Which of the following statements is true for triangle $LNM$ according to the Law of Sines?

• A. $\frac{\sin \left(L\right)}{m}=\frac{\sin \left(N\right)}{l}$
• B. The Law of Sines only applies to right triangles.
• C. $\frac{l}{n}=\frac{m}{l}$
• D. $\frac{\sin \left(L\right)}{l}=\frac{\sin \left(N\right)}{n}=\frac{\sin \left(M\right)}{m}$

Answer: (D)

Verified step by step guidance

1. Recall the Law of Sines, which states that in any triangle \( LNM \), the ratio of the sine of an angle to the length of the side opposite that angle is constant. This can be written as: \[ \frac{\sin(L)}{l} = \frac{\sin(N)}{n} = \frac{\sin(M)}{m} \]
2. Examine the first given statement: \( \frac{\sin(L)}{m} = \frac{\sin(N)}{l} \). Notice that the sides in the denominators do not correspond to the sides opposite the angles in the numerators, which contradicts the Law of Sines.
3. Consider the statement that the Law of Sines only applies to right triangles. This is incorrect because the Law of Sines applies to all types of triangles, not just right triangles.
4. Look at the third statement: \( \frac{l}{n} = \frac{m}{l} \). This is a ratio of side lengths only and does not involve sines of angles, so it does not represent the Law of Sines.
5. Therefore, the correct statement according to the Law of Sines is the one that equates the ratios of the sine of each angle to the length of its opposite side, as shown in step 1.