Question

In Fig., OD is the bisector of ∠AOC, OE is the bisector of ∠BOC and OD ⊥ OE. Show that the points A, O and B are collinear.

Answer 1

According to the question,

In figure,

OD ⊥ OE,

OD and OE are the bisector of ∠AOC and ∠BOC.

To prove: Points A, O and B are collinear

i.e., AOB is a straight line.

Proof:

Since, OD and OE bisect angles ∠AOC and ∠BOC respectively.

∠AOC = 2∠DOC …(eq.1)

And ∠COB = 2∠COE …(eq.2)

Adding (eq.1) and (eq.2), we get

∠AOC = ∠COB = 2∠DOC + 2∠COE

∠AOC +∠COB = 2(∠DOC +∠COE)

∠AOC + ∠COB = 2∠DOE

Since, OD⊥OE

We get,

∠AOC +∠COB = 2×90^o

∠AOC +∠COB =180^o

∠AOB =180^o

So, ∠AOC + ∠COB are forming linear pair.

Therefore, AOB is a straight line.

Hence, points A, O and B are collinear.