To find the angle $\angle PRS$, we can use the property of the cyclic quadrilateral and the known angles.

Given Information:

• $\angle SRP = 74^\circ$
• $\angle SPQ = 58^\circ$
• $\angle PQN = 27^\circ$

Step 1: Use the Cyclic Quadrilateral Property

In any cyclic quadrilateral, the opposite angles sum to $180^\circ$.

Let's use this property to find $\angle PRS$.

Since $PQRS$ is a cyclic quadrilateral, the opposite angles $\angle PRS$ and $\angle PSQ$ must satisfy: $\angle PRS + \angle PSQ = 180^\circ$

Step 2: Find $\angle PSQ$

Earlier, we calculated $\angle PSQ = 106^\circ$.

Step 3: Calculate $\angle PRS$

Now, using the cyclic quadrilateral property: $\angle PRS + 106^\circ = 180^\circ$ $\angle PRS = 180^\circ - 106^\circ = 74^\circ$

Thus, the angle $\angle PRS$ is $74^\circ$.

Would you like any further details or have any questions?

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Here are five related questions:

1. What other properties of cyclic quadrilaterals are useful in solving geometry problems?
2. How does the sum of angles in a cyclic quadrilateral compare to that in any quadrilateral?
3. How can you verify that the quadrilateral is cyclic based on its angles?
4. What role do the circle's tangents play in the angle relationships within the quadrilateral?
5. How can the inscribed angle theorem be applied to find other angles in the diagram?

Tip: When dealing with cyclic quadrilaterals, always check the sum of opposite angles, as they provide a straightforward way to find unknown angles.