show that AM = DN and BP = EQ, applying concepts like the Midpoint Theorem and the Angle Bisector Theorem.

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