Question

two plane mirrors are inclined at 70 deg. a ray incident on on one mirror at angle theta, after reflection falls on second mirror and is reflected from there parallel to the first mirror. What is theta

Answer 1

Lots of geometry!
See attached file. First see figure on left. (1,2,3 .... are angles, not numbers!)
We know 1 is equal to θ (angle of incidence)
2=1=θ (law of reflection)
4 =70 (angle of inclination of mirrors)

Now it can be proved easily (and i will prove later) that 4=8 =70
Since 7+8 =180, therefore ,7=110

Also 3+2=90 (because N1 is normal to mirror)
Hence 3=90-θ
Now, 3=5+6 (as second reflected ray is parallel to mirror 1)
Therefore 5+6=90-θ
Also 5=6 (law of reflection)

Hence 5=6=45-θ/2

I have shown all the angles in the right figure.
Now see triangle ABC, sum of all angle should be 180.

Hence 2+5+7 =180
θ+45-θ/2+110=180
giving θ=50

Answer 2

the proof that angle 4 is equal to angle 8

You can even state it as a theorem - angle between two lines is same as angle b/w their perpendiculars. N1 and N2 are perpendiculars to mirror 1 and mirror 2. So angle b/w N1 and N2 is same as the angle b/w mirror 1 and mirror 2. But which angle? There are 2 angles b/w N1 and N2 -one 70 and other 110. It may cause a confusion so it is better to find out geometrically as i have done above-I blindly applied the thm initially and wrongly persumed that angle 7=70!