Question

In the following figure, 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

Given In the following figure, OD ⊥ OE, OD and OE are the bisectors of ∠AOC and ∠BOC.

To show 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  ...(i)

And ∠COB = 2∠COE  ...(ii)

On adding equations (i) and (ii), we get

∠AOC + ∠COB = 2∠DOC + 2∠COE

⇒ ∠AOC + ∠COB = 2(∠DOC + ∠COE)

⇒ ∠AOC + ∠COB = 2∠DOE

⇒ ∠AOC + ∠COB = 2 × 90°  ...[∴ OD ⊥ OE]

⇒ ∠AOC + ∠COB = 180°

∴ ∠AOB = 180°

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

Also, AOB is a straight line.

Hence, points A, O and B are collinear.