Question

given: rectangle qprs,
\overline{qu} \cong \overline{su}
prove: \overline{pv} \cong \overline{rt}
3.

1. \angle pvq \cong \angle uqs,

\angle rts \cong \angle usq

1. \angle rts \cong \angle uqs

Explanation:

Step1: Identify parallel sides

Since QPRS is a rectangle (and thus a parallelogram), \(QP\parallel RS\).

Step2: Use alternate - interior angles

When two parallel lines \(QP\) and \(RS\) are cut by a transversal \(QS\), \(\angle PVQ\cong\angle UQS\) (alternate - interior angles) and \(\angle RTS\cong\angle USQ\) (alternate - interior angles). This is because of the Alternate - Interior Angles Theorem.

Step3: Use substitution

Since \(\angle PVQ\cong\angle UQS\) and \(\angle RTS\cong\angle USQ\), and we know that in \(\triangle VUQ\) and \(\triangle TUS\), \(\overline{QU}\cong\overline{SU}\) (given). By the Angle - Angle - Side (AAS) congruence criterion (if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, the triangles are congruent). But for the angle substitution \(\angle RTS\cong\angle UQS\) (substitution property of congruence: if \(a = b\) and \(b = c\), then \(a = c\)).

Answer:

1. \(QP\parallel RS\)
2. Alternate - Interior Angles Theorem
3. Substitution property of congruence