Mathematics
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
27 Likes
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.
Answered By
19 Likes
Related Questions
ABCD is a square. A is joined to a point P on BC and D is joined to a point Q on AB. If AP = DQ, prove that AP and DQ are perpendicular to each other.
Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
P and Q are points on opposite sides AD and BC of a parallelogram ABCD such that PQ passes through the point of intersection O of its diagonals AC and BD. Show that PQ is bisected at O.
In figure (2) given below, points P and Q have been taken on opposite sides AB and CD respectively of a parallelogram ABCD such that AP = CQ. Show that AC and PQ bisect each other.
