Which of the following conditions or set of conditions is sufficient for a parallelogram to be square?
A. Diagonals are perpendicular and diagonals are congruent.
B. Diagonals are congruent.
C. All sides are congruent.
D. The length of a diagonal is equal to the length of side.
A. Diagonals are perpendicular and diagonals are congruent is sufficient for a parallelogram to be a square.
Explanation:
- A parallelogram is a quadrilateral with two pairs of parallel sides.
- A square is a special type of parallelogram where all sides are congruent, and all angles are right angles (90 degrees).
- For a parallelogram to be a square, its diagonals must be congruent and perpendicular.
- Therefore, option A is the correct answer.
Option B (diagonals are congruent) is not sufficient because a rhombus also has diagonals that are congruent but not all angles are right angles.
Option C (all sides are congruent) is not sufficient because a rhombus also has all sides congruent but not all angles are right angles.
Option D (the length of a diagonal is equal to the length of side) is not sufficient because a parallelogram with this condition can be a rectangle (where opposite sides are congruent and all angles are right angles) but not necessarily a square.