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The diagram shows two polygons:

- **Pentagon M** has one axis of symmetry.
- **Polygon N** is a regular octagon (8 sides), as indicated by the note on the diagram.
- The angle between one side of pentagon M and octagon N is given as \( 130^\circ \).
- An unknown angle \( x \) is labeled between the two polygons.

### Objective:
We need to find the value of the angle \( x \).

### Key Information:
1. **Interior Angle Formula for Regular Polygons**: 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 an octagon (\( n = 8 \)):
   \[
   \text{Interior angle of octagon} = \frac{(8-2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = 135^\circ
   \]
2. **External Angles**: The external angle of a regular polygon is:
   \[
   \text{External angle} = 180^\circ - \text{Interior angle}
   \]
   For the octagon, the external angle is:
   \[
   180^\circ - 135^\circ = 45^\circ
   \]

### Step-by-Step Solution:
1. The external angle of the octagon at the point where the pentagon meets is \( 45^\circ \).
2. The sum of the angles at this junction must be \( 180^\circ \) (straight line). So:
   \[
   x + 45^\circ + 130^\circ = 180^\circ
   \]
3. Simplifying the equation:
   \[
   x + 175^\circ = 180^\circ
   \]
4. Solving for \( x \):
   \[
   x = 180^\circ - 175^\circ = 5^\circ
   \]

### Conclusion:
The value of angle \( x \) is \( 5^\circ \).

Would you like further clarification or details?

Here are 5 related questions you might explore:
1. What is the sum of interior angles of a regular octagon?
2. How do you calculate the interior angles of a pentagon?
3. What is the difference between internal and external angles in polygons?
4. How would the solution change if polygon N were a hexagon instead of an octagon?
5. Can an irregular polygon have an axis of symmetry?

**Tip:** Understanding the relationship between interior and exterior angles is essential for solving problems involving regular polygons.

The diagram consists of two polygons. Pentagon M has one axis of symmetry and polygon N is a regular polygon. Find the value of angle x.

Learn how to calculate the angle x between a regular pentagon and a regular octagon in a geometric diagram. This problem involves calculating interior and exterior angles of polygons, using symmetry, and applying the straight angle theorem. Suitable for students in Grades 9-11.

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