The statements that describe a parallelogram that must be a square are:

1. Parallelogram with congruent diagonals that are perpendicular
2. Parallelogram with four congruent sides and a right angle

Explanation:

• A square is a specific type of parallelogram where:
  • All four sides are congruent.
  • All angles are right angles (90°).
  • The diagonals are congruent and perpendicular to each other.

Here’s why each correct answer holds:

1. Parallelogram with congruent diagonals that are perpendicular: This condition is necessary and sufficient for a parallelogram to be a square. A parallelogram with perpendicular diagonals that are congruent must be a square.

2. Parallelogram with four congruent sides and a right angle: A parallelogram with four congruent sides and one right angle will have all four right angles (since opposite angles in a parallelogram are congruent), and it is therefore a square.

Other options:

• A parallelogram with a pair of congruent consecutive sides or a parallelogram with four congruent sides are not necessarily squares. These conditions may describe rhombuses (quadrilaterals with all sides congruent) or rectangles, but not necessarily squares, unless combined with other properties like right angles.

Do you need more explanation or further details on these properties?

Related Questions:

1. How does a rhombus differ from a square?
2. What properties distinguish a rectangle from a square?
3. What are the properties of a rhombus?
4. How do you prove a parallelogram is a square using the Pythagorean theorem?
5. What happens to the diagonals of a parallelogram when it becomes a square?

Tip:

Always remember that a square is a special type of parallelogram that satisfies three conditions: all sides are congruent, all angles are right angles, and the diagonals are congruent and perpendicular!