Question

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.

Answer 1

A Parallelogram with a right angle and diagonals that bisect the angles

Explanation:

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Answer 2

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 = ∠ D

Also, 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 = BC

Note 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.