Question

statementreason
1t is the midpoint of \\( \overline { p r } \\)given
2t is the midpoint of \\( \overline { q s } \\)given
3\\( \overline { p t } \cong \overline { r t } \\)definition of midpoint
4\\( \overline { q t } \cong \overline { s t } \\)definition of midpoint
5\\( \angle p t s \cong \angle q t r \\)vertical angle theorem
6

Explanation:

Step1: Recall the SAS (Side - Angle - Side) congruence criterion

Two triangles are congruent if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle.

Step2: Identify the triangles and their corresponding parts

In \(\triangle PTS\) and \(\triangle QTR\):

• \(\overline{PT}\cong\overline{RT}\) (from statement 3)
• \(\angle PTS\cong\angle QTR\) (from statement 5)
• \(\overline{ST}\cong\overline{QT}\) (from statement 4)

Answer:

\(\triangle PTS\cong\triangle QTR\) by the SAS (Side - Angle - Side) congruence criterion.