In a Rectangle the Diagonals are of Equal Lengths

Here we will prove that in a rectangle the diagonals are of equal lengths.

Given: PQRS is rectangle in which PQ ∥ SR, PS ∥ QR and ∠PQR = ∠QRP = ∠RSP = ∠SPQ = 90°.

To prove: The diagonals PR and QS are equal.

Proof:

            Statement

            Reason

In ∆PQR and ∆RSP,

1. ∠QPR = ∠SRP


1. PQ ∥ SR and PR is a transversal.

2. ∠QRP = ∠SPR

2. PS ∥ QR and PR is a transversal.

3. PR = PR

3. Common side.

4. ∆PQR ≅ ∆RSP

Therefore, PQ = RS and QR = SP

4. By AAS criterion of congruency. CPCTC

In ∆SPQ and ∆RQP,

5. SP = QR, PQ = PQ and ∠SPQ = ∠RQP


5. From Statement 4 and given.

6. ∆SPQ ≅ ∆RQP,

Therefore, QS = PR. (Proved)

6. By SAS criterion of congruency. CPCTC






9th Grade Math

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