# Diagonal AC of a parallelogram ABCD bisects ∠A (see Fig. 8.19). Show that(i) It bisects ∠ C also,(ii) ABCD is a rhombus.

(i) ABCD is a parallelogram.

DAC = BCA (Alternate interior angles) ... (1)

And,

BAC = DCA (Alternate interior angles) ... (2) However, it is given that AC bisects A

DAC = BAC ... (3)

From equations (1), (2), and (3), we obtain

DAC = BCA = BAC = DCA ... (4)

DCA = BCA

Hence, AC bisects C

(ii) From equation (4), we obtain

DAC = DCA

DA = DC (Side opposite to equal angles are equal)

However,

DA = BC and AB = CD (Opposite sides of a parallelogram)

AB = BC = CD = DA

Hence, ABCD is a rhombus

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