Question

Prove that the angle in a segment shorter than a semicircle is greater than a right angle.

Answer 1

 

\[ \stackrel\frown{QP}  \text{ is a major arc and } \angle PSQ \text{ is the angle formed by it in the alternate segment }  . \]
\[ \text{ We know that the angle subtended by an arc at the centre is twice the angle subtended by it at any point of the alternate segment of the circle } . \]

`=> 2angle "PSQ" = "m"`

`=> 2angle "PSQ" = 360^circ - "m"`

`=> 2 angle"PSQ" = 360^circ - 180^circ    ...(because angle "POQ" < 108^circ)`

`=> 2angle "PSQ" > 180^circ`

`=> angle "PSQ" > 90^circ`

Thus, the angle in a segment shorter than a semi-circle is greater than a right angle.