Question

q is the midpoint of \\(\overline{pr}\\), \\(\overline{qs} \cong \overline{rt}\\), and \\(\overline{qt} \cong \overline{ps}\\). complete the proof that \\(\triangle pqs \cong \triangle qrt\\).
(image of two triangles sharing vertex q, with vertices t, r, q and s, p, q)

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

Explanation:

Step1: Identify triangle sides

We have three pairs of sides: \(\overline{PQ} \cong \overline{QR}\) (from midpoint), \(\overline{QS} \cong \overline{RT}\) (given), \(\overline{QT} \cong \overline{PS}\) (given).

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 \(\triangle QRT\), we have \(\overline{PQ} \cong \overline{QR}\), \(\overline{QS} \cong \overline{RT}\), and \(\overline{PS} \cong \overline{QT}\) (since \(\overline{QT} \cong \overline{PS}\) implies \(\overline{PS} \cong \overline{QT}\) by the symmetric property of congruence). So by SSS, \(\triangle PQS \cong \triangle QRT\).

Answer:

SSS (Side - Side - Side) Congruence Postulate