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Question

# Find the relation between the angles ∠X and ∠Y.

A
Y=180oX
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B
Y=90oX
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C
They are supplementary angles.
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D
Nothing cannot be said without sufficient data.
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Solution

## The correct option is 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 180o, ∠ABW=180o−∠X WBV is a straight line. ∴∠ABV=180o−∠ABW ⇒∠ABV=180o−(180o−∠X)=∠X AYVB is a cyclic quadrilateral to the larger circle. ∠Y=180o−∠ABV ⇒∠Y=180o−∠X ⇒∠Y+∠X=180o ∴∠X and ∠Y are supplementary angles.

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