NCERT Solutions
Class 9 Maths
Quadrilaterals

Ex.8.1 Q.6
Diagonal AC of a parallelogram ABCD bisects ∠A (see Fig).
Show that
(1) it bisects ∠ C also,
(2) ABCD is a rhombus.
(1) ABCD is a parallelogram.
So, ∠DAC = ∠BCA………………………………... (1)
(Alternate interior angles)
And, ∠BAC = ∠DCA……………………………… (2)
(Alternate interior angles)
However, it is given that AC bisects ∠A.
So, ∠DAC = ∠BAC………………………………… (3)
From equations (1), (2), and (3), we obtain
∠DAC = ∠BCA = ∠BAC = ∠DCA ............ (4)
=> ∠DCA = ∠BCA
Hence, AC bisects ∠C.
(2) 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.