Question

determine if triangle pqr and triangle stu are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)

Explanation:

Step1: Calculate the third angle of triangle \(PQR\)

The sum of angles in a triangle is \(180^{\circ}\). For \(\triangle PQR\), if \(\angle P = 62^{\circ}\) and \(\angle Q=58^{\circ}\), then \(\angle R=180^{\circ}-(62^{\circ} + 58^{\circ})=180^{\circ}-120^{\circ} = 60^{\circ}\).

Step2: Calculate the third angle of triangle \(STU\)

For \(\triangle STU\), if \(\angle S = 62^{\circ}\) and \(\angle U = 60^{\circ}\), then \(\angle T=180^{\circ}-(62^{\circ}+60^{\circ})=180^{\circ}-122^{\circ}=58^{\circ}\).

Step3: Check the AA (Angle - Angle) similarity criterion

In \(\triangle PQR\) and \(\triangle STU\), \(\angle P=\angle S = 62^{\circ}\), \(\angle Q=\angle T = 58^{\circ}\), and \(\angle R=\angle U=60^{\circ}\). By the AA (Angle - Angle) similarity criterion (if two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar), \(\triangle PQR\sim\triangle STU\).

Answer:

The triangles \(PQR\) and \(STU\) are similar. We know this because in \(\triangle PQR\), \(\angle R=180^{\circ}-(62^{\circ}+58^{\circ}) = 60^{\circ}\), and in \(\triangle STU\), \(\angle T=180^{\circ}-(62^{\circ}+60^{\circ})=58^{\circ}\). Since \(\angle P=\angle S\), \(\angle Q=\angle T\), and \(\angle R=\angle U\), by the AA (Angle - Angle) similarity criterion, \(\triangle PQR\sim\triangle STU\).