Question

Seg $XZ$ is a diameter of a circle. Point Y lies in 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 a measure of $\angle XYZ$.

• A. Only one
• B. Only two
• C. Only three
• D. All

Answer 1

The correct option is C Only three

Let $P$ be any point on the arc $XZ$.
(Refer image)
$XZ$ is the diameter of the circle.
$\therefore \angle XPZ={90}^o$ (Angle in a semi-circle is ${90}^o$)
So, $\angle XYZ$ cannot be a right angle.
In $\Delta YPZ$,
$\angle XYZ>\angle YPZ$ (An exterior angle of a triangle is greater than its remote interior angle)
$\angle XYZ>{90}^o(\angle YPZ=\angle XPZ)$
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.