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.

Answer 1

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. ]