Question

given: \\( \angle tqr \cong \angle rst \\), \\( \overline{qr} \parallel \overline{ts} \\) prove: \\( \triangle qrt \cong \triangle str \\)

Explanation:

Step1: Find alternate interior angles

Since \( \overline{QR}\parallel\overline{TS}\), by the Alternate Interior Angles Theorem, \( \angle QRT=\angle STR\).

Step2: Check for common side

\( RT = TR\) (common side in \( \triangle QRT\) and \( \triangle STR\)).

Step3: Use AAS congruence criterion

We are given \( \angle TQR\cong\angle RST\), from Step1 \( \angle QRT=\angle STR\), and from Step2 \( RT = TR\). By the Angle - Angle - Side (AAS) congruence criterion, \( \triangle QRT\cong\triangle STR\).

Answer:

\(\triangle QRT\cong\triangle STR\) (by AAS congruence criterion)