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.