Certainly! Let's go through the solution step-by-step.
1. Understand the relationship:
- The ratio between the exterior angle and the interior angle of the polygon is given as [tex]\(1:5\)[/tex].
2. Define the angles:
- Let the exterior angle be [tex]\(x\)[/tex].
- Therefore, the interior angle will be [tex]\(5x\)[/tex] (since the ratio is 1:5).
3. Sum of angles at one vertex:
- The sum of the exterior and interior angles at one vertex of the polygon is [tex]\(180^\circ\)[/tex], because they form a linear pair.
- Hence, we have:
[tex]\[
x + 5x = 180^\circ
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
- Combine like terms:
[tex]\[
6x = 180^\circ
\][/tex]
- Divide both sides by 6:
[tex]\[
x = \frac{180^\circ}{6} = 30^\circ
\][/tex]
- So, the exterior angle [tex]\(x\)[/tex] is [tex]\(30^\circ\)[/tex].
5. Determine the interior angle:
- Since the interior angle is [tex]\(5x\)[/tex]:
[tex]\[
\text{Interior angle} = 5 \times 30^\circ = 150^\circ
\][/tex]
6. Find the number of sides:
- The sum of all the exterior angles of any polygon is always [tex]\(360^\circ\)[/tex].
- If each exterior angle is [tex]\(30^\circ\)[/tex], the number of sides [tex]\(n\)[/tex] of the polygon can be found using the formula for the exterior angle:
[tex]\[
\text{Exterior angle} = \frac{360^\circ}{n}
\][/tex]
- Plug [tex]\(30^\circ\)[/tex] into the formula:
[tex]\[
30^\circ = \frac{360^\circ}{n}
\][/tex]
- Solve for [tex]\(n\)[/tex]:
[tex]\[
n = \frac{360^\circ}{30^\circ} = 12
\][/tex]
So, the number of sides of the polygon is 12.
