Given two triangles, $△STU$ and $△XYZ$, if $△STU$ is congruent to $△XYZ$, which of the following statements about their corresponding sides and angles is true according to the Law of Sines?

• A. $\frac{a}{sin\left(A\right)}=\frac{x}{sin\left(X\right)}$
• B. $\frac{a}{sin\left(C\right)}=\frac{y}{sin\left(X\right)}$
• C. $\frac{a}{sin\left(B\right)}=\frac{z}{sin\left(Y\right)}$
• D. $\frac{a}{sin\left(A\right)}=\frac{y}{sin\left(Z\right)}$

Answer: (A)

Verified step by step guidance

1. Recall that if two triangles \( \triangle STU \) and \( \triangle XYZ \) are congruent, then their corresponding sides and angles are equal in measure. This means side \( ST \) corresponds to side \( XY \), angle \( S \) corresponds to angle \( X \), and so on.
2. The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Mathematically, for triangle \( ABC \), it is written as: \[\frac{a}{\sin(\!A)} = \frac{b}{\sin(\!B)} = \frac{c}{\sin(\!C)}\]
3. Apply the Law of Sines to both triangles \( \triangle STU \) and \( \triangle XYZ \). Since the triangles are congruent, the corresponding sides and angles match, so the ratios involving corresponding sides and their opposite angles should be equal.
4. Identify the corresponding sides and angles between the two triangles. For example, side \( a \) opposite angle \( A \) in \( \triangle STU \) corresponds to side \( x \) opposite angle \( X \) in \( \triangle XYZ \). Therefore, the ratio \( \frac{a}{\sin(\!A)} \) should equal \( \frac{x}{\sin(\!X)} \).
5. Conclude that the correct statement according to the Law of Sines and congruence is: \[\frac{a}{\sin(\!A)} = \frac{x}{\sin(\!X)}\] This reflects the equality of ratios of corresponding sides to the sines of their opposite angles.