State true or false:

The diagonal $B D$ of a parallelogram $A B C D$ bisects angles $B$ and $D$ . then $A B C D$ is a rhombus.

True

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False

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Solution

The correct option is A True
$N o w i t s g i v e n t h a t ∠ A B D = ∠ C B D . . . . . . .1 a n d ∠ A D B = ∠ B D C . . . . . . . . . .2 A s w e k n o w f o r a n y p a r a l l e l o g r a m A B C D ∠ B = ∠ D F r o m a b o v e 1 & 2 w e c a n w r i t e ∠ A B D + ∠ C B D = ∠ A D B + ∠ B D C \Rightarrow 2 ∠ A B D = 2 ∠ A D B \Rightarrow ∠ A B D = ∠ A D B S o w e c a n s t a t e t h a t △ A B D i s a i s o s c e l e s t r i a n g l e, i . e . A B = B D N o w f r o m t h e p r o p e r t y o f p a r a l l e l o g r a m A B C D, A B = D C = B D = B C . H e n c e w e c a n p r o v e t h a t p a r a l l e l o g r a m A B C D i s a r h o m b u s .$

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