Why is the angle ABO equal to the refraction angle in a glass sphere TIR problem?

[betaleonis]

Homework Statement

A ray of light incident on a glass sphere (refractive index sqrt 3) suffers total internal reflection before emerging out exactly parallel to the incident ray. What was the angle of incidence?

μ = refractive index of the glass sphere, i = angle of incidence, r = angle of refraction

Homework Equations

μ = sin i/sinr

The Attempt at a Solution

From the figure, it turns out that < AOE = i (since L1 is parallel to L2, the corresponding angles are equal)

i = 2r, which implies that sin i/ sinr = μ, or 2cosr = μ, or r = 30° and i = 60°, which is the answer.

I do not understand why < ABO = r. Why is it so? It'd be helpful if I someone could come up with a different way of approaching the problem.

 

[haruspex]

I do not understand why < ABO = r. Why is it so?

Isosceles triangle (two radii).

 

[Simon Bridge]

I do not understand why < ABO = r. Why is it so? It'd be helpful if I someone could come up with a different way of approaching the problem.

ΔABO is an isosceles triangle: |OA|=|OB|

[edit] beaten to it :)

 

[betaleonis]

Oops! I ought to have thought a bit more before posting that question. Thank you. :)

 

[Simon Bridge]

No worries. Everyone does it sometimes :)
You'd probably have noticed right away if the diagram was constructed rather than sketched, even though you noticed about the equal base-angles.