Question

use the 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 $\overline{ST} \parallel \overline{UR}$ and $\overline{RT}$ is a transversal, $\angle STR \cong \angle URT$ (Alternate Interior Angles Theorem).

Step2: Identify Another Pair of Angles

Since $\overline{TU} \parallel \overline{RS}$ and $\overline{RT}$ is a transversal, $\angle SRT \cong \angle UTR$ (Alternate Interior Angles Theorem).

Step3: Identify Common Side

$\overline{RT} \cong \overline{RT}$ (Reflexive Property of Congruence).

Step4: Apply ASA Congruence

By the Angle - Side - Angle (ASA) Congruence Postulate, $\triangle RST \cong \triangle TUR$ because we have two pairs of congruent angles and the included side congruent.

Answer:

$\triangle RST \cong \triangle TUR$ is proven using the ASA Congruence Postulate (after showing alternate interior angles congruent and the common side congruent).