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To prove that triangles \( \triangle DBC \) and \( \triangle APQ \) are congruent, let's analyze the given parallelogram \( ABCD \) and the lines \( P \) and \( Q \) that are drawn inside it. 

### Analyzing the Parallelogram and Lines:
1. **Parallelogram Properties**: Since \( ABCD \) is a parallelogram, opposite sides are equal and parallel:
   - \( AB \parallel CD \) and \( AB = CD \)
   - \( AD \parallel BC \) and \( AD = BC \)

2. **Triangles to Prove Congruence**:
   - We need to prove \( \triangle DBC \cong \triangle APQ \).

3. **Lines \( P \) and \( Q \)**:
   - \( P \) and \( Q \) are points on \( AD \) and \( BC \), respectively, suggesting that \( PQ \parallel AB \) and \( PQ \parallel CD \) because both \( PQ \) and \( AB \) are parallel to the base lines of the parallelogram.

### Proving Congruence:
To prove \( \triangle DBC \cong \triangle APQ \), we can use the following approach based on the properties of parallelograms and parallel lines:

1. **Parallel Lines**:
   - \( PQ \parallel AB \) and \( PQ \parallel CD \), implies \( \angle APQ = \angle DBC \) because they are corresponding angles.
   - Similarly, \( \angle PAQ = \angle BDC \) because they are corresponding angles formed by the transversal \( PQ \).

2. **Sides Equal Due to Parallelogram**:
   - \( AP = BD \) because \( AD = BC \) and \( AP \) and \( BD \) are parts of these equal lengths.
   - \( PQ = BC \) because they are between the same parallel lines \( AB \) and \( CD \).

3. **Angle-Side-Angle (ASA) Congruence**:
   - In \( \triangle APQ \) and \( \triangle DBC \):
     - \( \angle APQ = \angle DBC \) (corresponding angles due to parallel lines),
     - \( \angle PAQ = \angle BDC \) (corresponding angles),
     - \( PQ = BC \) (parallel lines property).
   
   With two angles and the included side equal, triangles \( \triangle APQ \) and \( \triangle DBC \) are congruent by the Angle-Side-Angle (ASA) postulate.

### Conclusion:
Thus, \( \triangle DBC \) and \( \triangle APQ \) are congruent based on ASA congruence criteria.

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Would you like more details or have any questions?

### Related Questions:
1. What other properties of parallelograms could help in proving triangle congruence?
2. How does the property of corresponding angles apply to non-parallel lines?
3. What are other ways to prove triangles congruent besides ASA?
4. Can you explain the properties of a parallelogram that make it a special type of quadrilateral?
5. What role do transversals play in the congruence of triangles?

### Tip:
In any geometric figure involving a parallelogram, always look for properties like equal opposite sides, parallel lines, and angle relationships to simplify proofs.

Learn how to prove triangles DBC and APQ congruent within parallelogram ABCD using the Angle-Side-Angle (ASA) congruence postulate. Detailed geometric analysis and step-by-step proof included.

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