Given triangle $△efg$, which equation could be used to find the measure of angle $e$ using the Law of Sines?

• A. $\frac{\sin \left(e\right)}{\mathrm{EF}}=\frac{\sin \left(f\right)}{\mathrm{EG}}$
• B. $\frac{\sin \left(e\right)}{\mathrm{FG}}=\frac{\sin \left(g\right)}{\mathrm{EF}}$
• C. $\frac{\sin \left(e\right)}{\mathrm{fg}}=\frac{\sin \left(f\right)}{\mathrm{eg}}$
• D. $\frac{\sin \left(e\right)}{\mathrm{fg}}=\frac{\sin \left(f\right)}{\mathrm{eg}}=\frac{\sin \left(g\right)}{\mathrm{ef}}$

Answer: (C)

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 \(\triangle efg\). The angle \(e\) is opposite side \(fg\), angle \(f\) is opposite side \(eg\), and angle \(g\) is opposite side \(ef\).
3. Set up the Law of Sines ratio involving angle \(e\) and its opposite side \(fg\), and angle \(f\) and its opposite side \(eg\). This gives the equation: \(\frac{\sin(e)}{fg} = \frac{\sin(f)}{eg}\).
4. Note that this equation allows you to solve for \(\sin(e)\) or the measure of angle \(e\) if the lengths of sides \(fg\) and \(eg\) and the measure of angle \(f\) are known.
5. Confirm that this matches the correct choice among the given options, which uses the Law of Sines correctly by pairing each angle with its opposite side.