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

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### 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).

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### 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.)

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### 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?  

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### 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?

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

Determine which condition/s is/are sufficient to prove that a quadrilateral is (a) a parallelogram, (b) a rectangle, (c) a rhombus, or (d) a square.

Explore sufficient conditions to prove a quadrilateral is a parallelogram, rectangle, rhombus, or square. Learn how properties such as congruent diagonals, opposite angles, and perpendicular diagonals define specific quadrilateral types. Suitable for Grades 8-10 students.

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