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

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by umakumar

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by umakumar

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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.

∠DAC =∠ BAC

∠DAC=∠BAC

but ∠DAC=∠BCA

so ∠DCA=∠CAB=∠DAC=∠BCA

so AB=BC=DA

so it is a rombus

∠DAC=∠BAC

but ∠DAC=∠BCA

so ∠DCA=∠CAB=∠DAC=∠BCA

so AB=BC=DA

so it is a rombus