Angles - Free Printable

Let's solve each problem step by step.

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Problem 1: The diagram shows a regular nonagon. Calculate \( x \).

#### Solution:
A nonagon is a polygon with 9 sides. For a regular polygon, all exterior angles are equal. The sum of the exterior angles of any polygon is always \( 360^\circ \).

For a regular nonagon:
\[
\text{Each exterior angle} = \frac{360^\circ}{9} = 40^\circ
\]

The variable \( x \) represents one of the exterior angles of the nonagon. Therefore:
\[
x = 40^\circ
\]

#### Final Answer:
\[
\boxed{40}
\]

---

Problem 2: The exterior angle of a regular polygon is \( 45^\circ \). What is the size of 1 of its interior angles?

#### Solution:
For any polygon, the relationship between an exterior angle and an interior angle is:
\[
\text{Interior angle} + \text{Exterior angle} = 180^\circ
\]

Given that the exterior angle is \( 45^\circ \):
\[
\text{Interior angle} = 180^\circ - 45^\circ = 135^\circ
\]

#### Final Answer:
\[
\boxed{135}
\]

---

Problem 3: Calculate \( x \).

#### Solution:
The given figure is a quadrilateral with three interior angles provided: \( 45^\circ \), \( 75^\circ \), and \( 95^\circ \). The fourth angle is labeled as \( x \).

The sum of the interior angles of a quadrilateral is:
\[
360^\circ
\]

Using this, we can write:
\[
45^\circ + 75^\circ + 95^\circ + x = 360^\circ
\]

Simplify:
\[
215^\circ + x = 360^\circ
\]

Solve for \( x \):
\[
x = 360^\circ - 215^\circ = 145^\circ
\]

#### Final Answer:
\[
\boxed{145}
\]

---

Problem 4: The diagram shows part of a regular polygon and its exterior angle. How many sides does the complete polygon have?

#### Solution:
The given exterior angle is \( 15^\circ \). For a regular polygon, the number of sides \( n \) can be calculated using the formula:
\[
\text{Exterior angle} = \frac{360^\circ}{n}
\]

Rearranging to solve for \( n \):
\[
n = \frac{360^\circ}{\text{Exterior angle}}
\]

Substitute the given exterior angle:
\[
n = \frac{360^\circ}{15^\circ} = 24
\]

#### Final Answer:
\[
\boxed{24}
\]

---

Problem 5: What is the exterior angle of a regular quadrilateral?

#### Solution:
A regular quadrilateral is a square. The sum of the exterior angles of any polygon is always \( 360^\circ \). For a regular polygon, all exterior angles are equal.

For a square (which has 4 sides):
\[
\text{Each exterior angle} = \frac{360^\circ}{4} = 90^\circ
\]

#### Final Answer:
\[
\boxed{90}
\]

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Problem 6: I'm thinking of a regular polygon. The interior angle is exactly half of the exterior angle. What polygon is it?

#### Solution:
Let the exterior angle of the polygon be \( E \). Then the interior angle is:
\[
\text{Interior angle} = 180^\circ - E
\]

According to the problem, the interior angle is half of the exterior angle:
\[
180^\circ - E = \frac{E}{2}
\]

Multiply through by 2 to eliminate the fraction:
\[
2(180^\circ - E) = E
\]
\[
360^\circ - 2E = E
\]

Combine like terms:
\[
360^\circ = 3E
\]

Solve for \( E \):
\[
E = \frac{360^\circ}{3} = 120^\circ
\]

Now, use the formula for the number of sides \( n \) of a regular polygon:
\[
E = \frac{360^\circ}{n}
\]

Substitute \( E = 120^\circ \):
\[
120^\circ = \frac{360^\circ}{n}
\]

Solve for \( n \):
\[
n = \frac{360^\circ}{120^\circ} = 3
\]

Thus, the polygon is a triangle (specifically, an equilateral triangle).

#### Final Answer:
\[
\boxed{3}
\]

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Summary of Answers:

1. \( \boxed{40} \)
2. \( \boxed{135} \)
3. \( \boxed{145} \)
4. \( \boxed{24} \)
5. \( \boxed{90} \)
6. \( \boxed{3} \)