ABCD is rhombus. Diagonal AC bisects ∠A and ∠C.
To Prove: Diagonal BD bisects ∠B and ∠D.
Proof: In a rhombus all sides are equal and opposite angles are equal to each other.
In ∆ABC, AB = BC
∴ ∠BAC = ∠BCA
∠B AC = ∠DCA (alternate angles)
∴ ∠BCA = ∠DCA …………… (i)
∴ AC bisects ∠C.
Now, ∠BCA = ∠DAC (alternate angles)
∠BAC = ∠DAC
∴ AC bisects ∠A .
In ∆ABD, AD = DB
∴ ∠ABD = ∠ADB
∠ABD = ∠CDB
∴ BD bisects ∠D.
∠ADB = ∠CBD
∠ABD = ∠CBD
∴ BD bisects ∠B.