To solve this problem, we need to understand the concepts of plane angle and solid angle.$\\$• A plane angle is defined as the angle subtended by two lines at a point. It is measured in radians.$\\$• A solid angle is defined as the angle subtended by a surface at a point. It is measured in steradians.$\\$Both plane angle and solid angle are dimensionless quantities.$\\$This is because they are defined as ratios:$\\$• Plane angle $(\theta)$ is the ratio of the arc length $(s)$ to the radius $(r): \\ \theta = \frac{s}{r} \\$• Solid angle $(\Omega)$ is the ratio of the area $(A)$ to the square of the radius $(r^2): \\ \Omega = \frac{A}{r^2} \\$Since both are ratios of similar quantities, they are dimensionless.$\\$However, they do have units:$\\$• Plane angle is measured in radians.$\\$• Solid angle is measured in steradians.$\\$Therefore, plane angle and solid angle have units but no dimensions.$\\$This corresponds to Option 1: Units but no dimensions.