To construct a regular hexagon by using equilateral triangles, follow these steps:

Construction Steps:

1. Draw an Equilateral Triangle:

   • Begin by drawing an equilateral triangle. An equilateral triangle has all three sides of equal length and all angles of $60^\circ$.
   • You can use a protractor to measure and ensure all angles are $60^\circ$ or use a ruler to ensure all three sides are equal in length.

2. Extend the Sides to Form More Equilateral Triangles:

   • Attach another equilateral triangle to one of the sides of the first triangle. Ensure the new triangle shares a side with the first.
   • Continue attaching equilateral triangles along each side, forming a total of six triangles around a central point.

3. Complete the Hexagon:

   • After attaching the six equilateral triangles, the outer vertices will form the vertices of the regular hexagon.
   • Connect these six outer points with straight lines. These lines are the sides of the hexagon.

4. Label the Vertices:

   • Label the six vertices as $A, B, C, D, E, F$ (or any other set of labels). These are the points where the sides of the hexagon meet.

5. Shade the Hexagon:

   • You can shade the interior of the hexagon to clearly indicate the shape.

Justification for Regularity:

• Equal Side Lengths: Since you constructed the hexagon by arranging six equilateral triangles around a central point, each side of the hexagon is formed by one side of an equilateral triangle. This ensures all six sides are equal in length.
• Equal Angles: Each interior angle of the hexagon is $120^\circ$. This can be derived from the fact that the sum of the interior angles of any polygon with $n$ sides is given by $180(n - 2)$, and for a hexagon ($n = 6$), this gives a total of 720°. Dividing by 6, each interior angle is $120^\circ$.
• Symmetry: The hexagon exhibits rotational symmetry, as rotating the hexagon by $60^\circ$ around its center brings it back to the same position, confirming that all sides and angles are congruent.

Thus, the hexagon you have constructed is regular, as it has equal side lengths, equal interior angles, and exhibits rotational symmetry.

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Would you like a more detailed explanation of any of these steps? Or do you have any other questions?

Here are 5 related questions you might find interesting:

1. How can you calculate the area of a regular hexagon?
2. What is the relationship between the radius of the circumscribed circle and the side length of a regular hexagon?
3. How does the number of sides in a polygon affect the sum of its interior angles?
4. What other regular polygons can be constructed using equilateral triangles?
5. How do you prove that the diagonals of a regular hexagon are all congruent?

Tip: Using a protractor or a set square can make your construction of equilateral triangles much more precise.