Given triangle $△\mathrm{jkl}$, which equation could be used to find $\angle j$ using the Law of Sines?

• A. $\frac{\sin (\angle j)}{\mathrm{JK}}=\frac{\sin (\angle k)}{\mathrm{JL}}$
• B. $\frac{\sin (\angle j)}{\mathrm{kl}}=\frac{\sin (\angle k)}{\mathrm{jl}}$
• C. $\frac{\sin (\angle j)}{\mathrm{KL}}=\frac{\sin (\angle k)}{\mathrm{JL}}$
• D. $\frac{\sin (\angle j)}{\mathrm{KL}}=\frac{\sin (\angle k)}{\mathrm{LJ}}=\frac{\sin (\angle l)}{\mathrm{JK}}$

Answer: (D)

Verified step by step guidance

1. Recall the Law of Sines, which states that in any triangle, the ratio of the sine of an angle to the length of the side opposite that angle is constant. Mathematically, this is expressed as: \(\frac{\sin(\angle A)}{a} = \frac{\sin(\angle B)}{b} = \frac{\sin(\angle C)}{c}\), where \(a\), \(b\), and \(c\) are the sides opposite angles \(A\), \(B\), and \(C\) respectively.
2. Identify the vertices and sides of triangle \(\triangle JKL\). The angle \(\angle j\) is at vertex \(J\), so the side opposite \(\angle j\) is the side \(KL\). Similarly, the side opposite \(\angle k\) is \(JL\), and the side opposite \(\angle l\) is \(JK\).
3. Set up the Law of Sines ratio for angles \(j\) and \(k\) using their opposite sides: \(\frac{\sin(\angle j)}{KL} = \frac{\sin(\angle k)}{JL}\).
4. Include the third angle \(l\) and its opposite side \(JK\) to complete the Law of Sines relationship: \(\frac{\sin(\angle j)}{KL} = \frac{\sin(\angle k)}{JL} = \frac{\sin(\angle l)}{JK}\).
5. Use this equation to solve for \(\sin(\angle j)\) or \(\angle j\) itself if the lengths of sides \(KL\), \(JL\), and \(JK\) and the measures of angles \(k\) or \(l\) are known.