Data: Diagonal AC of a parallelogram ABCD bisects ∠A.
To Prove: (i) Diagonal AC also bisects ∠C.
(ii) ABCD is a rhombus.
Proof: i) Diagonal AC also bisects ∠A.
∴ ∠DAC = ∠BAC …………… (i)
But, ∠DAC= ∠BCA (alternate angles)(ii)
∠BAC=∠DCA (alternate angles)(iii)
From (i), (ii) and (iii),
∠BCA = ∠DCA
∴ AC bisects ∠C.
(ii) ∠DAC = ∠DCA
∴ AD = DC
But, AD = BC
∴ DC = AB
∴ AB = BC = CD = DA
∴ ABCD is a rhombus.
