Question

rectangle pqrs has diagonals pr and qs. move options into the table to prove that diagonals pr and qs are congruent. statement: pqrs is a rectangle. reason: given. pr ≅ pr (reflexive property). ∠psr ≅ ∠qrs (interior angles of rectangles are congruent). ∠psr ≅ ∠rqp (interior angles of rectangles are congruent). pq ≅ rs (opposite sides of a rectangle are congruent). ps ≅ qr. pr ≅ qs. corresponding parts of congruent triangles are congruent.

Explanation:

Step1: Identify given information

Rectangle \(PQRS\) is given. In a rectangle, opposite sides are congruent, so \(PQ\cong RS\) and \(PS\cong QR\) (by the property that opposite sides of a rectangle are congruent). Also, \(\angle PSR\cong\angle RQP\) and \(\angle PSR\cong\angle QRS\) (interior angles of rectangles are congruent).

Step2: Consider the triangles

In \(\triangle PSR\) and \(\triangle QRS\), we have \(PS = QR\), \(SR=RS\) (by the reflexive property, a segment is congruent to itself), and \(\angle PSR=\angle QRS\).

Step3: Prove triangle - congruence

By the Side - Angle - Side (SAS) congruence criterion, \(\triangle PSR\cong\triangle QRS\).

Step4: Use corresponding parts of congruent triangles

Since \(\triangle PSR\cong\triangle QRS\), then the corresponding parts are congruent. In particular, the diagonals \(PR\) and \(QS\) (which are corresponding parts of \(\triangle PSR\) and \(\triangle QRS\)) are congruent, i.e., \(PR\cong QS\) (by corresponding parts of congruent triangles are congruent).

Statement	Reason
\(PQ\cong RS\), \(PS\cong QR\)	Opposite sides of a rectangle are congruent
\(\angle PSR\cong\angle QRS\), \(\angle PSR\cong\angle RQP\)	Interior angles of rectangles are congruent
\(PR\cong PR\), \(QS\cong QS\)	Reflexive property
\(\triangle PSR\cong\triangle QRS\)	SAS ( \(PS = QR\), \(\angle PSR=\angle QRS\), \(SR = RS\))
\(PR\cong QS\)	Corresponding parts of congruent triangles are congruent

Answer:

The diagonals \(PR\) and \(QS\) of rectangle \(PQRS\) are congruent because we can show that \(\triangle PSR\cong\triangle QRS\) using the properties of rectangles and the SAS congruence criterion, and then use the fact that corresponding parts of congruent triangles are congruent.