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 

Answer 1

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  .