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To determine the value of the angle \( z \) formed between the extensions of a regular hexagon and a regular octagon, we can follow these steps:

### Step-by-Step Solution
1. **Calculate the Interior Angles**:
   - A regular hexagon has \(6\) sides, and the formula for the interior angle of a regular polygon with \( n \) sides is:
     \[
     \text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}
     \]
   - For the hexagon (\(n = 6\)):
     \[
     \text{Hexagon interior angle} = \frac{(6-2) \times 180^\circ}{6} = \frac{4 \times 180^\circ}{6} = 120^\circ
     \]

   - For the octagon (\(n = 8\)):
     \[
     \text{Octagon interior angle} = \frac{(8-2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = 135^\circ
     \]

2. **Determine the Exterior Angles**:
   - The exterior angle of a polygon is the supplement of the interior angle, which is calculated by:
     \[
     \text{Exterior angle} = 180^\circ - \text{Interior angle}
     \]
   - For the hexagon:
     \[
     \text{Hexagon exterior angle} = 180^\circ - 120^\circ = 60^\circ
     \]

   - For the octagon:
     \[
     \text{Octagon exterior angle} = 180^\circ - 135^\circ = 45^\circ
     \]

3. **Calculate Angle \( z \)**:
   - The angle \( z \) is formed by the extension of a side of the hexagon and a side of the octagon.
   - Since \( z \) is the angle between these two extensions, it is equal to the sum of the two exterior angles:
     \[
     z = 60^\circ + 45^\circ = 105^\circ
     \]

### Conclusion
The value of \( z \) is:
\[
\boxed{105^\circ}
\]

Would you like more details or have any questions?

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Here are some related questions for further practice:
1. What is the sum of all interior angles of a regular octagon?
2. If a regular polygon has an exterior angle of \( 40^\circ \), how many sides does it have?
3. How would the angle \( z \) change if the polygons were a pentagon and a decagon?
4. Explain why the exterior angle formula works for any regular polygon.
5. How can we determine the number of sides in a regular polygon if we only know one interior angle?

**Tip**: For regular polygons, the interior and exterior angles can always be calculated with the same formulas, regardless of the number of sides.

The diagram shows a regular hexagon and a regular octagon. A quadrilateral is formed by extending sides of the two regular polygons. Find the value of z.

Learn how to determine the value of an angle formed by extending sides of a regular hexagon and a regular octagon. This problem covers geometric concepts related to polygon angles and includes a detailed, step-by-step solution suitable for students in Grades 9-10.

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