Interior Angles of Polygons Worksheet | PDF - Free Printable

Problem Analysis:

The worksheet involves two main tasks:
1. Finding the interior angles in given polygons.
2. Finding the value of \( x \) in various polygonal figures.

We will solve each part step by step.

---

Task 1: Find the interior angles in each given polygons

#### a) Rectangle
- A rectangle is a quadrilateral with 4 sides.
- The formula for the sum of interior angles of a polygon with \( n \) sides is:
\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\]
- For a rectangle (\( n = 4 \)):
\[
\text{Sum of interior angles} = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]
- All interior angles of a rectangle are right angles (\( 90^\circ \)).

#### b) Pentagon
- A pentagon has 5 sides.
- Using the formula for the sum of interior angles:
\[
\text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
- If the pentagon is regular, all interior angles are equal:
\[
\text{Each interior angle} = \frac{540^\circ}{5} = 108^\circ
\]

#### c) Octagon
- An octagon has 8 sides.
- Using the formula for the sum of interior angles:
\[
\text{Sum of interior angles} = (8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ
\]
- If the octagon is regular, all interior angles are equal:
\[
\text{Each interior angle} = \frac{1080^\circ}{8} = 135^\circ
\]

#### d) Triangle
- A triangle has 3 sides.
- Using the formula for the sum of interior angles:
\[
\text{Sum of interior angles} = (3 - 2) \times 180^\circ = 1 \times 180^\circ = 180^\circ
\]
- If the triangle is equilateral, all interior angles are equal:
\[
\text{Each interior angle} = \frac{180^\circ}{3} = 60^\circ
\]

---

Task 2: Find the value of \( x \)

#### a)
- The figure is a quadrilateral with one exterior angle marked as \( x^\circ \).
- The exterior angle is supplementary to the interior angle at that vertex.
- The sum of the interior angles of a quadrilateral is \( 360^\circ \).
- The given interior angles are \( 65^\circ \), \( 70^\circ \), and \( 75^\circ \). Let the fourth interior angle be \( y \):
\[
65^\circ + 70^\circ + 75^\circ + y = 360^\circ
\]
\[
210^\circ + y = 360^\circ
\]
\[
y = 150^\circ
\]
- The exterior angle \( x \) is supplementary to \( y \):
\[
x = 180^\circ - 150^\circ = 30^\circ
\]

#### b)
- The figure is a quadrilateral with one interior angle marked as \( x^\circ \).
- The sum of the interior angles of a quadrilateral is \( 360^\circ \).
- The given interior angles are \( 85^\circ \), \( 75^\circ \), and \( 105^\circ \). Let the fourth interior angle be \( x \):
\[
85^\circ + 75^\circ + 105^\circ + x = 360^\circ
\]
\[
265^\circ + x = 360^\circ
\]
\[
x = 95^\circ
\]

#### c)
- The figure is a triangle with one exterior angle marked as \( x^\circ \).
- The exterior angle is equal to the sum of the two non-adjacent interior angles.
- The given interior angles are \( 60^\circ \) and \( 30^\circ \):
\[
x = 60^\circ + 30^\circ = 90^\circ
\]

#### d)
- The figure is a pentagon with one interior angle marked as \( x^\circ \).
- The sum of the interior angles of a pentagon is \( 540^\circ \).
- The given interior angles are \( 55^\circ \), \( 95^\circ \), \( 280^\circ \), and \( 85^\circ \). Let the fifth interior angle be \( x \):
\[
55^\circ + 95^\circ + 280^\circ + 85^\circ + x = 540^\circ
\]
\[
515^\circ + x = 540^\circ
\]
\[
x = 25^\circ
\]

#### e)
- The figure is a quadrilateral with one interior angle marked as \( x^\circ \).
- The sum of the interior angles of a quadrilateral is \( 360^\circ \).
- The given interior angles are \( 125^\circ \), \( 117^\circ \), and \( 68^\circ \). Let the fourth interior angle be \( x \):
\[
125^\circ + 117^\circ + 68^\circ + x = 360^\circ
\]
\[
310^\circ + x = 360^\circ
\]
\[
x = 50^\circ
\]

#### f)
- The figure is a quadrilateral with one interior angle marked as \( x^\circ \).
- The sum of the interior angles of a quadrilateral is \( 360^\circ \).
- The given interior angles are \( 144^\circ \), \( 230^\circ \), and \( 106^\circ \). Let the fourth interior angle be \( x \):
\[
144^\circ + 230^\circ + 106^\circ + x = 360^\circ
\]
\[
480^\circ + x = 360^\circ
\]
\[
x = -120^\circ
\]
- This result is not possible for an interior angle. There might be an error in the problem setup or interpretation.

#### g)
- The figure is a quadrilateral with one interior angle marked as \( x^\circ \).
- The sum of the interior angles of a quadrilateral is \( 360^\circ \).
- The given interior angles are \( 118^\circ \), \( 124^\circ \), and \( 80^\circ \). Let the fourth interior angle be \( x \):
\[
118^\circ + 124^\circ + 80^\circ + x = 360^\circ
\]
\[
322^\circ + x = 360^\circ
\]
\[
x = 38^\circ
\]

---

Final Answers:

1. Interior Angles:
- a) Rectangle: \( 90^\circ \)
- b) Pentagon: \( 108^\circ \)
- c) Octagon: \( 135^\circ \)
- d) Triangle: \( 60^\circ \)

2. Value of \( x \):
- a) \( 30^\circ \)
- b) \( 95^\circ \)
- c) \( 90^\circ \)
- d) \( 25^\circ \)
- e) \( 50^\circ \)
- f) Not possible (error in setup)
- g) \( 38^\circ \)

\boxed{
\begin{aligned}
&\text{1. a) } 90^\circ, \text{ b) } 108^\circ, \text{ c) } 135^\circ, \text{ d) } 60^\circ \\
&\text{2. a) } 30^\circ, \text{ b) } 95^\circ, \text{ c) } 90^\circ, \text{ d) } 25^\circ, \text{ e) } 50^\circ, \text{ g) } 38^\circ
\end{aligned}
}