Where did this equation come from ∠I+ ∠E = ∠A+ ∠D?

Question

∠I+ ∠E= ∠A + ∠D Angle of incidence + angle of emergence = angle of prism (Normally 60°) + angle of deviation. If their sum is not equal,we made personal error in doing an experiment with prism. Please make sense of this equation.

Answer 1

The following diagram shows the prism with the incoming and outgoing light rays.

If you follow the incident light ray in, it gets bent by an angle $\theta_1 = i- r_1$. If you follow the light ray where it leaves the glass, it gets bent again by an angle $\theta_2 = e - r_2$, so the total deviation is:

$$ \begin{align} D &= \theta_1 + \theta_2 \\ &= i + e - (r_1 + r_2) \end{align} $$

For the next step look at the triangle formed by the top of the prism and the light ray, and note that the internal angles must add up to 180°. So:

$$ A + (90 - r_1) + (90 - r_2) = 180 $$

and a quick rearrangement gives:

$$ A = r_1 + r_2 $$

Now substitute for $r_1 + r_2$ in our first equation and we get:

$$ D = i + e - A $$

or:

$$ D + A = i + e $$