Q:

# Which statement describes a parallelogram that must be a square? A parallelogram with opposite sides that are congruent and diagonals that bisect the angles. A parallelogram with a right angle and diagonals that bisect the angles. A parallelogram with a right angle and opposites sides that are congruent. A parallelogram with all sides congruent.

Accepted Solution

A:
Consider a parallelogram with a right angle and diagonals that bisect the angles.In the figure, ABCD is a parallelogram with ∠ A = 90°.Since the opposite angles of a parallelogram are equal,∠ A = ∠ C = 90° and ∠ B = ∠ DAlso, since adjacent angles of a parallelogram are supplementary,∠ A + ∠ B = 180°But, since ∠ A = 90°, ∠ B = 90° and ∠ D = 90°Therefore, ∠ A = ∠ B = ∠ C = ∠ D.Now, it is given that the diagonals bisect the angles.Therefore, ∠ OAB = ∠ OBA = ∠ OBC = ∠ OCB = 45°Consider, triangles OBA and OBC.∠ OBA = ∠ OBC = 45°and OB = OB (common)Therefore, Δ OBA ≅ Δ OBC (SAS Rule)By corresponding parts of congruent triangles,AB = BCNote that in a parallelogram, the opposite angles are congruent.Therefore, AB = BC = CD = DA.Hence, in the parallelogram ABCD, we have,AB = BC = CD = DA and ∠ A = ∠ B = ∠ C = ∠ D = 90° Hence, ABCD is a square.