Practice worksheet for calculating interior angles of irregular polygons using the formula.

To solve the problem of finding the interior angles for each irregular polygon, we will follow these steps:

Step 1: Use the formula for the sum of the interior angles of a polygon

The formula to find the sum of the interior angles of a polygon is:
\[
\text{Sum of the interior angles} = (n - 2) \times 180^\circ
\]
where \( n \) is the number of sides of the polygon.

Step 2: Identify the number of sides (\( n \)) for each polygon

- Polygon 1: 5 sides (pentagon)
- Polygon 2: 6 sides (hexagon)
- Polygon 3: 5 sides (pentagon)
- Polygon 4: 6 sides (hexagon)
- Polygon 5: 5 sides (pentagon)
- Polygon 6: 5 sides (pentagon)
- Polygon 7: 7 sides (heptagon)
- Polygon 8: 5 sides (pentagon)
- Polygon 9: 4 sides (quadrilateral)

Step 3: Calculate the sum of the interior angles for each polygon

Using the formula \((n - 2) \times 180^\circ\):

1. Polygon 1 (Pentagon):
\[
n = 5 \implies \text{Sum} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]

2. Polygon 2 (Hexagon):
\[
n = 6 \implies \text{Sum} = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
\]

3. Polygon 3 (Pentagon):
\[
n = 5 \implies \text{Sum} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]

4. Polygon 4 (Hexagon):
\[
n = 6 \implies \text{Sum} = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
\]

5. Polygon 5 (Pentagon):
\[
n = 5 \implies \text{Sum} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]

6. Polygon 6 (Pentagon):
\[
n = 5 \implies \text{Sum} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]

7. Polygon 7 (Heptagon):
\[
n = 7 \implies \text{Sum} = (7 - 2) \times 180^\circ = 5 \times 180^\circ = 900^\circ
\]

8. Polygon 8 (Pentagon):
\[
n = 5 \implies \text{Sum} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]

9. Polygon 9 (Quadrilateral):
\[
n = 4 \implies \text{Sum} = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]

Step 4: Set up equations to find the missing interior angle(s)

For each polygon, the sum of all interior angles is equal to the calculated sum. We will use this to find the missing angle(s).

#### Polygon 1:
\[
\text{Sum} = 540^\circ
\]
Given angles: \(118^\circ, 105^\circ, 130^\circ, 25^\circ\)
\[
x + 118^\circ + 105^\circ + 130^\circ + 25^\circ = 540^\circ
\]
\[
x + 378^\circ = 540^\circ
\]
\[
x = 540^\circ - 378^\circ = 162^\circ
\]

#### Polygon 2:
\[
\text{Sum} = 720^\circ
\]
Given angles: \(130^\circ, 127^\circ, 130^\circ, 118^\circ, 126^\circ\)
\[
x + 130^\circ + 127^\circ + 130^\circ + 118^\circ + 126^\circ = 720^\circ
\]
\[
x + 631^\circ = 720^\circ
\]
\[
x = 720^\circ - 631^\circ = 89^\circ
\]

#### Polygon 3:
\[
\text{Sum} = 540^\circ
\]
Given angles: \(105^\circ, 108^\circ, 108^\circ, 105^\circ\)
\[
x + 105^\circ + 108^\circ + 108^\circ + 105^\circ = 540^\circ
\]
\[
x + 426^\circ = 540^\circ
\]
\[
x = 540^\circ - 426^\circ = 114^\circ
\]

#### Polygon 4:
\[
\text{Sum} = 720^\circ
\]
Given angles: \(86^\circ, 140^\circ, 115^\circ, 112^\circ, 135^\circ\)
\[
x + 86^\circ + 140^\circ + 115^\circ + 112^\circ + 135^\circ = 720^\circ
\]
\[
x + 588^\circ = 720^\circ
\]
\[
x = 720^\circ - 588^\circ = 132^\circ
\]

#### Polygon 5:
\[
\text{Sum} = 540^\circ
\]
Given angles: \(91^\circ, 120^\circ, 86^\circ, 135^\circ\)
\[
x + 91^\circ + 120^\circ + 86^\circ + 135^\circ = 540^\circ
\]
\[
x + 432^\circ = 540^\circ
\]
\[
x = 540^\circ - 432^\circ = 108^\circ
\]

#### Polygon 6:
\[
\text{Sum} = 540^\circ
\]
Given angles: \(120^\circ, 160^\circ, 150^\circ, 140^\circ, 120^\circ\)
\[
x + 120^\circ + 160^\circ + 150^\circ + 140^\circ + 120^\circ = 540^\circ
\]
\[
x + 790^\circ = 540^\circ
\]
\[
x = 540^\circ - 790^\circ = -250^\circ \quad \text{(This is incorrect; recheck given angles or problem setup)}
\]

#### Polygon 7:
\[
\text{Sum} = 900^\circ
\]
Given angles: \(140^\circ, 125^\circ, 110^\circ, 140^\circ, 125^\circ, 135^\circ\)
\[
x + 140^\circ + 125^\circ + 110^\circ + 140^\circ + 125^\circ + 135^\circ = 900^\circ
\]
\[
x + 895^\circ = 900^\circ
\]
\[
x = 900^\circ - 895^\circ = 5^\circ
\]

#### Polygon 8:
\[
\text{Sum} = 540^\circ
\]
Given angles: \(113^\circ, 95^\circ, 113^\circ, 130^\circ\)
\[
x + 113^\circ + 95^\circ + 113^\circ + 130^\circ = 540^\circ
\]
\[
x + 451^\circ = 540^\circ
\]
\[
x = 540^\circ - 451^\circ = 89^\circ
\]

#### Polygon 9:
\[
\text{Sum} = 360^\circ
\]
Given angles: \(125^\circ, 70^\circ, 60^\circ\)
\[
x + 125^\circ + 70^\circ + 60^\circ = 360^\circ
\]
\[
x + 255^\circ = 360^\circ
\]
\[
x = 360^\circ - 255^\circ = 105^\circ
\]

Final Answers:

\[
\boxed{
\begin{array}{ll}
1) & 162^\circ \\
2) & 89^\circ \\
3) & 114^\circ \\
4) & 132^\circ \\
5) & 108^\circ \\
6) & \text{Error in setup} \\
7) & 5^\circ \\
8) & 89^\circ \\
9) & 105^\circ \\
\end{array}
}
\]