Question

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1. can you determine whether △pqr and △tsr are similar? explain.

Explanation:

Step1: Analyze angles

In \(\triangle PQR\) and \(\triangle TSR\), \(\angle PQR=\angle TSR = 90^{\circ}\) (right angles). Also, \(\angle PRQ+\angle TRS = 180^{\circ}\) (linear - pair of angles), so \(\angle PRQ=\angle RTS\) (since \(\angle TRS + \angle RTS=90^{\circ}\) in \(\triangle TSR\) and \(\angle PRQ+\angle RTS = 90^{\circ}\) in \(\triangle PQR\)).

Step2: Apply 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. Here, in \(\triangle PQR\) and \(\triangle TSR\), \(\angle PQR=\angle TSR\) and \(\angle PRQ=\angle RTS\).

Answer:

Yes, \(\triangle PQR\sim\triangle TSR\) by the AA (Angle - Angle) similarity criterion.