Question

In fig OD is the bisector of angle AOC,OE is the bisector of angle BOC and OD is perpendicular to OE show that the points A,O and B are collinear​​
PlS answer

Answer 1

Solution :-

given that, OD is angle bisector of ∠AOC .

so,

→ ∠AOD = ∠DOC

then,

→ ∠AOC = ∠AOD + ∠DOC

→ ∠AOC = ∠DOC + ∠DOC

→ ∠AOC = 2∠DOC --------- Eqn.(1)

also, given that, OE is angle bisector of ∠COB .

so,

→ ∠BOE = ∠COE

then,

→ ∠COB = ∠BOE + ∠COE

→ ∠COB = ∠COE + ∠COE

→ ∠COB = 2∠COE --------- Eqn.(2)

also, given that, OD ⟂ OE .

→ ∠DOE = 90° ---------- Eqn.(3)

Now, adding Eqn. (1) and Eqn.(2) we get ,

→ ∠AOC + ∠COB = 2∠DOC + 2∠COE

→ ∠AOC +∠COB = 2(∠DOC + ∠COE)

→ ∠AOC + ∠COB = 2∠DOE

putting value from Eqn.(3) in RHS,

→ ∠AOC + ∠COB = 2 * 90°

→ ∠AOC + ∠COB = 180°

→ ∠AOB = 180°

therefore, we can conclude that, AOB is a straight line and

∠AOC and ∠COB are making linear pair angles .

Hence, we can conclude that, given points A , O and B are collinear .

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