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Cyclic Quadrilateral

A quadrilater...

Question

A quadrilateral ABCD will be a parallelogram for the following conditions.
$∠ D + ∠ B = 180^{0}$
If true then enter $1$ and if false then enter $0$

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Solution

$L e t u s c o n s i d e r f i g (i) & f i g (i i) . ∠ B + ∠ D = 180^{O} (g i v e n) ⟹ ∠ A + ∠ C = 180^{O} I t d o e s n o t m e a n t h a t ∠ B = ∠ D a s i n f i g (i) . e v e v e n i f ∠ B = ∠ D i t d o e s n o t m e a n t h a t ∠ A = ∠ C a s i n f i g (i) . F o r a q u a d r i l a t e r a l t o b e a p a r a l l e l o g r a m i t i s n e c e s s a r y t h a t b o t h t h e o p p o s i t e p a i r s o f a n g l e s a r e e q u a l a s i n f i g (i i) . H e r e t h e s u m o f t h e o p p o s i t e a n g l e s a r e 180^{O} . i . e A B C D t o b e a p a r a l l e l o g r a m i t i s n e c e s s a r y t h a t e a c h a n g l e s h o u l d b e = 90^{o} . I t m a y o r m a y n o t o c c u r i n e a c h q u a d r i l a t e r a l i n g e n e r a l . A n s - C a n b e b u t n e e d n o t b e .$

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