Answer :
SOLUTION
The number of diagonals and number of sides of a polygon are related with the formula
[tex]\begin{gathered} no\text{. of diagonals = }\frac{n(n-3)}{2} \\ \text{Where n = number of sides of the polygon } \end{gathered}[/tex]Substituting we have
[tex]\begin{gathered} no\text{. of diagonals = }\frac{n(n-3)}{2} \\ 20\text{ = }\frac{n(n-3)}{2} \\ 20=\frac{n^2-3n}{2} \\ \text{cross multiplying we have } \\ n^2-3n=20\times2 \\ n^2-3n=40 \\ n^2-3n-40=0 \end{gathered}[/tex]Solving the quadratic equation for n, we have
[tex]\begin{gathered} n^2-3n-40=0 \\ n^2-8n+5n-40=0 \\ n(n-8)+5(n-8)=0 \\ (n+5)(n-8)=0 \\ \text{Either } \\ n+5=0 \\ n=-5\text{ } \\ Or \\ n-8=0 \\ n=8 \end{gathered}[/tex]So, we will go with n = 8, since the number of sides cannot be a negative number.
Now, sum of interior angles in a regular polygon is given by
[tex]\begin{gathered} S=180(n-2) \\ \text{Where S = sum and n = numbers } \end{gathered}[/tex]So we have
[tex]\begin{gathered} S=180(n-2) \\ S=180(8-2) \\ S=180\times6 \\ S=1080\text{ } \end{gathered}[/tex]So, the measure of an interior angle becomes
[tex]\begin{gathered} \frac{S}{n} \\ =\frac{1080}{8} \\ =135\degree \end{gathered}[/tex]Hence the answer is 135 degrees