Let's analyze the problem and determine which conditions are sufficient to prove the given types of quadrilaterals:

Definitions for reference:

1. Parallelogram: A quadrilateral where both pairs of opposite sides are parallel.
2. Rectangle: A parallelogram with all angles congruent (each $90^\circ$).
3. Rhombus: A parallelogram where all sides are congruent.
4. Square: A rhombus with all angles congruent (a rhombus and a rectangle).

Analysis of each condition:

1. Diagonals are congruent:

   • Sufficient for: Rectangle, Square
     (For parallelograms, congruent diagonals indicate a rectangle. If all sides are also congruent, it is a square.)

2. Both pairs of opposite angles are congruent:

   • Sufficient for: Parallelogram
     (This is a defining property of a parallelogram but not specific enough for a rectangle, rhombus, or square.)

3. All angles are congruent:

   • Sufficient for: Rectangle, Square
     (All angles congruent means each angle is $90^\circ$, sufficient for a rectangle. If all sides are also congruent, it is a square.)

4. All sides are congruent:

   • Sufficient for: Rhombus, Square
     (All sides congruent is a defining property of a rhombus. If all angles are also $90^\circ$, it is a square.)

5. Diagonals are perpendicular and congruent:

   • Sufficient for: Square
     (Only a square has both perpendicular and congruent diagonals.)

Summary:

• (a) Parallelogram: Condition 2.
• (b) Rectangle: Conditions 1, 3.
• (c) Rhombus: Condition 4.
• (d) Square: Conditions 1, 3, 4, 5.

Would you like further clarifications or a detailed explanation?

Follow-up Questions:

1. How do congruent diagonals help identify specific quadrilaterals?
2. Why are perpendicular diagonals significant for a rhombus or square?
3. How does proving all angles are $90^\circ$ narrow down the shape to rectangles or squares?
4. What are the relationships between sides and diagonals in parallelograms?
5. Can a quadrilateral be a parallelogram without meeting any of the listed conditions?

Tip:

When proving quadrilaterals, always start by testing basic properties like congruency of sides or angles before moving to diagonals.