The diagram you've uploaded appears to involve a circle, a few inscribed angles, and some key angles marked. Let's break down the geometry.

Given Information:

• There is a circle with center $O$.
• Four points are labeled on the circle: $P$, $Q$, $R$, and $S$.
• Angle $\angle PRQ$ is given as $70^\circ$.
• Angle $\angle PSQ$ is marked as $x$, and this is what we need to find.
• Angle $\angle ROS$ is given as $30^\circ$.

Approach:

1. Inscribed Angle Theorem:

   • The inscribed angle subtended by an arc at the circumference is half of the angle subtended by the same arc at the center of the circle.

   From this theorem: $\angle PRQ = \frac{1}{2} \times \angle ROS$ Therefore, the angle at the center $\angle ROS$ subtended by arc $PS$ is: $\angle ROS = 2 \times \angle PRQ = 2 \times 70^\circ = 140^\circ$

2. Angle at the Circumference:

   • The angle $\angle PSQ$ subtended by the same arc $PR$ is half of the angle at the center. $\angle PSQ = \frac{1}{2} \times \angle ROS = \frac{1}{2} \times 140^\circ = 70^\circ$

Thus, the value of $x$ is $70^\circ$.

Summary:

• $x = 70^\circ$, based on the inscribed angle theorem.

Would you like further clarification on any of the steps, or more details?

Here are some follow-up questions:

1. What is the relationship between central and inscribed angles in a circle?
2. How would the result change if the center of the circle were not at $O$?
3. How can we use the angles in cyclic quadrilaterals for more complex problems?
4. What is the importance of the "Inscribed Angle Theorem" in solving circle geometry problems?
5. How would you solve this if another angle, such as $\angle PQR$, was given?

Tip: Always remember that the angle subtended at the center is double that subtended at the circumference when considering the same arc!