Question

Find the relation between the angles $\angle X$ and $\angle Y.$

• A. $\angle Y={180}^o-\angle X$
• B. $\angle Y={90}^o-\angle X$
• C. They are supplementary angles.
• D. Nothing cannot be said without sufficient data.

Answer 1

Answer: (A), (C)

They are supplementary angles.

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 ${180}^o,$
$\angle ABW={180}^o-\angle X$

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

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

$\therefore \angle X$ and $\angle Y$ are supplementary angles.