In the figure, XR and YQ are tangents to the circle YZXP if ZXR = 45° and YZX = 55°, Find ZYQ
Given: ∠ZXR = 45º (tangent XR), ∠YZX = 55º (inscribed angle).
∠YZX subtends arc YX, so arc YX = 2 × 55° = 110º.
Major arc YZX = 360º - 110º = 250º.
By the alternate segment theorem, ∠ZXR subtends arc XZ, so arc XZ = 2 × 45º = 90º.
Arc YZ = 250º - 90º = 160º.
Major arc YZ = 360º - 160º = 200º.
∠ZYQ (tangent YQ) = \(\frac{1}{2}\) × major arc YZ = \(\frac{1}{2}\) × 200º = 100º (by alternate segment theorem, larger arc for obtuse angle).
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