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In figure (1) given below, ABCD is a parallelogram and X is mid-point of BC. The line AX produced meets DC produced at Q. The parallelogram ABPQ is completed. Prove that

(i) the triangles ABX and QCX are congruent.

(ii) DC = CQ = QP

In figure (1) given below, ABCD is a parallelogram and X is mid-point of BC. The line AX produced meets DC produced at Q. The parallelogram ABPQ is completed. Prove that (i) the triangles ABX and QCX are congruent (ii) DC = CQ = QP. Rectilinear Figures, ML Aggarwal Understanding Mathematics Solutions ICSE Class 9.

Rectilinear Figures

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Answer

(i) Considering △ABX and △QCX we have,

⇒ ∠XAB = ∠XQC (Alternate angles are equal)

⇒ XB = XC (As X is mid-point of BC)

⇒ ∠AXB = ∠CXQ (Vertically opposite angles are equal)

Hence, △ABX ≅ △QCX by ASA axiom.

(ii) Since, △ABX ≅ △QCX

∴ AB = CQ (By C.P.C.T.) ……….(i)

AB = CD and AB = QP (Opposite sides of parallelogram are equal) ………(ii)

From (i) and (ii) we get,

⇒ AB = DC = CQ = QP

⇒ DC = CQ = QP

Hence, proved that DC = CQ = QP.

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