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Question
Prove that the angle in a segment shorter than a semicircle is greater than a right angle.
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Solution
\[ \stackrel\frown{QP} \text{ is a major arc and } \angle PSQ \text{ is the angle formed by it in the alternate segment } . \]
\[ \text{ We know that the angle subtended by an arc at the centre is twice the angle subtended by it at any point of the alternate segment of the circle } . \]
`=> 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.
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