Question

ABCD is a rectangle in which diagonal BD bisects ∠B. Show that ABCD is a square.

Answer 1

Let ABCD be a rectangle.
Here, AB = CD and BC = DA and ∠A = ∠B = ∠C= ∠D = 90^o
Now, we have:
∠​ABD = ∠​DBC [ ∵BD bisects ∠B ]
​∠​ADB = ​∠​DBC [ Alternate interior angles]
​∠​ABD = ∠​ADB
As the two opposite angles of ∆ABD are equal, the opposite sides must also be equal.
i.e., AB = DA
∴ AB = CD = DA = BC
So, when all the sides are equal and all the angles are of 90^o, the quadrilateral is a square.
Hence, ABCD is a square.