To solve the problem of finding the angle formed at each vertex of the star-shaped figure created by extending the sides of a regular octagon, we need to approach it step by step.

Step 1: Interior Angle of a Regular Octagon

The sum of the interior angles $S$ of any polygon is given by the formula: $S = 180(n - 2)$ where $n$ is the number of sides. For an octagon ($n = 8$): $S = 180(8 - 2) = 180 \times 6 = 1080^\circ$ The interior angle $A$ of a regular octagon is: $A = \frac{1080^\circ}{8} = 135^\circ$

Step 2: Exterior Angle of the Octagon

The exterior angle of a regular polygon is related to its interior angle as follows: $\text{Exterior Angle} = 180^\circ - \text{Interior Angle}$ Thus, the exterior angle of the regular octagon is: $\text{Exterior Angle} = 180^\circ - 135^\circ = 45^\circ$

Step 3: Star Shaped Figure Angle

When the sides of the octagon are extended, they intersect at the vertices of the star. These angles formed at the vertices of the star are external angles, which means they are equal to the exterior angle of the octagon. Therefore, each angle at the vertex of the star is $45^\circ$.

Thus, the correct answer is: $\boxed{45^\circ}$

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Relative Questions:

1. What is the relationship between interior and exterior angles in polygons?
2. How would the angles change if the polygon were a regular decagon instead of an octagon?
3. What would the sum of the exterior angles of a polygon always equal, regardless of the number of sides?
4. How can you derive the number of sides of a regular polygon given its interior or exterior angle?
5. Can you explain how to find the area of a regular octagon using only its side length?

Tip:

For any regular polygon, the sum of its exterior angles is always $360^\circ$, regardless of the number of sides!