According to the $LawofSines$, which triangles can be mapped onto one another through a sequence of rigid transformations?

• A. Triangles that have proportional sides but different angles
• B. Triangles that are congruent
• C. Triangles that have the same area but different shapes
• D. Triangles that have two equal angles but different side lengths

Answer: (B)

Verified step by step guidance

1. Recall the Law of Sines, which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).
2. Understand that rigid transformations (translations, rotations, reflections) preserve the size and shape of a figure, meaning the triangles must be congruent to be mapped onto one another through such transformations.
3. Recognize that triangles with proportional sides but different angles are similar but not congruent, so they cannot be mapped onto each other by rigid transformations alone.
4. Note that triangles with the same area but different shapes or triangles with two equal angles but different side lengths are not congruent, so they also cannot be mapped onto each other by rigid transformations.
5. Conclude that only triangles that are congruent (having exactly the same side lengths and angles) can be mapped onto one another through a sequence of rigid transformations.