Question

Diagonal AC of a parallelogram ABCD bisects ∠A (see the given figure). Show that

1. It bisects ∠C also,
2. 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.