Question

regular hexagon abcdef is divided into six equilateral triangles, as shown. explain how to determine the sum of the interior angles of the hexagon using these triangles. use the drop - down menus to complete the statements. image of a regular hexagon abcdef divided into six equilateral triangles click the arrows to choose an answer from each menu. the sum of the interior angles of each triangle is choose... degrees, so the measure of each interior angle of the equilateral triangles in the figure is choose... degrees. the interior angle at each vertex of the hexagon is composed of choose... interior

Explanation:

Step1: Recall triangle angle sum

The sum of interior angles of any triangle is \(180^\circ\). So for each equilateral triangle, the sum is \(180\) degrees.

Step2: Find angle of equilateral triangle

In an equilateral triangle, all angles are equal. Let each angle be \(x\). Then \(3x = 180^\circ\), so \(x=\frac{180^\circ}{3}=60^\circ\). Each interior angle of the equilateral triangles is \(60\) degrees.

Step3: Analyze hexagon vertex angle

Looking at the regular hexagon divided into six equilateral triangles, each interior angle of the hexagon is formed by two angles of the equilateral triangles (since the center angles are around a point, and the hexagon's vertex angle is made by two triangle angles). So the interior angle at each vertex of the hexagon is composed of \(2\) interior angles of the equilateral triangles. Then, to find the sum of interior angles of the hexagon: each triangle has \(180^\circ\), six triangles have \(6\times180^\circ = 1080^\circ\), but we subtract the \(360^\circ\) around the center (since those are not part of the hexagon's interior angles). So \(1080^\circ - 360^\circ=720^\circ\). But for the drop - down parts:

• The sum of the interior angles of each triangle is \(180\) degrees.
• Each interior angle of the equilateral triangles in the figure is \(60\) degrees.
• The interior angle at each vertex of the hexagon is composed of \(2\) interior angles of the equilateral triangles. Then the measure of each interior angle of the hexagon is \(2\times60^\circ = 120^\circ\), and the sum of interior angles of the hexagon is \(6\times120^\circ=720^\circ\) (or using the triangle method: \(6\times180 - 360 = 720\)).

Answer:

For the first "Choose...": \(180\)
For the second "Choose...": \(60\)
For the third "Choose...": \(2\) (and then to find the sum of interior angles of hexagon, we can use these to get \(720^\circ\) as the sum)