Question

1. given \\( \overline { q r } \cong \overline { s t }, \overline { p q } \cong \overline { q s }, \overline { q r } \parallel \overline { s t } \\) prove: \\( \triangle p q r \cong \triangle q s t \\) \\( \overline { q r } \cong \overline { s t } \\) given \\( \overline { p q } \cong \overline { q s } \\) given \\( \overline { q r } \parallel \overline { s t } \\) given \\( \angle r q p \cong \angle q t s \\) alt. interior

Explanation:

Step1: Use the property of parallel lines

Since \( \overline{QR}\parallel\overline{ST}\), by the alternate - interior angles theorem, \( \angle RQP=\angle QTS\) (alternate interior angles are congruent when two parallel lines are cut by a transversal).

Step2: Apply the Side - Angle - Side (SAS) congruence criterion

We are given that \( \overline{QR}\cong\overline{ST}\), \( \overline{PQ}\cong\overline{QS}\), and we just proved that \( \angle RQP\cong\angle QTS\).
In \(\triangle PQR\) and \(\triangle QST\), we have:

• \(PQ = QS\) (given)
• \(\angle RQP=\angle QTS\) (proved from parallel lines)
• \(QR = ST\) (given)

By the SAS (Side - Angle - Side) congruence criterion, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Answer:

\(\triangle PQR\cong\triangle QST\) by the SAS (Side - Angle - Side) congruence criterion.