In the figure, given below, triangle ABC is right-angled at B. ABPQ and ACRS are squares. Prove that :

(i) △ ACQ and △ ASB are congruent.

(ii) CQ = BS.

From figure,

⇒ ∠QAC = ∠QAB + ∠BAC = 90° + ∠BAC.

⇒ ∠BAS = ∠CAS + ∠BAC = 90° + ∠BAC.

∴ ∠QAC = ∠BAS.

In △ QAC and △ BAS,

⇒ QA = AB (Since, ABPQ is a square)

⇒ ∠QAC = ∠BAS (Proved above)

⇒ AC = AS (Since, ACRS is a square)

∴ △ QAC ≅ △ BAS (By S.A.S. axiom)

Hence, proved that △ ACQ and △ ASB are congruent.

(ii) Since, △ QAC ≅ △ BAS

We know that,

Corresponding parts of congruent triangles are equal.

∴ CQ = BS.

Hence, proved that CQ = BS.