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"19. In the wiven firure, \\( A B C D \\) is a square and \\( P , Q , R \\) are points\nan \\( A B \\), BC and \\( C D \\) respectively such that \\( A P = B Q = C R \\) and\n\\( \\angle P Q R = 90 ^ { \\circ } \\).\n\\( \\begin{array} { l l } { \\text { Prove that: } } & { \\text { (ii) } P Q = Q R } \\\\ { \\text { (i) } P B = Q C \\text { and } A P = B Q \\Rightarrow A B - A P = B C - B Q \\Rightarrow } & { Q C } \\\\ { \\text { Aint. } A B = B C \\text { and } A P = B C R + \\angle Q R C = 90 ^ { \\circ } + \\angle Q R C \\text { . } } \\\\ { \\text { Also, } \\angle R Q B = 90 ^ { \\circ } + \\angle Q R C = 90 ^ { \\circ } + \\angle Q R C \\text { . } } \\\\ { \\therefore \\angle Q R C = \\angle P Q B } \\end{array} \\)\n\\( \\therefore \\angle Q R = \\angle P Q B \\)\n\\( \\therefore \\angle Q R = \\angle P Q B \\)\n\\( \\therefore \\angle Q R = \\angle P Q B \\)"
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