Question In \( \triangle Q R S, \overline{S Q} \cong \overline{R S} \) and \( \mathrm{m} \angle R=48^{\circ} \). Find \( \mathrm{m} \angle S \). Answer Attempt 2 out of 3 Submit Answer

In triangle \( \triangle QRS \), we know that \( \overline{SQ} \cong \overline{RS} \). This indicates that triangle \( QRS \) is isosceles with \( SQ = RS \). Given that \( \mathrm{m} \angle R = 48^{\circ} \), we can denote the angles as follows: - Let \( \mathrm{m} \angle S = x \). - Since \( SQ = RS \), we have \( \mathrm{m} \angle Q = x \) as well (the angles opposite the equal sides are equal). Now, we can use the fact that the sum of the angles in a triangle is \( 180^{\circ} \): \[ \mathrm{m} \angle Q + \mathrm{m} \angle R + \mathrm{m} \angle S = 180^{\circ} \] Substituting the known values: \[ x + 48^{\circ} + x = 180^{\circ} \] This simplifies to: \[ 2x + 48^{\circ} = 180^{\circ} \] Now, we can isolate \( x \): \[ 2x = 180^{\circ} - 48^{\circ} \] \[ 2x = 132^{\circ} \] \[ x = \frac{132^{\circ}}{2} \] \[ x = 66^{\circ} \] Thus, \( \mathrm{m} \angle S = 66^{\circ} \). The final answer is: \[ \mathrm{m} \angle S = 66^{\circ} \]