Question

△qst and △qpr are shown below. which statement is true? △qst is similar to △qpr. △qst is not similar to △qpr. there is not enough information to determine whether the triangles are similar.

Explanation:

Step1: Recall the AA (Angle - Angle) similarity criterion

Two triangles are similar if two pairs of corresponding angles are congruent.

Step2: Analyze the angles in \(\triangle QST\) and \(\triangle QPR\)

In \(\triangle QST\) and \(\triangle QPR\), \(\angle Q\) is common (one pair of congruent angles). Also, \(\angle QST+\angle TSP = 90^{\circ}\) and \(\angle QPR = 90^{\circ}\). Since \(\angle QTS+\angle QST = 90^{\circ}\) (because the sum of angles in \(\triangle QST\) is \(180^{\circ}\) and \(\angle Q\) is common), and \(\angle QPR = 90^{\circ}\), we can find that \(\angle QTS=\angle QSP\) (by angle - angle relationships in right - angled and non - right - angled parts of the figure). So, two pairs of angles are congruent.

Answer:

\(\triangle QST\) is similar to \(\triangle QPR\).