Question

given \\( \triangle qrs \cong \triangle xzy \\)\\( \angle s \cong \angle z \\)\\( m\angle s=m\angle z \\)\\( m\angle s = 42 ^ { \circ } \\)describe the error in finding \\( m\angle s \\).

Explanation:

Step1: Analyze the congruent triangles

When \(\triangle QRS\cong\triangle XZY\), the corresponding angles are equal. That is \(\angle Q=\angle X = 90^{\circ}\), \(\angle R=\angle Y\), \(\angle S=\angle Z\).

Step2: Identify the error

The error is that the wrong pair of angles was used. \(\angle S\) is congruent to \(\angle Z\), but in the right - angled triangle \( \triangle XZY\), the angle of \(42^{\circ}\) is not \(\angle Z\). In a right - angled triangle, the sum of the two non - right angles is \(90^{\circ}\). If one non - right angle is \(42^{\circ}\), then the other non - right angle (the one corresponding to \(\angle S\)) is \(90^{\circ}- 42^{\circ}=48^{\circ}\).

Answer:

The error is using the non - corresponding angle. \(\angle S\cong\angle Z\), but in \(\triangle XZY\), \(m\angle Z = 90^{\circ}-42^{\circ}=48^{\circ}\), not \(42^{\circ}\) as wrongly assumed.