Question

problem 24
given: rectangle qprs,
\\( \overline { q u } \cong \overline { s u } \\)
prove: \\( \overline { p v } \cong \overline { r t } \\)

1. transitive property

7.

1. \\( \angle p \cong \angle r \\)

Explanation:

Step1: Use properties of rectangle

In rectangle \( QPRS\), \( PQ = SR\) (opposite sides of a rectangle are equal). Also, \(\angle P=\angle R = 90^{\circ}\) (angles of a rectangle are right angles).

Step2: Use given congruence

Given \(\overline{QU}\cong\overline{SU}\), so \( \triangle QUS\) is isosceles. Then \(\angle UQS=\angle USQ\). Since \(PQ\parallel SR\) (property of a rectangle), \(\angle PQU=\angle SRU\) (alternate - interior angles).

Step3: Prove triangles congruent

In \(\triangle PVQ\) and \(\triangle RTS\), \(\angle P=\angle R\) (right angles), \(\angle PQU=\angle SRU\) (from above), \(PQ = SR\) (rectangle property). By \(AAS\) (Angle - Angle - Side) congruence criterion, \(\triangle PVQ\cong\triangle RTS\).

Step4: Use congruent triangle property

If \(\triangle PVQ\cong\triangle RTS\), then \(\overline{PV}\cong\overline{RT}\) (corresponding parts of congruent triangles are congruent).

Answer:

1. \(\triangle PVQ\cong\triangle RTS\)
2. \(\overline{PV}\cong\overline{RT}\)
3. All angles of a rectangle are right angles (so \(\angle P\) and \(\angle R\) are right angles and thus congruent)