Question

in the given figure ray OC is the bisector of angle AOB and OD is the ray opposite to OC. Show that angle AOD =angle BOD ​

Answer 1

Concept:

The sum of all angles on a line is equal to 180 degrees. Also, the sum of all angles around a point is 360 degrees.

The bisector of an angles divides the given angle in two equal parts.

Given:

OC is the bisector of angles AOB.

To prove:

angle AOD =angle BOD ​

Solution:

We know that,

The sum of all angles on a line is equal to 180 degrees.

So, Angle COB + angle BOD = 180

Similarly,

angle COA + angle AOD= 180

Therefore,

Angle COB + angle BOD = angle COA + angle AOD

But,

Angle COB = Angle COA

This is because the bisector of an angles divides the given angle in two equal parts.

Therefore, Angle BOD = Angle AOD

Hence, proved.

Answer 2

∠AOC  =∠BOD

Given:

The ray OC is the bisector of angle AOB and OD is the ray opposite to OC.

To Show:

∠AOD=∠BOD

Solution:

When a ray bisects an angle will equally divide the angle into half of the angle exactly.

Ray OC is the bisector of ∠AOB

⇒ ∠AOC =∠BOC --------------(a)

The angle made by a straight line is 180°

From the figure,

∠AOD+∠AOC = 180° ---------------(1)

Similarly,

∠BOD +∠BOC = 180°---------------(2)

From equations (1) and (2)

∠AOD+∠AOC  =∠BOD +∠BOC

∠AOC =∠ BOC (From equation (a))

∵ ∠AOD+∠AOC  =∠BOD + ∠AOC

⇒ ∠AOC  =∠BOD

Thus showed.

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