Mad Mirrors

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Note that if we reflect the triangle about line $AB$ , then one reflection along line $AB$ corresponds to one intersection with line $AB$ (see the diagram if this is not clear to you.)

We can continue reflecting the triangle like this until it forms a circle:

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Thus the problem becomes finding the maximum number of intersections of this light ray line with $AB$ or $AC$ . Clearly, since $\angle BAC = 1^{\circ}$ , that there are $360$ total reflected lines $AB$ and $AC$ . However, any drawn line can at most travel through half of them (shown above), so the answer is $360\div 2=\boxed{180}$ .

Since angle between mirrors is 1 degree, after each reflection the angle of incidence decreases by 1 degree (draw any triangle and see by yourself). When the angle of incidence is 0 it traces back the bath it came. The maximum angle of incidence is 90 degree (when light is parallel to one mirror). It will be reflected 90 times before angle of incidence becomes 0 and then it will trace back its path (again 90 reflections) so total reflections are 180.