Question

In the picture below, O is the centre of the circle and the line OD is parallel to the line CA.

Prove that OD bisects ∠AOB.

Answer 1

Given: CA is parallel to OD.

CB acts as the transversal.

∴ ∠ACO = ∠DOB (Corresponding angles) … (1)

We know that angle made by an arc at any point on the alternate arc is equal to half the angle made at the centre.

∴ ∠AOB = 2∠ACB

Using equation (1):

∠AOB = 2∠DOB

⇒ OD acts as the bisector of ∠AOB.

Therefore, OD bisects ∠AOB.

Yes, we can use this concept to draw the bisector of a given angle.

Suppose, in the given figure, we are given only ∠AOB and CB as the diametre of the circle.

Let us join one end point, C of the diametre CB with point A of ∠AOB.

We get that 2∠ACB = ∠AOB.

If we draw a line segment passing through point O and parallel to AC such that it intersects the circle at point D, then OD acts as the bisector of ∠AOB.