Q. 13.8( 20 Votes )

In Fig. 6.9, OD i

Answer :

From the figure, it is given to us


ODOE


DOE = 90° - - - - (i)


OD is the bisector of AOC.


OE is the bisector of BOC.


We have to show that the points A, O and B are collinear, i.e.,


To show that AOB is a straight line. - - - - (ii)


OD is the bisector of AOC


AOD = COD


AOC = 2 × COD - - - - (ii)
Similarly, OE is the bisector of
BOC


BOE = COE


BOC = 2 × COE - - - - (iii)


Adding equation (ii) and equation (iii),


AOC + BOC = 2 × COD + 2 × COE


AOC + BOC = 2 × (COD + COE)


From the figure, we can see that


COD + COE = DOE - - - - (iv)


Substituting equation (iv) in the above equation,


AOC + BOC = 2 × (COD + COE)


AOC + BOC = 2 × DOE


AOC + BOC = 2 × 90° (From equation (i), we have DOE = 90°)


AOC + BOC = 180°


AOB = 180° (From the figure, AOC + BOC = AOB) - - - - (iv)


From (iv), we can say that


AOC and BOC are forming linear pair of angles.


Since, AOC and BOC are two adjacent angles and their sum is 180°, where OC is a ray standing on AOB,


It is true that A, O and B are collinear, thus making AOB a straight line.


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