To determine how many sides a regular polygon has when each interior angle measures 157.5°, we can use the formula for the interior angle of a regular polygon.
The formula for the interior angle of a regular polygon with [tex]\( n \)[/tex] sides is:
[tex]\[ \text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n} \][/tex]
Given that the interior angle is 157.5°, we can set up the equation:
[tex]\[ \frac{(n-2) \times 180}{n} = 157.5 \][/tex]
Now we solve for [tex]\( n \)[/tex]:
1. Start by multiplying both sides by [tex]\( n \)[/tex] to get rid of the denominator:
[tex]\[ (n-2) \times 180 = 157.5n \][/tex]
2. Distribute 180 on the left side:
[tex]\[ 180n - 360 = 157.5n \][/tex]
3. Isolate [tex]\( n \)[/tex] by moving all [tex]\( n \)[/tex]-terms to one side and constants to the other:
[tex]\[ 180n - 157.5n = 360 \][/tex]
4. Combine like terms:
[tex]\[ 22.5n = 360 \][/tex]
5. Solve for [tex]\( n \)[/tex] by dividing both sides by 22.5:
[tex]\[ n = \frac{360}{22.5} \][/tex]
6. Calculate the value:
[tex]\[ n = 16 \][/tex]
Therefore, the polygon has [tex]\( 16 \)[/tex] sides.
The correct answer is:
[tex]\[ \boxed{16 \text{ sides}} \][/tex]
So the answer is (B) 16 sides.
