Question

describe the error in finding ( mangle s ).
given ( \triangle qrscong\triangle xzy )

Explanation:

Step1: Analyze the congruence of triangles

When two triangles are congruent, their corresponding parts are congruent. In \(\triangle QRS\) and \(\triangle XZY\), if \(\triangle QRS\cong\triangle XZY\), then \(\angle S\) corresponds to \(\angle Y\) (not \(\angle Z\)).

Step2: Use the Triangle Sum Theorem

In a right - triangle (both \(\triangle QRS\) and \(\triangle XZY\) are right - triangles as they have a right angle). The sum of the interior angles of a triangle is \(180^{\circ}\). For \(\triangle XZY\), \(\angle X = 90^{\circ}\), \(\angle Z=42^{\circ}\), then \(\angle Y=180^{\circ}-\angle X - \angle Z=180^{\circ}-90^{\circ}-42^{\circ} = 48^{\circ}\). Since \(\angle S\) corresponds to \(\angle Y\) (because of the congruence \(\triangle QRS\cong\triangle XZY\)), the error is that \(\angle S\) was wrongly matched with \(\angle Z\) instead of the correct corresponding angle \(\angle Y\).

Answer:

The error is that when using the congruence \(\triangle QRS\cong\triangle XZY\), \(\angle S\) was incorrectly matched with \(\angle Z\). In congruent triangles \(\triangle QRS\) and \(\triangle XZY\), \(\angle S\) corresponds to \(\angle Y\). Using the Triangle Sum Theorem in \(\triangle XZY\) (\(\angle X = 90^{\circ}\), \(\angle Z = 42^{\circ}\)), \(\angle Y=180^{\circ}-90^{\circ}-42^{\circ}=48^{\circ}\), so \(m\angle S = 48^{\circ}\) (not \(42^{\circ}\) as wrongly assumed by matching \(\angle S\) with \(\angle Z\)).