Q4 of 13 Page 96

E and F are twp points on two parallel straight-line AB and CD respectively. O is the midpoint of line segment EF. We draw a straight line passing through O which intersect AB and CD at P and Q respectively. Let’s prove that O bisects the line segment PQ.


E and F are two points on straight lines AB and CD respectively.


Also, O is the midpoint of EF and a straight line is drawn passing through O which intersect AB and CD at P and Q respectively.


OE=OF (O is the midpoint)


AB||CD


∠PEO=∠QFO (alternate interior angles) …………… (1)


∠POE=∠QOF (vertically opposite angles) …………. (2)


Now, in ΔPOE and ΔQOF,


OE=OF (O is the midpoint)


∠PEO=∠QFO (from (1))


∠POE=∠QOF (from (2))


∴ ΔPOE ≅ ΔQOF (by ASA rule)


So, OP=OQ (by cpct)


Hence, O bisects PQ


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