Question

For the given figure, determine the angle ∠P.

degrees

Answer 1

XYVW is a quadrilateral formed by intersections of two circles.
Connect the intersection points of the two circles.
XABW is a cyclic quadrilateral to the smaller circle.
As opposite angles of a cyclic quadrilateral have a sum of ${180}^o,$
$\angle ABW={180}^o-\angle AXW$
$\Rightarrow \angle ABW={180}^o-{115}^o={65}^o$

WBV is a straight line.
$\therefore \angle ABV={180}^o-\angle ABW$
$\Rightarrow \angle ABV={180}^o-{65}^o={115}^o$

AYVB is a cyclic quadrilateral to the larger circle.
$\angle Y={180}^o-\angle ABV$
$\Rightarrow \angle Y={180}^o-{115}^o=={65}^o$

For $△PYV,$
$\angle P+\angle Y+\angle V={180}^o$
$\Rightarrow \angle P+{65}^o+{80}^o={180}^o$
$\Rightarrow \angle P={180}^o-{65}^o-{80}^o={35}^o$