$I n a Δ A B C, i f s i n^{2} A + s i n^{2} B = s i n^{2}, C show that the triangle is right angled.$

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Solution

$L e t s i n A = a k, s i n B = b k, s i n C = c k s i n^{2} A + s i n^{2} B = s i n^{2} C \Rightarrow k^{2} a^{2} + k^{2} b^{2} = k^{2} c^{2} [Using sine rule] \Rightarrow a^{2} + b^{2} = c^{2} Since the triangle satisfies the Pythagoras theoram, therefore it is right angled.$

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