Question

△pqs is equilateral, \\( \overline { pt } \cong \overline { qr } \\), and \\( \angle spt \cong \angle rqs \\). complete the proof that \\( \triangle qrs \cong \triangle pts \\).

statement
1 △pqs is equilateral
2 \\( \overline { pt } \cong \overline { qr } \\)
3 \\( \angle spt \cong \angle rqs \\)
4 \\( \overline { ps } \cong \overline { qs } \\)
5 \\( \triangle qrs \cong \triangle pts \\)

reason
given
given
given
sas

Explanation:

Step1: Properties of equilateral triangle

Since \(\triangle PQS\) is equilateral, all its sides are equal. So, \(PS = QS\) (by the definition of an equilateral triangle).

Step2: Using SAS (Side - Angle - Side) criterion

We are given \(\overline{PT}\cong\overline{QR}\), \(\angle SPT\cong\angle RQS\), and from step 1 \(PS = QS\).
In \(\triangle QRS\) and \(\triangle PTS\), we have:

• \(QR = PT\) (given)
• \(\angle RQS=\angle SPT\) (given)
• \(QS = PS\) (from the equilateral triangle \(\triangle PQS\))

By the Side - Angle - Side (SAS) congruence criterion, \(\triangle QRS\cong\triangle PTS\)

Answer:

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