Angles in Irregular Polygons Worksheet | PDF Printable Geometry ... - Free Printable

Problem Analysis:

The task involves calculating the sum of interior angles for irregular polygons and finding missing angles. The key formula to use is:

\[
\text{Sum of interior angles of an } n\text{-sided polygon} = (n-2) \times 180^\circ
\]

We will apply this formula to each polygon and solve for the missing angle \( x \).

---

Step-by-Step Solution:

#### 1. Write an expression in terms of \( n \) for the sum of the interior angles in any \( n \)-sided polygon.

The formula for the sum of the interior angles of an \( n \)-sided polygon is:

\[
\boxed{(n-2) \times 180^\circ}
\]

---

#### 2. Triangle (3 sides)

- Name of Shape: Triangle
- Number of Sides: 3
- Sum of all Angles: For a triangle, the sum of the interior angles is always \( 180^\circ \).
- Given Angles: \( 33^\circ \) and \( 71^\circ \)
- Missing Angle \( x \):
\[
x = 180^\circ - (33^\circ + 71^\circ) = 180^\circ - 104^\circ = 76^\circ
\]

| Name of Shape | Number of Sides | Sum of all Angles | Value of Missing Angle \( x \) |
|---------------|------------------|-------------------|--------------------------------|
| Triangle | 3 | \( 180^\circ \) | \( 76^\circ \) |

---

#### 3. Quadrilateral (4 sides)

- Name of Shape: Quadrilateral
- Number of Sides: 4
- Sum of all Angles: Using the formula:
\[
(4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
- Given Angles: \( 74^\circ \), \( 96^\circ \), and \( 112^\circ \)
- Missing Angle \( x \):
\[
x = 360^\circ - (74^\circ + 96^\circ + 112^\circ) = 360^\circ - 282^\circ = 78^\circ
\]

| Name of Shape | Number of Sides | Sum of all Angles | Value of Missing Angle \( x \) |
|---------------|------------------|-------------------|--------------------------------|
| Quadrilateral | 4 | \( 360^\circ \) | \( 78^\circ \) |

---

#### 4. Pentagon (5 sides)

- Name of Shape: Pentagon
- Number of Sides: 5
- Sum of all Angles: Using the formula:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- Given Angles: \( 101^\circ \), \( 168^\circ \), \( 126^\circ \), and \( 89^\circ \)
- Missing Angle \( x \):
\[
x = 540^\circ - (101^\circ + 168^\circ + 126^\circ + 89^\circ) = 540^\circ - 484^\circ = 56^\circ
\]

| Name of Shape | Number of Sides | Sum of all Angles | Value of Missing Angle \( x \) |
|---------------|------------------|-------------------|--------------------------------|
| Pentagon | 5 | \( 540^\circ \) | \( 56^\circ \) |

---

#### 5. Hexagon (6 sides)

- Name of Shape: Hexagon
- Number of Sides: 6
- Sum of all Angles: Using the formula:
\[
(6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ
\]
- Given Angles: \( 155^\circ \), \( 163^\circ \), \( 112^\circ \), \( 92^\circ \), and \( 163^\circ \)
- Missing Angle \( x \):
\[
x = 720^\circ - (155^\circ + 163^\circ + 112^\circ + 92^\circ + 163^\circ) = 720^\circ - 685^\circ = 35^\circ
\]

| Name of Shape | Number of Sides | Sum of all Angles | Value of Missing Angle \( x \) |
|---------------|------------------|-------------------|--------------------------------|
| Hexagon | 6 | \( 720^\circ \) | \( 35^\circ \) |

---

#### 6. Heptagon (7 sides)

- Name of Shape: Heptagon
- Number of Sides: 7
- Sum of all Angles: Using the formula:
\[
(7-2) \times 180^\circ = 5 \times 180^\circ = 900^\circ
\]
- Given Angles: \( 95^\circ \), \( 209^\circ \), \( 77^\circ \), \( 84^\circ \), \( 250^\circ \), and \( 106^\circ \)
- Missing Angle \( x \):
\[
x = 900^\circ - (95^\circ + 209^\circ + 77^\circ + 84^\circ + 250^\circ + 106^\circ) = 900^\circ - 821^\circ = 79^\circ
\]

| Name of Shape | Number of Sides | Sum of all Angles | Value of Missing Angle \( x \) |
|---------------|------------------|-------------------|--------------------------------|
| Heptagon | 7 | \( 900^\circ \) | \( 79^\circ \) |

---

Final Answer:

\[
\boxed{
\begin{array}{|c|c|c|c|}
\hline
\text{Name of Shape} & \text{Number of Sides} & \text{Sum of all Angles} & \text{Value of Missing Angle } x \\
\hline
\text{Triangle} & 3 & 180^\circ & 76^\circ \\
\hline
\text{Quadrilateral} & 4 & 360^\circ & 78^\circ \\
\hline
\text{Pentagon} & 5 & 540^\circ & 56^\circ \\
\hline
\text{Hexagon} & 6 & 720^\circ & 35^\circ \\
\hline
\text{Heptagon} & 7 & 900^\circ & 79^\circ \\
\hline
\end{array}
}
\]