(1) In \( \triangle X Y Z \), if \( \frac{\sin X}{3}=\frac{\sin Y}{4}=\frac{\sin Z}{5} \), then the greatest angle measure in the triangle equals \( \begin{array}{llll}\text { (a) } 60^{\circ} & \text { (b) } 75^{\circ} & \text { (c) } 90^{\circ} & \text { (d) } 120^{\circ}\end{array} \)

To solve this problem, we can use 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. In other words, for any triangle \( \triangle XYZ \), we have: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] where \( a, b, \) and \( c \) are the lengths of the sides opposite angles \( A, B, \) and \( C \), respectively. Given that \( \frac{\sin X}{3} = \frac{\sin Y}{4} = \frac{\sin Z}{5} \), we can set up the following equations: \[ \frac{\sin X}{3} = \frac{\sin Y}{4} \quad \text{and} \quad \frac{\sin X}{3} = \frac{\sin Z}{5} \] From these equations, we can find the ratios of the sides of the triangle: \[ \frac{a}{3} = \frac{b}{4} \quad \text{and} \quad \frac{a}{3} = \frac{c}{5} \] Let's denote the common value of these ratios as \( k \). Then we have: \[ a = 3k, \quad b = 4k, \quad \text{and} \quad c = 5k \] Now, we need to find the greatest angle measure in the triangle. According to the Law of Sines, the largest angle in a triangle is opposite the longest side. Therefore, angle \( Z \) is the largest angle because side \( c \) is the longest side. To find the measure of angle \( Z \), we can use the inverse sine function (sin^-1) to solve for \( \sin Z \): \[ \sin Z = \frac{c}{k} \cdot \frac{1}{5} = \frac{5k}{k} \cdot \frac{1}{5} = 1 \] The sine of an angle is 1 when the angle is \( 90^\circ \). Therefore, the greatest angle measure in the triangle is \( 90^\circ \). So the correct answer is: \[ \boxed{\text{(c) } 90^\circ} \]