# Diagonals of a Square are Equal in Length and they Meet at Right Angles

Here we will prove that in a square, the diagonals are equal in length and they meet at right angles.

Given: PQRS is a square in which PQ = QR = RS = SP, and ∠QPS = ∠PQR = ∠QRS = ∠RSP = 90°.

To prove: PR = QS and PR ⊥ QS

Proof:

 Statement Reason 1. In ∆SPQ and ∆RQP,(i) SP = QR (i) Given (ii) PQ = PQ (ii) Common side (iii) ∠SPQ = ∠PQR (iii) Given (iv) ∆SPQ ≅ ∆RQPTherefore, QS = PR    (Proved) (iv) By SAS criterion of congruency. CPCTC. 2.(v) ∠PQS = ∠PSQ (v) In ∆PQS, PQ = PS (vi) ∠PQS + ∠PSQ = 90°. (vi) In ∆QPS, ∠QPS = 90° and sum of three angles of a triangle is 180°. (vii) ∠PQS = $$\frac{90°}{2}$$ = 45° (vii) By statements (v) and (vi). (viii) ∠QPR = 45° (viii) Similarly as (vi) and (vii) for the ∆PQR. (ix) ∠POQ = 180° - (PQO + ∠QPO)               = 180° - (45° + 45°)               = 180° - 90°                = 90°Therefore, OP ⊥ OQTherefore, ∠POQ = 90° Therefore, PR ⊥ QS       (Proved) (ix) By statements (vii), (viii) and the sum of the angles of ∆POQ is 180°.

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