In the image, we see a cyclic quadrilateral $PQRS$ inscribed in a circle, and a few angles labeled. The following angles and relationships are important:

• $\angle QPR = y^\circ$
• $\angle TSR = x^\circ$
• $\angle QRS = 130^\circ$

Key properties:

1. Cyclic Quadrilateral Property: Opposite angles in a cyclic quadrilateral are supplementary, i.e., they sum to $180^\circ$.

   • This means $\angle QPR + \angle QRS = 180^\circ$.
   • Given $\angle QRS = 130^\circ$, we can find $y$ using the equation: $\angle QPR + 130^\circ = 180^\circ$ $y^\circ = 50^\circ$

2. Angle in the Same Segment: Angles subtended by the same arc in a circle are equal. Since $\angle QRS$ and $\angle RTS$ are subtended by the same arc $QS$, we have: $\angle RTS = \angle QRS = 130^\circ$ Thus, $x^\circ = 130^\circ$.

Summary:

• $y = 50^\circ$
• $x = 130^\circ$

Would you like more details or have any further questions?

Related Questions:

1. What are the properties of angles in a cyclic quadrilateral?
2. How do you find the angles subtended by the same arc?
3. What is the importance of the supplementary angle rule in cyclic quadrilaterals?
4. How does the inscribed angle theorem apply to this diagram?
5. Can we use alternate methods to verify the values of $x$ and $y$?

Tip:

Always look for symmetrical properties in cyclic quadrilaterals—they can simplify complex angle calculations!