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To find the value of \( x \) in the given circle diagram, we need to apply some geometric principles, particularly related to cyclic quadrilaterals and angles subtended by the same arc.

### Given information:
- \( \angle RQP = 70^\circ \)
- \( \angle OPS = 30^\circ \)
- \( O \) is the center of the circle.

### Step-by-step solution:

1. **Angle at the Center and Angle at the Circumference:**
   The angle subtended by the same arc at the center of a circle is twice the angle subtended at the circumference. Here, \( \angle RQS \) subtends the same arc as \( \angle ROS \), and:
   \[
   \angle ROS = 2 \times \angle RQS
   \]
   Thus, \( \angle ROS = 2 \times 70^\circ = 140^\circ \).

2. **Angle Sum in a Triangle:**
   In triangle \( ROS \), the sum of all angles should equal \( 180^\circ \). We know two angles:
   - \( \angle ROS = 140^\circ \)
   - \( \angle OPS = 30^\circ \)

   Using the sum of angles in triangle \( ROS \):
   \[
   \angle ORS + \angle OPS + \angle ROS = 180^\circ
   \]
   Substituting the known values:
   \[
   \angle ORS + 30^\circ + 140^\circ = 180^\circ
   \]
   Simplifying:
   \[
   \angle ORS = 180^\circ - 170^\circ = 10^\circ
   \]

3. **Finding \( x \):**
   Since \( x \) is the external angle at \( S \), it is equal to the opposite internal angle \( \angle ORS \). Thus:
   \[
   x = 10^\circ
   \]

### Final Answer:
\[
x = 10^\circ
\]

Would you like more details on any specific steps or have further questions?

Here are five related questions to consider:
1. What are cyclic quadrilaterals, and how do their angle properties apply here?
2. Why is the external angle at \( S \) equal to the opposite interior angle \( ORS \)?
3. Can you explain the relationship between angles at the center and circumference of a circle?
4. How does the sum of angles in a triangle help solve problems like this?
5. What other geometric theorems apply to circles and angles?

**Tip:** Remember that angles subtended by the same arc on the circumference are always equal, which is a crucial property for solving circle-related problems.

Learn how to find the value of x in a circle diagram using the properties of cyclic quadrilaterals and circle theorems. This problem includes step-by-step calculations and explanations, focusing on geometric principles and theorems related to angles subtended by the same arc. Suitable for students in Grades 9-10.

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