Proof:

Given: Ray OC is the bisector of angle AOB.

To prove: Angle AOD = Angle BOD.

Proof:

Step 1: Draw the figure

Draw a line segment AB and mark a point O on it.
From point O, draw ray OC that bisects angle AOB.
Draw a line segment OD opposite to ray OC.

Step 2: Identify the angles

In the figure, we have the following angles:
- Angle AOB: This is the angle formed by line segment AB.
- Angle AOD: This is the angle formed by line segments AO and OD.
- Angle BOD: This is the angle formed by line segments BO and OD.

Step 3: Use the angle bisector property

Since ray OC is the bisector of angle AOB, it divides the angle into two equal parts. This property states that the angles formed by the bisector are equal.

Therefore, we can say that angle AOC = angle BOC.

Step 4: Prove angle AOD = angle BOD

To prove that angle AOD = angle BOD, we need to show that the angles formed by the rays AO and OD, and BO and OD are equal.

Using the transitive property of equality, we can rewrite the above statement as:
angle AOC = angle BOC = angle AOD = angle BOD.

This implies that angle AOD is equal to angle BOD.

Hence, we have proved that angle AOD = angle BOD using the angle bisector property.