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Question
Four alternative answers for the following question is given. Choose the correct alternative.
Seg XZ is a diameter of a circle. Point Y lies in its interior. How many of the following statements are true ? (i) It is not possible that ∠XYZ is an acute angle. (ii) ∠XYZ can’t be a right angle. (iii) ∠XYZ is an obtuse angle. (iv) Can’t make a definite statement for measure of ∠XYZ.
Options
Only one
Only two
Only three
All
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Solution
Let P be any point on the arc XZ. 
XZ is the diameter of the circle.
∴ ∠XPZ = 90º (Angle in a semi-circle is 90º)
So, ∠XYZ cannot be a right angle.
In ∆YPZ,
∠XYZ > ∠YPZ (An exterior angle of a triangle is greater than its remote interior angle)
⇒ ∠XYZ > 90º (∠YPZ = ∠XPZ)
So, ∠XYZ is an obtuse angle. Therefore, it is not possible that ∠XYZ is an acute angle.
Thus, three of the following statements are true.
Hence, the correct answer is Only three .
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