First make a diagram to clear the steps done below.

Assume ABCD to be the glass plate with sides AB parallel to CD.

Let the refractive index of air and glass be n1and n2.

Consider a ray of light EF incident on surface AB of the glass slab with angle of incidence, ∠i.
The light ray is refracted into the glass slab along the path FG and towards the normal, NN’ . ∠ r1 is the angle of refraction.
The light ray is refracted out of the glass slab at the surface CD along the path GH. The angle of incidence at this surface is ∠r2.

The light ray emerges out of the slab forming ∠e as angle of emergence.

The surfaces AB and CD are parallel and FG is the transversal,
Therefore,  ∠r​1= ∠r1

Applying Snell’s Law at the surface AB,
sin isin r1=n2n1 .......(1)
Similarly, at surface CD, according to Snell’s Law
sin r2sin e=n1n2 .......(2)
Multiplying equation 1 and 2,

sin isin r1×sin r2sin e=n2n1×n1n2sin isin r1×sin r2sin e=1

As AB is parallel to CD, the perpendiculars to the AB and CD, NN’ and MM’ are also parallel
Therefore, alternate interior angles,  r1 and r2are congruent i.e. r1 =  r2 
Hence,
sin isin r1×sin r1sin e=1sin isin e=1Therefore, sin i=sin eor, i=e

Hence, angle of incidence equals angle of emergence.

That's all🙂

Proof that Angle of Incidence is Equal to Angle of Emergence in a Glass Slab



When light travels from one medium to another, it changes its speed and direction due to the change in the refractive index of the medium. The angle of incidence, angle of refraction, and the refractive index of the medium are related to each other through Snell's law. However, when light travels through a glass slab, it undergoes both refraction and reflection, and the angle of incidence is equal to the angle of emergence.



When a light ray is incident on a glass slab, it undergoes both reflection and refraction. The reflected ray obeys the law of reflection, which states that the angle of incidence is equal to the angle of reflection. The refracted ray, on the other hand, obeys Snell's law, which states that:

sin(i)/sin(r) = n

where i is the angle of incidence, r is the angle of refraction, and n is the refractive index of the glass slab.

Since the angles of incidence and reflection are equal, we can write:

sin(i)/sin(r) = n

sin(i)/sin(i) = n

1 = n

This means that the refractive index of the glass slab is 1, which is the same as the refractive index of air. Therefore, the angle of refraction is equal to the angle of incidence.

Now, when the refracted ray emerges from the glass slab, it undergoes both refraction and reflection again. The refracted ray obeys Snell's law, which states that:

sin(r)/sin(e) = n

where e is the angle of emergence.

Since the refractive index of the glass slab is 1, we can write:

sin(r)/sin(e) = 1

sin(r) = sin(e)

This means that the angle of refraction is equal to the angle of emergence.



In conclusion, when light travels through a glass slab, it undergoes both refraction and reflection. However, the angle of incidence is equal to the angle of emergence, which means that the angle of refraction is also equal to the angle of incidence. This is because the refractive index of the glass slab is 1, which is the same as the refractive index of air.