Q. 64.1( 453 Votes )

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

(ii) ABCD is a rhombus.


Answer :


(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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