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 $\angle$XYZ is an acute angle. (ii) $\angle$XYZ can’t be a right angle. (iii) $\angle$XYZ is an obtuse angle. (iv) Can’t make a definite statement for measure of $\angle$XYZ.

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KollegeApply · Accepted Answer

Let P be any point on the arc XZ. XZ is the diameter of the circle. $\therefore$ $\angle$XPZ = 90º (Angle in a semi-circle is 90º) So, $\angle$XYZ cannot be a right angle. In ∆YPZ, $\angle$XYZ > ​$\angle$YPZ (An exterior angle of a triangle is greater than its remote interior angle) $\Rightarrow$ $\angle$XYZ > ​90º ($\angle$YPZ = $\angle$XPZ) So, $\angle$XYZ is an obtuse angle. Therefore, it is not possible that $\angle$XYZ is an acute angle. Thus, three of the following statements are true. Hence, the correct answer is Only three .

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