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In the given diagram, we are tasked with finding the values of angles \( x \) and \( y \). The diagram shows a cyclic quadrilateral \( PQRS \), where points \( P, Q, R, S \) lie on a circle. There is an angle of \( 80^\circ \) at \( \angle Q \), and \( x \) and \( y \) represent unknown angles at points \( P \) and \( R \), respectively.

### To solve this:
1. **Angle Sum Property of a Cyclic Quadrilateral:** The sum of opposite angles in a cyclic quadrilateral is \( 180^\circ \). This means:
   \[
   \angle PQR + \angle PSR = 180^\circ
   \]
   With \( \angle PQR = 80^\circ \), we find:
   \[
   \angle PSR = 180^\circ - 80^\circ = 100^\circ
   \]
   Thus, \( y = 100^\circ \).

2. **Use of Inscribed Angle Theorem:** An inscribed angle subtended by the same chord is half the central angle. In this case, if we know that the angle at \( P \) is subtended by the chord \( PR \), then:
   \[
   \angle P = \frac{1}{2} \times \text{arc} PR
   \]
   Since \( \angle PSR = 100^\circ \) is opposite to \( \angle P \), we find:
   \[
   x = 100^\circ
   \]

Thus, the values of the angles are:
- \( x = 100^\circ \)
- \( y = 100^\circ \)

Would you like more details on any part of the solution? Or is there something else you’d like to clarify?

### Follow-up questions:
1. How do we prove that the sum of opposite angles in a cyclic quadrilateral is \( 180^\circ \)?
2. Can we apply the same approach if the quadrilateral is not cyclic?
3. How do we calculate other angles in cyclic quadrilaterals if additional angles or properties are known?
4. What is the difference between inscribed angles and central angles in a circle?
5. How can we use the inscribed angle theorem in different geometric problems?

**Tip**: When working with cyclic quadrilaterals, remember that the opposite angles always add up to \( 180^\circ \), which can be a helpful shortcut for solving many geometry problems.

Find the values of x and y in the given cyclic quadrilateral PQRS where angle PQR = 80 degrees.

Learn how to find the values of angles x and y in a cyclic quadrilateral PQRS with angle PQR = 80 degrees using key geometric theorems. Explore the sum property of opposite angles and the inscribed angle theorem for accurate solutions.

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