Question

Given: Quadrilateral PQRS is a rectangle. Prove: PR = QS Reason Statement 1. Quadrilateral PQRS is a rectangle. given 2. Rectangle PQRS is a parallelogram. definition of a rectangle 3.QP ≅ RS QR ≅ PS 4. m∠QPS = m∠RSP = 90° definition of a rectangle 5. Δ PQS ≅ ΔSRP SAS criterion for congruence 6. PR ≅ QS Corresponding sides of congruent triangles are congruent. 7. PR = QS Congruent line segments have equal measures. What is the reason for the third step in this proof?

Answer 1

Since, the opposite sides of parallelogram are always congruent.

With using this property, the proof is mentioned below,

Given : Quadrilateral PQRS is a rectangle.

To Prove: PR = QS

Quadrilateral PQRS is a rectangle ( Given )

Rectangle PQRS is a parallelogram. ( Definition of a rectangle )

QP ≅ RS QR ≅ PS ( By the definition of parallelogram)

m∠QPS = m∠RSP = 90° ( Definition of a rectangle )

Δ PQS ≅ ΔSRP (SAS criterion for congruence )

PR ≅ QS ( Corresponding sides of congruent triangles are congruent)

PR = QS ( Congruent line segments have equal measures )

Hence proved.

Answer 2

Answer: opposite sides in rectangle are congruent.

Explanation:

It is given that Quadrilateral PQRS is a rectangle.

Since opposite sides of rectangle are congruent.

Therefore , QP ≅ RS, QR ≅ PS

Therefore, the reason for "QP ≅ RS, QR ≅ PS" in this proof is "opposite sides in rectangle are congruent."

hence, the reason for the third step in this proof is "opposite sides in rectangle are congruent."