Question

Diagonal AC of a parallelogram ABCD bisects ∠ A (see Fig. 8.11). Show that

(i) it bisects ∠C also,

(ii) ABCD is a rhombus.

Answer 1

(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.