Given two triangles with corresponding angles and side lengths related by the $Law of Sines$, which of the following pairs of triangles can be mapped onto each other using a single translation?

• A. Two triangles with proportional side lengths but different sizes
• B. Two triangles with equal corresponding angles but different side lengths
• C. Two congruent triangles with the same orientation and side lengths
• D. Two triangles with the same side lengths but mirrored orientation

Answer: (C)

Verified step by step guidance

1. Understand that a translation is a rigid motion that slides a figure without rotating or reflecting it, so the figure's size, shape, and orientation remain exactly the same.
2. Recall that congruent triangles have exactly the same side lengths and angles, and if they have the same orientation, one can be mapped onto the other by a translation alone.
3. Recognize that triangles with proportional side lengths but different sizes are similar but not congruent, so a translation cannot map one onto the other without resizing.
4. Note that triangles with equal corresponding angles but different side lengths are similar, not congruent, so a translation alone cannot map one onto the other.
5. Understand that triangles with the same side lengths but mirrored orientation are congruent but require a reflection (or rotation plus reflection) to map onto each other, not just a translation.