# Two mirrors m and n are perpendicular to each other at O .Ray AB is incident to the mirror m at point B and reflected back in the direction BC which is again incident o the mirror n at point C Ray CD is the finally reflected ray.Prove that ray AB is parallel to Ray CD

$Mirror\;m\;and\;n\;are\;\perp \;to \;each\;other.\\*\because\;AB\;is\;incident\;ray\;and\;CD \;is\;reflected\;ray.\\*\therefore \;\angle 1=\angle2\;and\;\angle3=\angle4\\* Now,\;construct\;EA\perp m\;and\;EC\perp n.\\* \therefore \angle AEC=90^{\circ}\\*In\;\triangle AEC,\\*\indent \angle AEC+\angle EAC+\angle ACE=180^{\circ}\\*\Rightarrow 90^{\circ}+\angle 2 +\angle 3=180^{\circ}\\*\Rightarrow \angle 2 +\angle 3=90^{\circ}\\*Now, \;multiply\;it\;by\;2\\*2\times \angle 2 +2\times \angle 3=180^{\circ}\\*\Rightarrow \angle BAC +\angle ACD=180^{\circ}\\* \because \angle BAC \and\; \angle ACD,\;forms\;a\;pair\;of\;consecutive\;interior\;\\*angles\;and\;are\;supplementary\;angles.\\* \therefore AB || CD,\;Hence\;proved.$

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