Question

1. in the figure below, ∠pqs = ∠tqr, ∠ptq = ∠rsq, and tq = sq. which congruence relationship must be true? qs = ts pt = sr ts = pt qs = pq

Explanation:

Step1: Recall congruence - criteria

We know that if two triangles have two - angles and the included side equal, they are congruent (ASA - Angle - Side - Angle congruence criterion). Given \(\angle PQS=\angle TQR\), \(\angle PTQ = \angle RSQ\), and \(TQ = SQ\).

Step2: Determine congruent triangles

In \(\triangle PTQ\) and \(\triangle RSQ\), we have two angles and the included side equal. So, \(\triangle PTQ\cong\triangle RSQ\) by ASA.

Step3: Use congruent - triangle properties

If two triangles are congruent, their corresponding sides are equal. So, \(PT = SR\).

Answer:

\(PT = SR\)