Given:
- Figure 6.17 shows a line POQ, where O is a point on the line.
- Ray OR is perpendicular to line PQ.
- Ray OS lies between rays OP and OR.

To prove:
Angle ROS = 1/2 * (angle QOS - angle POS)

Proof:

Step 1: Draw a diagram as per the given information.

Step 2: Identify the angles in the diagram.

- Angle ROS: The angle formed between rays OR and OS.
- Angle QOS: The angle formed between rays OQ and OS.
- Angle POS: The angle formed between rays OP and OS.

Step 3: Observe the relationship between the angles.

- Since ray OR is perpendicular to line PQ, angle ROS is a right angle (90 degrees).
- From the diagram, we can see that angle QOS can be divided into two parts: angle QOR and angle ROS.
- Similarly, angle POS can also be divided into two parts: angle POR and angle ROS.

Step 4: Apply the angle addition property.

- According to the angle addition property, angle QOS = angle QOR + angle ROS.
- Similarly, angle POS = angle POR + angle ROS.

Step 5: Substitute the values in the given equation.

- We are given the equation: Angle ROS = 1/2 * (angle QOS - angle POS).
- Substitute the values of angle QOS and angle POS obtained from step 4.
- Angle ROS = 1/2 * ((angle QOR + angle ROS) - (angle POR + angle ROS)).
- Simplify the equation: Angle ROS = 1/2 * (angle QOR + angle ROS - angle POR - angle ROS).
- Angle ROS = 1/2 * (angle QOR - angle POR).

Step 6: Simplify the equation further.

- Angle ROS = 1/2 * (angle QOR - angle POR).
- Angle ROS = 1/2 * angle QOR - 1/2 * angle POR.

Step 7: Conclusion

- Angle ROS = 1/2 * angle QOR - 1/2 * angle POR.
- Hence, angle ROS is equal to 1/2 * angle QOR minus angle POR, as required to prove.