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This diagram appears to show a quadrilateral inscribed in a circle. From the problem, it asks to find the value of the sum of angles \( x + y + z + w \).

In a quadrilateral inscribed in a circle, the opposite angles are supplementary, meaning that their sum is \( 180^\circ \).

We can apply this property:

1. \( x + z = 180^\circ \)
2. \( y + w = 180^\circ \)

Adding both equations together:

\[
(x + z) + (y + w) = 180^\circ + 180^\circ = 360^\circ
\]

Thus, \( x + y + z + w = 360^\circ \).

Let me know if you need further details or have any questions!

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Here are 5 related questions for practice:

1. What is the sum of the interior angles of any quadrilateral?
2. Can a quadrilateral that isn't inscribed in a circle have supplementary opposite angles?
3. How do you prove that a quadrilateral is cyclic (inscribed in a circle)?
4. If one of the angles is \( 90^\circ \), what are the values of the other angles in this cyclic quadrilateral?
5. What is the difference between cyclic and non-cyclic quadrilaterals?

**Tip:** Always remember, the sum of opposite angles in any cyclic quadrilateral is always \( 180^\circ \)!

Find the value of x + y + z + w in this diagram.

Understand how to calculate the sum of angles in a cyclic quadrilateral. This problem demonstrates the application of the Cyclic Quadrilateral Theorem, where the sum of opposite angles equals 180°. Suitable for students in Grades 9-10.

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