This is a geometric proof problem where you are tasked to prove the congruence of angles ∠WQP and ∠LQX using the given information, the diagram, and the structure of the proof provided. Let’s analyze the steps:

Given:

1. PQ bisects YZ.
2. Q is the midpoint of WX.
3. ∠Y ≅ ∠Z, WZ ≅ XY.

Goal:

Prove ∠WQP ≅ ∠XQP.

Statements and Reasons:

1. PQ bisects YZ
   Reason: Given.

2. ΔWQP ≅ ΔXQP
   Reason: SAS Postulate - Side-Angle-Side congruence. PQ is shared (reflexive property), WX is bisected, and the base angles are congruent.

3. WP ≅ PX
   Reason: SSS Postulate - ΔWQP ≅ ΔXQP implies corresponding sides are congruent.

4. ∠Z ≅ ∠Y
   Reason: Given.

5. WZ ≅ XY
   Reason: Given.

6. ∠Z ≅ ∠Y
   Reason: Given (restating for clarity).

7. WQ ≅ QX
   Reason: Reflexive property (segment bisector).

8. Q is the midpoint of WX
   Reason: Given.

9. ∠WQP ≅ ∠XQP
   Reason: CPCTC - Corresponding Parts of Congruent Triangles are Congruent.

Would you like a breakdown of any specific statement, further clarification on the proof structure, or help with a related question? Here are five related questions for further practice:

1. How can you identify when to use the SAS Postulate versus the SSS Postulate?
2. Why is it necessary to establish congruent triangles before using CPCTC?
3. How does defining midpoints simplify proving segment congruence?
4. What additional properties of bisectors can be useful in other geometric proofs?
5. How does this proof align with the properties of isosceles triangles?

Tip: When solving geometric proofs, always identify shared segments or angles first—they often serve as the foundation for congruence relationships.