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.