To solve the problem, let's break down the structure of the hexagon and the geometry of the triangles formed.

Step 1: Understand the Hexagon and Its Symmetry

• A regular hexagon has six equal sides and six equal angles.
• The interior angle of a regular hexagon is calculated by: $\text{Interior angle} = \frac{(n - 2) \times 180^\circ}{n} = \frac{(6 - 2) \times 180^\circ}{6} = 120^\circ$
• The hexagon is divided into three equal regions by lines drawn from the center to the vertices. These lines create six triangles at the center.

Step 2: Understanding the Triangles

• The lines from the center to the vertices divide the hexagon into six isosceles triangles.
• These triangles have two equal sides (the radius of the hexagon) and an angle at the center of the hexagon, which is the central angle.
• Since there are six triangles and the sum of all central angles around the point must be $360^\circ$, each central angle will be: $\frac{360^\circ}{6} = 60^\circ$ So, each triangle has a central angle of $60^\circ$.

Step 3: Determine the Base Angles of the Triangles

Each of the six triangles is isosceles, with the equal sides being the radii of the hexagon. The base angles (denoted as $\theta$) of these isosceles triangles can be calculated as follows:

• The sum of all angles in any triangle is $180^\circ$.
• In our isosceles triangle, the central angle is $60^\circ$, and the two base angles are equal. Let $\theta$ represent each base angle.

Thus, we have the equation for the angles in the triangle: $\theta + \theta + 60^\circ = 180^\circ$ $2\theta = 180^\circ - 60^\circ = 120^\circ$ $\theta = \frac{120^\circ}{2} = 60^\circ$

Final Answer:

The base angles of each triangle are 60°.

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Would you like more details or have any questions? Here are some related questions for further exploration:

1. How does the geometry of a regular polygon change when dividing it into more or fewer regions?
2. What is the relationship between the central angle and the base angles of isosceles triangles in polygons?
3. How can we generalize this method to find base angles in other regular polygons (e.g., heptagons, octagons)?
4. What other types of triangles can be formed within regular polygons and how do their angles relate to the polygon's properties?
5. How does the symmetry of a regular hexagon help in dividing it into equal regions?

Tip: For regular polygons, the base angles of the isosceles triangles formed by connecting the center to the vertices can always be found by dividing the central angle equally.