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Step-by-Step Solution
Step 1: Identify the mirrors and the angles
Let the two plane mirrors be M1 and M2, inclined at an angle of 70°. Let the normal to mirror M1 at the point of incidence be N1 and the normal to mirror M2 at the point of reflection be N2. The angle between N1 and N2 is also 70° because the normals make the same angle as the mirrors themselves.
Step 2: Define the angle of incidence on the first mirror
Let $\\theta$ be the angle of incidence on mirror M1, measured from the normal N1. By the law of reflection, the angle of reflection from M1 is also $\\theta$. Thus, the reflected ray leaves mirror M1 making an angle $\\theta$ with N1 (but on the other side of the normal).
Step 3: Determine the incidence on the second mirror
The reflected ray from mirror M1 now strikes mirror M2 at some angle. Because the normals N1 and N2 differ by 70°, the angle of incidence on mirror M2, call it $i_{2}$, can be expressed as either $70° - \\theta$ or $70° + \\theta$ depending on orientation. In this scenario, it is $70° - \\theta$ (since $\\theta$ is measured from N1 and we are 'subtracting' that from the 70° difference between the normals). By the law of reflection, the angle of reflection from mirror M2 is also $70° - \\theta$.
Step 4: Impose the condition for the final ray to be parallel to the first mirror
The problem states that after reflection from the second mirror, the ray is parallel to mirror M1. For the ray to be parallel to M1, its direction must be perpendicular to the normal N1. That means the angle between the final reflected ray and N1 should be 90°.
Step 5: Relate the final direction to the normal of the first mirror
After reflection from mirror M2, the ray makes an angle $70° - \\theta$ with N2. The total angle between N1 and the final ray can be shown (by adding angles properly around the vertex where the normals meet) to be $70° + (70° - \\theta) = 140° - \\theta$. For the ray to be perpendicular to N1, we set:
$140° - \\theta = 90°$
Solving for $\\theta$ gives:
$\\theta = 140° - 90° = 50°$
Step 6: Conclude the value of $\\theta$
The angle of incidence on the first mirror must therefore be 50° to ensure that after two reflections, the final ray is parallel to the first mirror.
Answer: 50°
