Interior Angles, Exterior Angles and the Sum | Polygon Worksheets - Free Printable

To solve the problem of finding the measures of the indicated angles in polygons, we need to use some fundamental geometric principles. Let's break it down step by step.

Key Concepts:

1. Sum of Interior Angles of a Polygon:
The sum of the interior angles of an \( n \)-sided polygon is given by:
\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\]

2. Regular Polygon:
If the polygon is regular (all sides and angles are equal), then each interior angle can be calculated as:
\[
\text{Each interior angle} = \frac{(n - 2) \times 180^\circ}{n}
\]

3. Exterior Angle:
The exterior angle of a polygon is supplementary to its corresponding interior angle:
\[
\text{Exterior angle} = 180^\circ - \text{Interior angle}
\]
For a regular polygon, the measure of each exterior angle is:
\[
\text{Each exterior angle} = \frac{360^\circ}{n}
\]

Step-by-Step Solution:

#### Problem 1: Pentagon
- Number of sides (\( n \)): 5
- Sum of interior angles:
\[
(n - 2) \times 180^\circ = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- Each interior angle (regular pentagon):
\[
\frac{540^\circ}{5} = 108^\circ
\]
- Each exterior angle:
\[
\frac{360^\circ}{5} = 72^\circ
\]

#### Problem 2: Hexagon
- Number of sides (\( n \)): 6
- Sum of interior angles:
\[
(n - 2) \times 180^\circ = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
\]
- Each interior angle (regular hexagon):
\[
\frac{720^\circ}{6} = 120^\circ
\]
- Each exterior angle:
\[
\frac{360^\circ}{6} = 60^\circ
\]

#### Problem 3: Heptagon
- Number of sides (\( n \)): 7
- Sum of interior angles:
\[
(n - 2) \times 180^\circ = (7 - 2) \times 180^\circ = 5 \times 180^\circ = 900^\circ
\]
- Each interior angle (regular heptagon):
\[
\frac{900^\circ}{7} \approx 128.57^\circ
\]
- Each exterior angle:
\[
\frac{360^\circ}{7} \approx 51.43^\circ
\]

#### Problem 4: Octagon
- Number of sides (\( n \)): 8
- Sum of interior angles:
\[
(n - 2) \times 180^\circ = (8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ
\]
- Each interior angle (regular octagon):
\[
\frac{1080^\circ}{8} = 135^\circ
\]
- Each exterior angle:
\[
\frac{360^\circ}{8} = 45^\circ
\]

Final Answers:

1. Pentagon:
- Sum of interior angles: \( 540^\circ \)
- Each interior angle: \( 108^\circ \)
- Each exterior angle: \( 72^\circ \)

2. Hexagon:
- Sum of interior angles: \( 720^\circ \)
- Each interior angle: \( 120^\circ \)
- Each exterior angle: \( 60^\circ \)

3. Heptagon:
- Sum of interior angles: \( 900^\circ \)
- Each interior angle: \( \approx 128.57^\circ \)
- Each exterior angle: \( \approx 51.43^\circ \)

4. Octagon:
- Sum of interior angles: \( 1080^\circ \)
- Each interior angle: \( 135^\circ \)
- Each exterior angle: \( 45^\circ \)

\[
\boxed{
\begin{array}{c|c|c|c}
\text{Polygon} & \text{Sum of Interior Angles} & \text{Each Interior Angle} & \text{Each Exterior Angle} \\
\hline
\text{Pentagon} & 540^\circ & 108^\circ & 72^\circ \\
\text{Hexagon} & 720^\circ & 120^\circ & 60^\circ \\
\text{Heptagon} & 900^\circ & \approx 128.57^\circ & \approx 51.43^\circ \\
\text{Octagon} & 1080^\circ & 135^\circ & 45^\circ \\
\end{array}
}
\]