Question

problem 24
given: rectangle qprs,
\\( \overline { q u } \cong \overline { s u } \\)
prove: \\( \overline { p v } \cong \overline { r t } \\)
9.
9.

1. \\( \triangle p v q \cong \triangle r t s \\)

10.

1. \\( \overline { p v } \cong \overline { r t } \\)

11.

Explanation:

Step1: Identify properties of rectangle

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

Step2: Use given \( \overline{QU}\cong\overline{SU}\)

Since \( \overline{QU}\cong\overline{SU}\), \( \angle UQS=\angle USQ\). And \( \angle PQV = 90^{\circ}-\angle UQS\), \( \angle RST=90^{\circ}-\angle USQ\), so \( \angle PQV=\angle RST\).

Step3: Prove \( \triangle PVQ\cong\triangle RTS\) (AAS)

We have \( \angle P=\angle R\), \( \angle PQV=\angle RST\), \(PQ = RS\). By the Angle - Angle - Side (AAS) congruence criterion, \( \triangle PVQ\cong\triangle RTS\).

Step4: Conclude \( \overline{PV}\cong\overline{RT}\)

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

Answer:

1. \( \angle P=\angle R = 90^{\circ}, PQ = RS,\angle PQV=\angle RST\) (using rectangle properties and angle - subtraction from right angles due to \( \overline{QU}\cong\overline{SU}\))
2. \( \triangle PVQ\cong\triangle RTS\) (AAS: \( \angle P=\angle R\), \( \angle PQV=\angle RST\), \(PQ = RS\))
3. \( \overline{PV}\cong\overline{RT}\) (Corresponding Parts of Congruent Triangles are Congruent - CPCTC)