Q9 of 16 Page 88

O is any point inside two parallel lines AB and CD. OP and OQ are two perpendiculars on AB and CD respectively. Let’s prove that P,O and Q are collinear.


Given that AB||CD and O is a point inside AB and CD.


Also, P and Q are two points on AB and CD respectively such that


OPꞱAB and OQꞱCD.


So, ∠OPA=∠OPB=90°


And ∠OQC=∠OQD=90°


Draw XY through O such that AB||XY||CD


Since, AB||XY


So, ∠YOQ=∠BPO=90° (corresponding angles)


And ∠YOP=∠DQO=90° (corresponding angles)


∴ ∠POQ=∠YOP+∠YOQ=90°+90°=180°


So, PQ is s straight line.


∴ P,O and Q are collinear.


Proved.


More from this chapter

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7

In the figure beside AB||CD; let’s write the measurement of ∠PQR.


 8

In the figure beside PQ||RS, ∠BPQ=40°, ∠BPR=155° and ∠CRS=70°; let’s write the measurement of all the angles of APR


 10

Each pair of arms of two angles is parallel to each other. Let’s prove either these angles are equal in measurement or they are supplementary.

 11

The diagonal AC of the parallelogram ABCD bisects ∠BAD. Let’s prove that the diagonal AC also bisects ∠BCD.