Given: PQ || UR, QT || RS, QT ≅ RS. Evidence: Rule: Then: ΔRSU ≅ ________

Understand the Problem

The question requires providing logical evidence and rules to demonstrate the congruence of triangles based on given conditions and a geometric diagram.

Answer

$\triangle RUS \cong \triangle QUT$

Steps to Solve

Identify Given Information

The given information states that:

$PQ \parallel UR$
$QT \parallel RS$
$QT \cong RS$

Establish Angle Relationships

Since $PQ \parallel UR$ and lines are cut by transversal $QU$, alternate interior angles are formed. Thus, we have:

$\angle PQU \cong \angle RUS$

Apply Parallel Line Theorems

From $QT \parallel RS$, the alternate interior angles formed are also equal. Thus, we have:

$\angle QTU \cong \angle RUS$

Conclude with Side-Side-Side (SSS) Congruence

We have shown:

$QT \cong RS$
$\angle PQU \cong \angle RUS$
$\angle QTU \cong \angle RUS$

This establishes the corresponding sides and angles are congruent.

Use the Rule for Triangle Congruence

Since the triangles have equal sides and angles, the triangles $\triangle QUT$ and $\triangle RUS$ are congruent by the Side-Angle-Side (SAS) postulate.

$\triangle RUS \cong \triangle QUT$

More Information

The congruence of triangles allows us to deduce that if two triangles are congruent, all their corresponding sides and angles are equal. This principle is crucial in geometry and can be used in various proofs.

Tips

Failing to identify corresponding angles correctly when dealing with parallel lines.
Using the wrong congruence rule (e.g., using SSS when SAS should have been applied).
Mislabeling the triangles during congruence proofs.

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