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In the given figure, O is the center of the circle. The line UTV is a tangent to the circle at T. ∠VTR = 52° and ∆ PTR is an isosceles triangle such that TP = TR. What is ∠x + ∠y + ∠z equal to? 

(a) 175° 

(b) 208° 

(c) 218° 

(d) 250°

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Answer : (C) 218º 

x = ∠VTR = 52° (Angles in alternate segment are equal) 

x + z = 180° 

(PTMR is a cyclic quad.)

⇒ z = 180° – x = 180° – 52° = 128°. 

In ∆ PTR, PT = TR ⇒ a = x = 52° 

⇒ ∠PTU = a = 52° (Angles in alternate segment are equal) 

∠QTU = y + ∠PTU

⇒ y + ∠PTU = 90° (∵ ∠QTU = 90°, rad. OT ⊥ tangent UV) 

⇒ y = 90° – 52° = 38° 

∴ x + y + z = 52° + 38° + 128° = 218°.

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