ABCD is a rectangle in which diagonal BD bisects ∠B. Show that ABCD is a square.
Let there be a rectangle ABCD with AB = CD and BC = AD and ∠A = ∠B = ∠C= ∠D = 90o
Since, BD bisects ∠B
∠ABD = ∠DBC (i)
And, ∠ADB = ∠DBC [Alternate interior angles]
∠ABD = ∠ADB. [From (i)]
AB = DA. (Sides opposite to equal angles)
∴ AB = CD = DA = BC
Since, all the sides are equal and all the angles are equal to 90o, thus the quadrilateral is a square.
Hence, ABCD is a square.