Answer :
To solve this problem, we need to understand some key properties of polygons and the specific details provided about the decagon in question.
1. Understanding Exterior Angles of a Polygon:
- A regular polygon (a polygon with all sides and angles equal) with \( n \) sides has an exterior angle sum that is always equal to \( 360 \) degrees, regardless of the number of sides.
- Each exterior angle of a regular polygon is given by the formula:
[tex]\[ \frac{360}{n} \][/tex]
2. Identifying the Number of Sides:
- In this problem, we are dealing with a regular decagon. A decagon has \( 10 \) sides.
3. Calculating Each Exterior Angle:
- The measure of each exterior angle in a regular decagon can be calculated using the formula mentioned above:
[tex]\[ \text{Each exterior angle} = \frac{360}{10} = 36 \text{ degrees} \][/tex]
4. Given the Exterior Angle Expression:
- According to the problem, each exterior angle is expressed as \( (3x + 6) \) degrees.
5. Setting Up the Equation:
- We know that the exterior angle measure \( 36 \) degrees is equal to the given expression \( 3x + 6 \). Thus, we can set up the following equation:
[tex]\[ 3x + 6 = 36 \][/tex]
6. Solving the Equation for \( x \):
- To find the value of \( x \), we solve the equation:
[tex]\[ 3x + 6 = 36 \][/tex]
Subtract \( 6 \) from both sides:
[tex]\[ 3x = 30 \][/tex]
Divide both sides by \( 3 \):
[tex]\[ x = 10 \][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{10} \)[/tex].
1. Understanding Exterior Angles of a Polygon:
- A regular polygon (a polygon with all sides and angles equal) with \( n \) sides has an exterior angle sum that is always equal to \( 360 \) degrees, regardless of the number of sides.
- Each exterior angle of a regular polygon is given by the formula:
[tex]\[ \frac{360}{n} \][/tex]
2. Identifying the Number of Sides:
- In this problem, we are dealing with a regular decagon. A decagon has \( 10 \) sides.
3. Calculating Each Exterior Angle:
- The measure of each exterior angle in a regular decagon can be calculated using the formula mentioned above:
[tex]\[ \text{Each exterior angle} = \frac{360}{10} = 36 \text{ degrees} \][/tex]
4. Given the Exterior Angle Expression:
- According to the problem, each exterior angle is expressed as \( (3x + 6) \) degrees.
5. Setting Up the Equation:
- We know that the exterior angle measure \( 36 \) degrees is equal to the given expression \( 3x + 6 \). Thus, we can set up the following equation:
[tex]\[ 3x + 6 = 36 \][/tex]
6. Solving the Equation for \( x \):
- To find the value of \( x \), we solve the equation:
[tex]\[ 3x + 6 = 36 \][/tex]
Subtract \( 6 \) from both sides:
[tex]\[ 3x = 30 \][/tex]
Divide both sides by \( 3 \):
[tex]\[ x = 10 \][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{10} \)[/tex].
