Question

ABCD is a parallelogram in which diagonal AC bisect angles A as well as C . Show that ABCD is rhombus

Answer 1

Angle DAC = Angle BAC
Angle DAC = Angle BAC
But, Angle DAC = Angle BCA (Alternate angles)
Therefore , Angle DCA = Angle CAB = Angle DAC = Angle BCA
Therefore, AB=BC=DC=DA (Sides opposite to equal angles are equal)

Since all sides are equal ABCD is a rhombus.

Answer 2

∠DAC =∠ BAC
∠DAC=∠BAC
but ∠DAC=∠BCA
so ∠DCA=∠CAB=∠DAC=∠BCA
so AB=BC=DA
so it is a rombus