Question

1. title: \the angle secrets of polygons\ or another creative title. 2. interior angle sum theorem section: include formula (n - 2)×180°, define n, show two examples, and include a diagram dividing the polygon into triangles. 3. exterior angle sum theorem section: include formula 360°, explain why its constant, and

Explanation:

Step1: Define n in interior - angle formula

In the formula $(n - 2)\times180^{\circ}$ for the sum of interior angles of a polygon, $n$ represents the number of sides of the polygon.

Step2: Provide interior - angle examples

Example 1: For a triangle ($n = 3$), using the formula $(n - 2)\times180^{\circ}=(3 - 2)\times180^{\circ}=180^{\circ}$.
Example 2: For a quadrilateral ($n = 4$), $(n - 2)\times180^{\circ}=(4 - 2)\times180^{\circ}=360^{\circ}$.

Step3: Diagram for dividing polygon into triangles

To show how to divide a polygon into triangles, for a pentagon ($n = 5$), we can start from one vertex and draw diagonals to non - adjacent vertices. We can draw 2 diagonals from one vertex of a pentagon, dividing it into 3 triangles. Since the sum of angles in a triangle is $180^{\circ}$, the sum of interior angles of a pentagon is $3\times180^{\circ}=(5 - 2)\times180^{\circ}=540^{\circ}$.

Step4: Explain exterior - angle sum

The sum of exterior angles of any polygon is always $360^{\circ}$. At each vertex of a polygon, the sum of an interior angle and its corresponding exterior angle is $180^{\circ}$. Let the interior angles of a polygon be $I_1,I_2,\cdots,I_n$ and the exterior angles be $E_1,E_2,\cdots,E_n$. We know that $(I_1 + E_1)+(I_2 + E_2)+\cdots+(I_n+E_n)=n\times180^{\circ}$. And the sum of interior angles $S_{int}=(n - 2)\times180^{\circ}$. So the sum of exterior angles $S_{ext}=n\times180^{\circ}-(n - 2)\times180^{\circ}=n\times180^{\circ}-n\times180^{\circ}+ 360^{\circ}=360^{\circ}$.

Answer:

The above steps explain the interior and exterior angle sum theorems for polygons as required.