Question

find ( mangle r ).
if ( \triangle pqrcong\triangle stu ),

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle PQR\cong\triangle STU\), corresponding angles are equal. So \(\angle P=\angle S\), \(\angle Q=\angle T = 65^{\circ}\), \(\angle R=\angle U\).

Step2: Apply the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). For \(\triangle PQR\), we know that \(\angle P+\angle Q+\angle R = 180^{\circ}\). Given \(\angle P = 41^{\circ}\) and \(\angle Q=65^{\circ}\).
Substitute the values into the formula: \(41^{\circ}+65^{\circ}+\angle R=180^{\circ}\).

Step3: Solve for \(\angle R\)

First, add \(41^{\circ}\) and \(65^{\circ}\): \(41 + 65=106\). Then the equation becomes \(106^{\circ}+\angle R=180^{\circ}\).
Subtract \(106^{\circ}\) from both sides: \(\angle R=180^{\circ}- 106^{\circ}\).

Answer:

\(74^{\circ}\)