# 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 ≅ ∆RSPTherefore, 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

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