ABCD is a cyclic quadrilateral.The tangents at A and C meet at P. If angle APC = 50 degree, Then what is the value of angle ADC?

Answer :

We form our diagram from given information , As  :

Here

ABCD is a cyclic quadrilateral and $\angle$ APC  =  50$°$

We know " A tangent to a circle is perpendicular to the radius at the point of tangency . "
So,
$\angle$ OAP  =  $\angle$ OCP  =  90$°$

From angle sum property of quadrilateral we get in AOCP 

$\angle$ OAP  +  $\angle$ OCP  + $\angle$ APC +  $\angle$ AOC  =  360$°$ , Substitute all values we get

90$°$  + 90$°$  + 50$°$  + $\angle$ AOC  =  360$°$

$\angle$ AOC  =  130$°$

And we know " The angle formed at the centre of the circle by lines originating from two points on the circle's circumference is double the angle formed on the circumference of the circle by lines originating from the same points. "
So,
$\angle$ AOC  =  2 $\angle$  ABC 

So,

2$\angle$  ABC  =  130$°$

$\angle$  ABC  =  65$°$

We know in cyclic quadrilateral opposite angles are supplementary , So

$\angle$ ABC  +  $\angle$ ADC  = 180$°$  , So

65$°$   + $\angle$ ADC  = 180$°$ 

$\angle$ ADC  = 115$°$                                                 ( Ans )