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.
Answer:
A Parallelogram with a right angle and diagonals that bisect the angles
Step-by-step explanation:
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