The measure of the arcs are ;
From the diagram given ;
XY = VW (opposite angles)
VW = 55°
XW = UV (opposite angles)
XW = 108°
XY + UY + UV + VW + XW = 360° (sum of reflex angles)
55 + UY + 108 + 55 + 108 = 360
UY + 326 = 360
UY = 360 - 326
UY = 34°
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