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Reflections and Rotations

Any rotation can be expressed as the composition of two reflections. In the sketch below, triangle A'B'C' is the result of rotating triangle ABC around point O by angle . Triangle A''B''C'' is the result of reflecting triangle ABC over m followed by n. Play with the sketch, and decide how to compose  and  to produce the desired rotation by dragging lines m and n until triangle A''B''C'' coincides with triangle A'B'C'. Is the composition unique?