Let's sketch the problem to understand better:
Since the angle of reflection is congruent to the angle of incidence, we have r1 = i1 = 38.2°.
Now, let's calculate angle x, knowing that it is complementary to the angle r1:
[tex]\begin{gathered} x+r1=90\\ \\ x+38.2=90\\ \\ x=90-38.2\\ \\ x=51.8° \end{gathered}[/tex]To calculate angle y, let's add all angles in the triangle and equate the sum to 180°:
[tex]\begin{gathered} x+y+53.7=180\\ \\ 51.8+y+53.7=180\\ \\ y+105.5=180\\ \\ y=180-105.5\\ \\ y=74.5° \end{gathered}[/tex]Angles i2 and y are complementary, so we have:
[tex]\begin{gathered} i2+y=90\\ \\ i2+74.5=90\\ \\ i2=90-74.5\\ \\ i2=15.5° \end{gathered}[/tex]Since r2 and i2 are congruent, so the angle of reflection in the second mirror is 15.5°.
