To determine if a polygon can have interior angles that add up to \( 423^\circ \), we need to consider the general formula for the sum of the interior angles of a polygon. For a polygon with \( n \) sides, the sum of the interior angles is given by:
[tex]\[ (n-2) \times 180^\circ \][/tex]
We need to check if this sum can be \( 423^\circ \). Therefore, we set up the equation:
[tex]\[ (n-2) \times 180 = 423 \][/tex]
First, solve this equation for \( n \):
[tex]\[ (n-2) \times 180 = 423 \][/tex]
Divide both sides by 180:
[tex]\[ n-2 = \frac{423}{180} \][/tex]
Simplify the fraction:
[tex]\[ n-2 = \frac{423}{180} = \frac{87}{20} \][/tex]
So we have:
[tex]\[ n-2 = \frac{87}{20} \][/tex]
To find \( n \), add 2 to both sides:
[tex]\[ n = \frac{87}{20} + 2 \][/tex]
[tex]\[ n = \frac{87}{20} + \frac{40}{20} \][/tex]
[tex]\[ n = \frac{127}{20} \][/tex]
Now, calculate \( \frac{127}{20} \):
[tex]\[ n = 6.35 \][/tex]
Since \( n \) must be a positive integer (because the number of sides of a polygon must be a whole number), and \( 6.35 \) is not an integer, we conclude that it is not possible for a polygon to have interior angles that add up to \( 423^\circ \).
Thus, a polygon cannot have interior angles that sum to [tex]\( 423^\circ \)[/tex] because the number of sides calculated from this sum is not an integer.
