Question
Diagonal $AC$ of a parallelogram $\text{ABCD}$ bisects $\angle A ($See figure$).$ Show that:
$i.$ It bisects $\angle C$ also.
$ii. \text{ABCD}$ is a rhombus.
$i.$ It bisects $\angle C$ also.
$ii. \text{ABCD}$ is a rhombus.
✓
Answer
Diagonal $AC$ bisects $\angle A$ of the parallelogram $\text{ABCD}.$
$i.$ Since $A B \| D C$ and $A C$ intersects them.
$\therefore \angle 1=\angle 3 [$Alternate angles$] ...(i)$
Similarly $\angle 2=\angle 4 \ldots (ii)$
But $\angle 1=\angle 2 [$Given$]$
Thus $A C$ bisects $\angle C$.
$ii. \angle 2=\angle 3=\angle 4=\angle 1$
$\Rightarrow A D=C D [$Sides opposite to equal angles$]$
$\therefore A B=C D=A D=B C$
Hence $\text{ABCD}$ is a rhombus.
$i.$ Since $A B \| D C$ and $A C$ intersects them.
$\therefore \angle 1=\angle 3 [$Alternate angles$] ...(i)$
Similarly $\angle 2=\angle 4 \ldots (ii)$
But $\angle 1=\angle 2 [$Given$]$
Thus $A C$ bisects $\angle C$.
$ii. \angle 2=\angle 3=\angle 4=\angle 1$
$\Rightarrow A D=C D [$Sides opposite to equal angles$]$
$\therefore A B=C D=A D=B C$
Hence $\text{ABCD}$ is a rhombus.
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