Question

q is the midpoint of \\( \overline{pr} \\). complete the proof that \\( \triangle qrt \cong \triangle pqs \\).

statement	reason
2	\\( \overline{qt} \cong \overline{ps} \\)	given
3	\\( \overline{rt} \cong \overline{qs} \\)	given
4	\\( \overline{pq} \cong \overline{qr} \\)
5	\\( \triangle qrt \cong \triangle pqs \\)

Explanation:

Step1: Midpoint definition

Since \(Q\) is the midpoint of \(\overline{PR}\), by the definition of a midpoint, \(PQ = QR\), so \(\overline{PQ}\cong\overline{QR}\).

Step2: SSS congruence criterion

We have \(\overline{QT}\cong\overline{PS}\), \(\overline{RT}\cong\overline{QS}\), and \(\overline{PQ}\cong\overline{QR}\). By the Side - Side - Side (SSS) congruence criterion, if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. So \(\triangle QRT\cong\triangle PQS\).

Answer:

1. Reason: Definition of midpoint
2. Reason: SSS (Side - Side - Side) congruence criterion