Let's solve each part of the question step-by-step.
### Part (a)
To determine how many sides a polygon has given the sum of its interior angles, we use the formula for the sum of the interior angles of an [tex]\( n \)[/tex]-sided polygon:
[tex]\[ \text{Sum of interior angles} = (n-2) \times 180^\circ \][/tex]
Given the sum of the interior angles is 4680 degrees, we set up the equation:
[tex]\[ (n-2) \times 180^\circ = 4680^\circ \][/tex]
Solving for [tex]\( n \)[/tex]:
1. Divide both sides of the equation by 180:
[tex]\[ n-2 = \frac{4680}{180} \][/tex]
2. Simplify the division:
[tex]\[ n-2 = 26 \][/tex]
3. Add 2 to both sides to solve for [tex]\( n \)[/tex]:
[tex]\[ n = 26 + 2 \][/tex]
[tex]\[ n = 28 \][/tex]
So, the polygon has 28 sides.
### Part (b)
To find the number of sides of a regular polygon given one interior angle, we use the relationship between the interior angle and the number of sides. The formula for the interior angle of a regular [tex]\( n \)[/tex]-sided polygon is:
[tex]\[ \text{Interior angle} = \frac{(n-2) \times 180^\circ}{n} \][/tex]
Given the interior angle is 140 degrees, we set up the equation:
[tex]\[ \frac{(n-2) \times 180^\circ}{n} = 140^\circ \][/tex]
Solving for [tex]\( n \)[/tex]:
1. Multiply both sides by [tex]\( n \)[/tex] to clear the fraction:
[tex]\[ (n-2) \times 180 = 140n \][/tex]
2. Distribute the 180 on the left side:
[tex]\[ 180n - 360 = 140n \][/tex]
3. Move all terms involving [tex]\( n \)[/tex] to one side:
[tex]\[ 180n - 140n = 360 \][/tex]
4. Simplify the equation:
[tex]\[ 40n = 360 \][/tex]
5. Divide both sides by 40 to solve for [tex]\( n \)[/tex]:
[tex]\[ n = \frac{360}{40} \][/tex]
[tex]\[ n = 9 \][/tex]
So, the regular polygon has 9 sides.
### Part (c)
To find the measure of each exterior angle in a regular 18-sided polygon (18-gon), we use the formula for the exterior angle of a regular polygon:
[tex]\[ \text{Exterior angle} = \frac{360^\circ}{n} \][/tex]
Given [tex]\( n = 18 \)[/tex]:
[tex]\[ \text{Exterior angle} = \frac{360^\circ}{18} \][/tex]
Simplify the division:
[tex]\[ \text{Exterior angle} = 20^\circ \][/tex]
So, each exterior angle of a regular 18-gon is 20 degrees.
