Given:
- Line ray POQ is perpendicular to line PQ.
- Ray OS lies between rays OP and OR.

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

Proof:

Step 1: Draw a diagram

It is always helpful to draw a diagram to visualize the given information and better understand the problem.

Diagram:
```
O
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/_______________\
P Q
|
|
S
```

Step 2: Identify the key points and angles

- Angle QOS: the angle between rays OP and OS.
- Angle POS: the angle between rays OS and OR.
- Angle ROS: the angle between rays OR and OS.
- Angle ROQ: the right angle formed by line POQ.

Step 3: Determine the relationship between angles QOS, POS, and ROS

- Since rays OP and OR lie on the same line, the sum of angles QOS and POS must be equal to angle ROQ (180 degrees).
- Therefore, QOS + POS = 180 degrees - ROQ.

Step 4: Rewrite the equation

- We want to prove that angle ROS is equal to 1/2(QOS - POS).
- We can rewrite this as QOS - POS = 2(ROS).

Step 5: Substitute the value of QOS + POS from step 3

- From step 3, we know that QOS + POS = 180 degrees - ROQ.
- Substituting this value into the equation QOS - POS = 2(ROS), we get:
180 degrees - ROQ = 2(ROS).

Step 6: Simplify the equation

- Distributing the 2 on the right side of the equation, we get:
180 degrees - ROQ = 2ROS.
- Solving for ROS, we divide both sides of the equation by 2:
ROS = (180 degrees - ROQ) / 2.

Step 7: Simplify further

- We know that ROQ is a right angle (90 degrees), so we can substitute this value into the equation:
ROS = (180 degrees - 90 degrees) / 2.
- Simplifying, we get:
ROS = 90 degrees / 2.
ROS = 45 degrees.

Step 8: Conclusion

- We have proved that angle ROS is equal to 45 degrees.
- Therefore, angle ROS is equal to 1/2(QOS - POS).

Final Answer:
- Angle ROS = 45 degrees
- Angle ROS = 1/2(QOS - POS)