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Choice D is correct. Two triangles are similar if they have three pairs of congruent corresponding angles. It’s given that angles $L$ and $R$ each measure $60°$, and so these corresponding angles are congruent. If angle $M$ is $70°$, then angle $N$ must be $50°$ so that the sum of the angles in triangle $LMN$ is $180°$. If angle $T$ is $50°$, then angle $S$ must be $70°$ so that the sum of the angles in triangle $RST$ is $180°$. Therefore, if the measures of angles $M$ and $T$ are $70°$ and $50°$, respectively, then corresponding angles $M$ and $S$ are both $70°$, and corresponding angles $N$ and $T$ are both $50°$. It follows that triangles $LMN$ and $RST$ have three pairs of congruent corresponding angles, and so the triangles are similar. Therefore, the additional piece of information that is sufficient to prove that triangle $LMN$ is similar to triangle $RST$ is that the measures of angles $M$ and $T$ are $70°$ and $50°$, respectively.

Choice A is incorrect. If the measures of two sides in one triangle are proportional to the corresponding sides in another triangle and the included angles are congruent, then the triangles are similar. However, the two sides given are not proportional and the angle given is not included by the given sides.

Choice B is incorrect. If the measures of two sides in one triangle are proportional to the corresponding sides in another triangle and the included angles are congruent, then the triangles are similar. However, the angle given is not included between the proportional sides.

Choice C is incorrect and may result from conceptual or calculation errors.