Question

State true or false

A diagonal of a parallelogram bisects one of its angle. Then it is a rhombus .

• A. True
• B. False

Answer 1

The correct option is A True

Let $ABCD$ is a parallelogram and diagonal $AC$ bisects the angle $A.$

$\therefore$ $\angle CAB=\angle CAD$

$AB\parallel CD$ and $AC$ is a trasnversal.

$\therefore$ $\angle CAB=\angle ACD$ [ Alternate interior angles ]

Again, $AD\parallel BC$ and $AC$ is a transversal.

$\therefore$ $\angle CAD=\angle ACB$ [ Alternate interior angles ]

$\Rightarrow$ So, $\angle ACD=\angle ACB$ [ Since, $\angle CAB=\angle CAD$ ] ---- ( 2 )

$\Rightarrow$ $\angle A=\angle C$ [ Opposite angels of parallelogram are equal ]

$\Rightarrow$ $\frac{1}{2}\angle A=\frac{1}{2}\angle C$ [ Dividing both sides by $2$ ]

$\Rightarrow$ $\angle DAC=\angle DCA$ [ From ( 1 ) and ( 2 ) ]

$\Rightarrow$ $CD=AD$ [ Sides opposite to the equal angles are equal ]

$\Rightarrow$ $AB=CD$ and $AD=BC$ [ Opposite sides of parallelogram are equal ]

$\therefore$ $AB=BC=CD=AD$

Thus, all sides are equal. So, $ABCD$ is a rhombus.

$\therefore$ The statement , diagonal of a parallelogram bisects one of its angle, then it is a rhombus it is correct.