Q4 of 33 Page 108

The two diagonals of a parallelogram ABCD intersect each other at the point O. A straight line passing through O intersects the sides AB and DC at the points P and Q respectively. Let us prove that OP = OQ.

Figure according to given information:



Consider ΔPOB and ΔQOD


∠POB = ∠QOD … vertically opposite angles


As AB || DC line k is the transversal


∠BPO = ∠DQO … alternate interior angles


OD = OB … in a parallelogram diagonals bisect each other


Therefore, ΔPOB ≅ ΔQOD … AAS test for congruency


⇒ PO = QO … corresponding sides of congruent triangles


Hence proved


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