Question

given: qr ⊥ pt and ∠qpr ≅ ∠str
prove: △pqr ~ △tsr

Explanation:

Step1: Identify right angles

Since \(QR\perp PT\), by the definition of perpendicular, \(\angle PRQ = 90^{\circ}\) and \(\angle TRS=90^{\circ}\). So \(\angle PRQ\cong\angle TRS\) (all right angles are congruent).

Step2: Use AA similarity theorem

We are given that \(\angle QPR\cong\angle STR\). In \(\triangle PQR\) and \(\triangle TSR\), we have two pairs of congruent angles (\(\angle QPR\cong\angle STR\) and \(\angle PRQ\cong\angle TRS\)). By the AA (Angle - Angle) similarity theorem, \(\triangle PQR\sim\triangle TSR\).

Answer:

\(\triangle PQR\sim\triangle TSR\) by AA similarity theorem (because \(\angle QPR\cong\angle STR\) (given) and \(\angle PRQ\cong\angle TRS\) (right angles from \(QR\perp PT\))).