\begin{tabular}{l} Question \\ In \( \triangle W X Y, \overline{X Y} \cong \overline{W X} \) and \( \mathrm{m} \angle W=61^{\circ} \). Find \( \mathrm{m} \angle X \) \\ Answer Attempt 2 out of 3 \\ \hline\end{tabular}

To solve the problem, we need to analyze the given triangle \( \triangle WXY \) with the following known conditions: 1. \( \overline{XY} \cong \overline{WX} \) (which means that sides \( XY \) and \( WX \) are equal in length). 2. \( \mathrm{m} \angle W = 61^{\circ} \). Since \( \overline{XY} \cong \overline{WX} \), triangle \( WXY \) is an isosceles triangle with \( WX = XY \). In an isosceles triangle, the angles opposite the equal sides are also equal. Therefore, we can denote: - \( \mathrm{m} \angle X = \mathrm{m} \angle Y \). Let \( \mathrm{m} \angle X = \mathrm{m} \angle Y = x \). Now, we can use the fact that the sum of the angles in any triangle is \( 180^{\circ} \): \[ \mathrm{m} \angle W + \mathrm{m} \angle X + \mathrm{m} \angle Y = 180^{\circ} \] Substituting the known values: \[ 61^{\circ} + x + x = 180^{\circ} \] This simplifies to: \[ 61^{\circ} + 2x = 180^{\circ} \] Now, we can isolate \( x \): \[ 2x = 180^{\circ} - 61^{\circ} \] Calculating the right side: \[ 2x = 119^{\circ} \] Now, divide both sides by 2 to find \( x \): \[ x = \frac{119^{\circ}}{2} = 59.5^{\circ} \] Thus, the measure of angle \( X \) is: \[ \mathrm{m} \angle X = 59.5^{\circ} \] Therefore, the final answer is: \[ \mathrm{m} \angle X = 59.5^{\circ} \]