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Question
In the figure, given below, triangle ABC is right-angled at B. ABPQ and ACRS are squares. 
Prove that:
(i) ΔACQ and ΔASB are congruent.
(ii) CQ = BS.
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Solution
Given: A(Δ ABC) is right-angled at B.
ABPQ and ACRS are squares
To Prove:
(i) ΔACQ ≅ ΔASB
(ii) CQ = BS
Proof:
(i)
∠ QAB = 90° ...[ ABPQ is a square ] ...(1)
∠ CAS = 90° ...[ ACRS is a square ] ...(2)
From (1) and (2) , We have
∠ QAB = ∠CAS ...(3)
Adding ∠BAC to both sides of (3), We have
∠ QAB + ∠BAC = ∠CAS+ ∠BAC
⇒ ∠QAC = ∠BAS ...(4)
In ΔACQ ≅ ΔASB, (by SAS)
QA = AB ...[ Sides of a square ABPQ ]
∠QAC = ∠SAB ...[ From(4) ]
AC = AS ...[ sides of a square ACRS ]
∴ By Side -Angle-Side criterion of congruence,
ΔACQ ≅ ΔASB
(ii)
The corresponding parts of the congruent triangles are congruent,
∴ CQ = SB ...[ c.p.c.t. ]
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