# In Fig. 6.9, OD i

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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