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Regular and Irregular Polygons

1. Polygons may be regular or irregular.

2. Regular polygons are equilateral and equiangular.

3. When polygons are regular, we just add the term "regular" to its name, except for regular triangles (we call it equilateral triangle) and for regular quadrilaterals (we call it square).

4. Measure of each interior angle:
\begin{tabular}{|c|c|c|c|}
\hline a. Regular & b. Regular & c. Regular & d. Regular \\
Pentagon & Hexagon & \begin{tabular}{l}
Octagon \\
Decagon
\end{tabular} \\
\hline [tex]$108^{\circ}$[/tex] each & [tex]$120^{\circ}$[/tex] each & [tex]$135^{\circ}$[/tex] each & [tex]$144^{\circ}$[/tex] each \\
\hline
\end{tabular}
(They will need this information when drawing regular polygons.)

5. The sum of the interior angles of a polygon is given by the formula: [tex](n-2) \times 180^{\circ}[/tex], where [tex]n[/tex] is the number of sides.



Answer :

GinnyAnswer
Sure! Let's explore the sum of the interior angles of regular polygons step-by-step.

The sum of the interior angles of a polygon can be determined using the formula:
[tex]\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \][/tex]
where [tex]\( n \)[/tex] represents the number of sides of the polygon.

### Given Polygons:
1. Pentagon ([tex]\(n = 5\)[/tex])
2. Hexagon ([tex]\(n = 6\)[/tex])
3. Octagon ([tex]\(n = 8\)[/tex])
4. Decagon ([tex]\(n = 10\)[/tex])

Let's calculate the sum of the interior angles for each of these polygons:

1. Pentagon (5 sides):
[tex]\[ \text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \][/tex]

2. Hexagon (6 sides):
[tex]\[ \text{Sum of interior angles} = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ \][/tex]

3. Octagon (8 sides):
[tex]\[ \text{Sum of interior angles} = (8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ \][/tex]

4. Decagon (10 sides):
[tex]\[ \text{Sum of interior angles} = (10 - 2) \times 180^\circ = 8 \times 180^\circ = 1440^\circ \][/tex]

### Results:
- Pentagon: The sum of the interior angles is [tex]\(540^\circ\)[/tex].
- Hexagon: The sum of the interior angles is [tex]\(720^\circ\)[/tex].
- Octagon: The sum of the interior angles is [tex]\(1080^\circ\)[/tex].
- Decagon: The sum of the interior angles is [tex]\(1440^\circ\)[/tex].

These sums are important as they help in determining the individual angle measures in regular polygons and assist in drawing polygons accurately.

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