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


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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