Question

Question 6 (ii)
Diagonal AC of parallelogram ABCD bisects $\angle$ A ( see the given figure). Show that
ABCD is a rhombus

Answer 1

ABCD is a parallelogram
$\angle DAC=\angle BCA$ (Alernate interior angles) ….(1)
And, $\angle BAC=\angle DCA$ (Alternate interior angles) ...(2)
However, it is given that AC bisects $\angle A$
$\angle DAC=\angle BAC$ …(3)

From equations (1) ,(2) and (3), we obtain
$\angle DAC=\angle BCA=\angle BAC=\angle DCA$ …(4)
$\therefore \angle DAC=\angle DCA$
$\Rightarrow DA=DC$
(Side opposite to equal angles are equal)

However,
$DA=BCandAB=CD$
(Opposite sides of a parallelogram)

$\therefore AB=BC=CD=DA$
Hence, ABCD is a rhombus.