Question

1. modeling real life the image printed on a chinese checkers board is shown. without measuring, classify each of the six triangles surrounding the hexagon by sides and angle measures. explain your reasoning.

Explanation:

Step1: Analyze the angles

Since the sum of angles around a point is \(360^{\circ}\), and the central angle of the hexagon is \(120^{\circ}\), then for each triangle, the two non - adjacent angles at the vertex (where the triangle meets the hexagon) can be found. Let's consider the angles at the vertex of the triangle. The angle adjacent to the \(120^{\circ}\) angle in the hexagon: \(\frac{360 - 120}{6}=40^{\circ}\) is not correct. Wait, actually, for a regular hexagon - star (Chinese checkers - like) pattern. The triangles are equilateral. Because the figure is symmetric. In a regular star - hexagon combination (as in Chinese checkers), the triangles have all sides equal (by symmetry of the board design). For angles, since the triangles are part of a symmetric pattern, each angle of the triangle is \(60^{\circ}\).

Step2: Classify by sides

A triangle with all three sides equal is an equilateral triangle. In the Chinese checkers board image (due to the symmetry of the design), the six triangles surrounding the hexagon have all sides of equal length.

Step3: Classify by angles

A triangle with all angles equal to \(60^{\circ}\) is an equilateral triangle (since the sum of angles in a triangle is \(180^{\circ}\), and if \(a = b=c\), then \(a + b + c=180^{\circ}\) gives \(a=b = c=60^{\circ}\)). In the symmetric Chinese checkers - board pattern, the angles of each of the six surrounding triangles are equal.

Answer:

By sides: equilateral triangles (all sides are equal due to the symmetry of the Chinese checkers - board design). By angles: equilateral triangles (all angles are \(60^{\circ}\) because of the symmetry of the pattern and the angle - sum property of a triangle (\(180^{\circ}\))). So, each of the six triangles is an equilateral triangle.