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The problem asks to find the sum of the angles \(\angle a + \angle b + \angle c + \angle d + \angle e + \angle f\) in the given geometric figure.

### Step-by-Step Solution:

1. The diagram represents a star-like shape, which is a **concave polygon**. Specifically, it looks like a hexagram made up of two overlapping triangles.

2. **External angle property of polygons**: The sum of the external angles of any polygon is always \(360^\circ\).

3. Since this figure is symmetric and is composed of two overlapping triangles, the sum of the angles at the vertices (denoted as \(a\), \(b\), \(c\), \(d\), \(e\), \(f\)) represents the sum of the **exterior angles** of this concave polygon.

4. By the external angle theorem, the sum of these exterior angles is always \(360^\circ\).

Therefore, the sum of \(\angle a + \angle b + \angle c + \angle d + \angle e + \angle f = 360^\circ\).

The correct answer is:

\[
\boxed{360^\circ} \ (\text{Option D})
\]

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### Related Questions:
1. What is the formula for the sum of exterior angles of a polygon?
2. How do you calculate the sum of interior angles for a polygon?
3. What are the properties of concave polygons?
4. How does symmetry help in solving geometric problems?
5. How do overlapping polygons affect angle sums?

### Tip:
When dealing with polygons, always remember that the sum of exterior angles is always \(360^\circ\), regardless of the number of sides!

求∠a + ∠b + ∠c + ∠d + ∠e + ∠f in the given geometric figure.

This problem involves calculating the sum of exterior angles in a concave polygon. The total of exterior angles for any polygon is always 360°. Learn how to apply this property to solve geometric problems, suitable for high school geometry students.

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