Exercise:
Show that when light reflects from two mirrors that meet each other at a right angle the outgoing ray is parallel to the incoming ray.

Solution:
center tikzpictureLatex line width.pt % Mirrors drawvery thick - -- noderight Mirror ; drawvery thick - -- nodeabove Mirror ; % Pos coordinate O at ; coordinate A at ; coordinate B at ; coordinate P at ; coordinate C at ; coordinate N at .; % normal helper po at A coordinate N at .; % normal helper po at B % Normals dashed drawdashed gray -. -- ; drawdashed gray -. -- ; % Auxiliary triangle O-A-B drawdotted O -- A; drawdotted O -- B; % Rays draw- thick blue P -- A nodemidway above right incoming; draw- thick blue A -- B; draw- thick blue B -- C nodemidway above right outgoing; % Angle marks pic"theta" draw angle radius.cm angle eccentricity. angle P--A--N; pic"theta" draw angle radius.cm angle eccentricity. angle N--A--B; pic"varphi" draw angle radius.cm angle eccentricity. angle A--B--N; pic"varphi" draw angle radius.cm angle eccentricity. angle N--B--C; pic"^circ" draw angle radius.cm angle A--O--B; nodebelow at A A; nodeleft at B B; nodebelow left at O O; tikzpicture center medskip textbfStep -- Law of reflection. Let the incoming ray strike mirror at po A with angle of incidence theta measured from the normal to mirror . By the law of reflection the reflected angle at A is also theta. This reflected ray travels to mirror and strikes it at po B with angle of incidence varphi measured from the normal to mirror reflecting again at angle varphi. medskip textbfStep -- Triangle OAB. Let O be the corner where the two mirrors meet so angle AOB ^circ. Since the normal at A is perpicular to mirror the angle between segment AB and mirror itself is angle OAB ^circ - theta . Similarly the angle between AB and mirror at B is angle OBA ^circ - varphi . Since the angles of triangle OAB to ^circ: ^circ + ^circ - theta + ^circ - varphi ^circ quadLongrightarrowquad theta + varphi ^circ . medskip textbfStep -- Total deviation of the ray. Each reflection deviates the ray's direction of travel by ^circ - theta at A and ^circ - varphi at B. The total deviation after both reflections is ^circ - theta + ^circ - varphi ^circ - theta + varphi ^circ - ^circ ^circ . A total deviation of exactly ^circ means the outgoing ray pos in exactly the opposite direction to the incoming ray. Hence the outgoing ray is parallel in fact antiparallel to the incoming ray. qed bigskip noindent textbfAlternative proof vector form. Place mirror along the x-axis and mirror along the y-axis. Reflection off a horizontal mirror negates the y-component of a ray's direction vector; reflection off a vertical mirror negates the x-component. If the incoming ray has direction d_x d_y then d_x d_y ;xrightarrowtextmirror ; d_x -d_y ;xrightarrowtextmirror ; -d_x -d_y -d_x d_y. The outgoing direction is exactly - times the incoming direction so the two rays are parallel antiparallel -- regardless of the angle of incidence. This is the general two-dimensional corner-reflector result. qed