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. Here's how we will approach each part:

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#### Step 1: General Formula for the Sum of Interior Angles
The sum of the interior angles of an \( n \)-sided polygon is given by the formula:
\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\]

#### Step 2: Solve Each Polygon Individually
We will use the formula above to find the sum of the interior angles for each polygon and then solve for the missing angle \( x \).

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Solution for Each Question

#### Question 1: Write an expression in terms of \( n \) for the sum of the interior angles in any \( n \)-sided polygon.
Using the formula:
\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\]
So, the expression is:
\[
\boxed{(n - 2) \times 180^\circ}
\]

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#### Question 2: Triangle
- Name of Shape: Triangle
- Number of Sides: 3
- Sum of All Angles: For a triangle, the sum of the interior angles is always:
\[
(3 - 2) \times 180^\circ = 180^\circ
\]
- 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° | 76° |

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#### Question 3: Quadrilateral
- Name of Shape: Quadrilateral
- Number of Sides: 4
- Sum of All Angles: For a quadrilateral, the sum of the interior angles is:
\[
(4 - 2) \times 180^\circ = 360^\circ
\]
- Given Angles: \( 74^\circ \), \( 112^\circ \), and \( 96^\circ \)
- Missing Angle \( x \):
\[
x = 360^\circ - (74^\circ + 112^\circ + 96^\circ) = 360^\circ - 282^\circ = 78^\circ
\]

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

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#### Question 4: Pentagon
- Name of Shape: Pentagon
- Number of Sides: 5
- Sum of All Angles: For a pentagon, the sum of the interior angles is:
\[
(5 - 2) \times 180^\circ = 540^\circ
\]
- 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° | 56° |

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#### Question 5: Hexagon
- Name of Shape: Hexagon
- Number of Sides: 6
- Sum of All Angles: For a hexagon, the sum of the interior angles is:
\[
(6 - 2) \times 180^\circ = 720^\circ
\]
- 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° | 35° |

---

#### Question 6: Heptagon
- Name of Shape: Heptagon
- Number of Sides: 7
- Sum of All Angles: For a heptagon, the sum of the interior angles is:
\[
(7 - 2) \times 180^\circ = 900^\circ
\]
- 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° | 79° |

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Final Answer Summary

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