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 |
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