Question

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

	statement	reason
1	q is the midpoint of \\(\overline{pr}\\)	given
2	\\(\overline{rt} \cong \overline{ps}\\)	given
3	\\(\overline{qs} \cong \overline{qt}\\)	given
4	\\(\overline{pq} \cong \overline{qr}\\)	definition of midpoint
5	\\(\triangle pqs \cong \triangle rqt\\)

Explanation:

Step1: Identify triangle sides

We have three pairs of congruent sides: $\overline{RT} \cong \overline{PS}$ (Step 2), $\overline{QS} \cong \overline{QT}$ (Step 3), and $\overline{PQ} \cong \overline{QR}$ (Step 4, from midpoint definition).

Step2: Apply SSS congruence

The Side - Side - Side (SSS) Congruence Postulate states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. For $\triangle PQS$ and $\triangle RQT$, we have $\overline{PQ} \cong \overline{QR}$, $\overline{QS} \cong \overline{QT}$, and $\overline{PS} \cong \overline{RT}$. So by SSS, $\triangle PQS \cong \triangle RQT$.

Answer:

SSS (Side - Side - Side) Congruence Postulate