# How to Understand and Master Polygons and Angles

Polygons are shapes made up of straight lines, like triangles, rectangles, and pentagons. The corners where these lines meet create angles. These angles can tell us a lot about the polygon. For instance, in a square, all angles are the same and measure $$90$$ degrees. When we delve deeper into studying polygons, understanding these angles becomes key. They help in drawing, designing, and even in real-life tasks like building or sewing. In short, by exploring polygons and their angles, we uncover a mix of math, art, and everyday practicality.

## Step-by-step Guide: Polygons and Angles

1. Definition of a Polygon:
A polygon is a closed two-dimensional shape with straight sides.
Polygons can be classified based on the number of sides they have, e.g., triangle ($$3$$ sides), quadrilateral ($$4$$ sides), pentagon ($$5$$ sides), and so on.
2. Sum of Interior Angles: The sum of interior angles in any polygon can be determined using the formula:
Sum of Interior Angles $$= (n-2) \times 180^\circ$$
where $$n$$ is the number of sides in the polygon.
3. Each Interior Angle of a Regular Polygon: A regular polygon has all its sides and angles equal. The measure of each interior angle can be found using:
Each Interior Angle $$= \frac{\text{Sum of Interior Angles}}{n}$$
where $$n$$ is the number of sides.

### Examples

Example 1:
Find the sum of the interior angles of a pentagon.

Solution:
For a pentagon, $$n = 5$$.
Sum of Interior Angles $$= (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$$

Example 2:
Calculate the sum of the interior angles of an octagon.

Solution:
For an octagon, $$n = 8$$.
Sum of Interior Angles $$= (8-2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ$$

Example 3:
Determine the measure of each interior angle of a regular hexagon.

Solution:
For a hexagon, $$n = 6$$.
Sum of Interior Angles $$= (6-2) \times 180^\circ = 720^\circ$$.
Each interior angle:
$$= \frac{720^\circ}{6} = 120^\circ$$

### Practice Questions:

1. What is the sum of the interior angles of a triangle?
2. If a regular polygon has an interior angle of $$150^\circ$$, how many sides does it have?
3. Determine the sum of the interior angles for a decagon ($$10$$ sides).

Answers:

1. $$180^\circ$$.
2. $$12$$ sides.
3. $$1440^\circ$$.

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