Question

he given information to prove that $\triangle rst\cong\triangle tur$.
given: $\overline{st}\parallel\overline{ur}$
$\overline{tu}\parallel\overline{rs}$
prove: $\triangle rst\cong\triangle tur$

Explanation:

Step1: Identify alternate interior angles

Since \(ST\parallel UR\), \(\angle RST=\angle TUR\) (alternate - interior angles).
Since \(TU\parallel RS\), \(\angle URT = \angle STR\) (alternate - interior angles).

Step2: Use the ASA (Angle - Side - Angle) criterion

We know that \(RT = TR\) (common side).
In \(\triangle RST\) and \(\triangle TUR\):
\(\angle RST=\angle TUR\) (from parallel lines \(ST\parallel UR\)), \(RT = TR\) (common side), \(\angle URT=\angle STR\) (from parallel lines \(TU\parallel RS\)).
By the ASA (Angle - Side - Angle) congruence criterion, \(\triangle RST\cong\triangle TUR\).

Answer:

\(\triangle RST\cong\triangle TUR\) by the ASA (Angle - Side - Angle) congruence criterion.