Question

given: \\( \overline{pq}\cong\overline{st},\overline{qr}\cong\overline{tr},r \\) is the midpoint of \\( \overline{ps} \\) prove: \\( \triangle pqr\cong\triangle str \\)

Explanation:

Step1: Use the mid - point definition

Since \(R\) is the mid - point of \(\overline{PS}\), by the mid - point definition, \(PR = SR\), so \(\overline{PR}\cong\overline{SR}\) (Reason: Definition of mid - point).

Step2: Use the SSS (Side - Side - Side) congruence criterion

We are given \(\overline{PQ}\cong\overline{ST}\), \(\overline{QR}\cong\overline{TR}\), and from Step 1 \(\overline{PR}\cong\overline{SR}\).
In \(\triangle PQR\) and \(\triangle STR\), we have:

• \(PQ = ST\) (given)
• \(QR = TR\) (given)
• \(PR = SR\) (from mid - point property)

By the SSS congruence criterion (\(SSS\): If three sides of one triangle are equal to three sides of another triangle, then the two triangles are congruent), \(\triangle PQR\cong\triangle STR\).

Answer:

The proof is completed as above.