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The image you uploaded contains multiple geometric problems. Let’s solve each one step by step. The problems involve proving geometric relationships and making constructions.

### Problem 21:
**Given:** \( AB \parallel CD \)
- **Prove:** \( \angle ABD = \angle CDB \)

**Solution:**
Since \( AB \parallel CD \), and \( BD \) is a transversal, we can use the alternate interior angles theorem.
- \( \angle ABD \) and \( \angle CDB \) are alternate interior angles.
- Therefore, \( \angle ABD = \angle CDB \).

### Problem 22:
**Given:** \( \angle ABD = \angle ACD \)
- **Prove:** \( AB \parallel CD \)

**Solution:**
The converse of the alternate interior angles theorem states that if two lines are cut by a transversal and the alternate interior angles are equal, then the lines are parallel.
- Since \( \angle ABD = \angle ACD \), by the converse of the alternate interior angles theorem, \( AB \parallel CD \).

### Problem 23:
**Given:** \( AD = BD = CD \)
- **Prove:** \( \triangle ABD \cong \triangle CBD \)

**Solution:**
Use the Side-Side-Side (SSS) congruence criterion:
- \( AD = CD \) (given)
- \( BD = BD \) (common side)
- \( AB = AB \) (common side)
- Therefore, \( \triangle ABD \cong \triangle CBD \) by the SSS criterion.

### Problem 24:
**Given:** \( RS = RT \), \( \angle S = 90^\circ \), and \( TU \) is a line.
- **Prove:** \( RU \) is a height.

**Solution:**
In a right-angled triangle, the altitude to the hypotenuse is the height. Since \( \angle S = 90^\circ \) and \( RS = RT \), \( \triangle RST \) is isosceles.
- \( RU \) is perpendicular to \( ST \) (since \( \angle S = 90^\circ \)), so \( RU \) is a height.

### Problem 25:
**Given:** \( \angle R = \angle S \), \( \angle T = \angle U \), and \( RS = RT \)
- **Prove:** \( \triangle RST \cong \triangle RUT \)

**Solution:**
Use the Angle-Side-Angle (ASA) congruence criterion:
- \( \angle R = \angle R \) (common angle)
- \( RS = RT \) (given)
- \( \angle S = \angle U \) (given)
- Therefore, \( \triangle RST \cong \triangle RUT \) by ASA.

### Problem 26:
**Given:** \( AB = CD \), \( \angle A = \angle C \)
- **Prove:** \( \triangle ABD \cong \triangle CDB \)

**Solution:**
Use the Side-Angle-Side (SAS) congruence criterion:
- \( AB = CD \) (given)
- \( \angle A = \angle C \) (given)
- \( BD = BD \) (common side)
- Therefore, \( \triangle ABD \cong \triangle CDB \) by SAS.

### Problem 27:
**Given:** \( QR = SR \), \( \angle PQR = \angle PSR \)
- **Prove:** \( PR \) is perpendicular to \( QR \)

**Solution:**
Since \( QR = SR \) and the angles \( \angle PQR = \angle PSR \) are equal, \( PR \) is the perpendicular bisector of \( QS \). By the definition of a perpendicular bisector, \( PR \) is perpendicular to \( QR \).

### Problem 28:
**Given:** \( PQ = QR = RS = ST \)
- **Prove:** \( PS = QT \)

**Solution:**
In this quadrilateral, since the sides are equal and the opposite angles are equal, the diagonals \( PS \) and \( QT \) are equal due to the properties of a parallelogram (or an isosceles trapezoid if it's cyclic).

**Would you like detailed explanations on any specific problem or further assistance with any related concepts?**

### Questions:
1. How does the alternate interior angles theorem help in proving parallel lines?
2. What are the key criteria for triangle congruence, and how are they applied?
3. Can you explain how the perpendicular bisector theorem works?
4. What are the properties of isosceles triangles?
5. How do you identify and prove a height in a triangle?
6. What distinguishes an isosceles trapezoid from a regular trapezoid?
7. How do you determine if a quadrilateral is a parallelogram?
8. Can the SSS criterion be applied if only two sides and an angle are known?

**Tip:** When solving geometric proofs, it's helpful to sketch the diagram and label all given information before proceeding with the proof. This visual aid can clarify relationships and make it easier to apply theorems.

Learn how to solve geometric problems involving parallel lines and triangle congruence with detailed step-by-step solutions. This problem set covers key theorems such as the Alternate Interior Angles Theorem and criteria like SSS, ASA, and SAS for triangle congruence. Suitable for high school students studying geometry.

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