The number of diagonals of a regular polygon is 27. Then, find the measure of each of the interior angles of the polygon.

Explanation

Let number of sides of a polygon be n.
`implies" ""Number of diagonals"=(n(n-3))/(2)`
`therefore" "(n(n-3))/(2)=27`
`" "n(n-3)=54`
`{:(implies,n^(2)-3n-54=0,),(implies,(n-9)(n+6)=0,),(therefore,n-9=0orn+6=0,),(implies,n=9orn= -6,("no.of sides connot be negative")),(therefore,n=9,):}`
`therefore` It is a 9-sided polygon.
`therefore` Each interior angle `=((n-2))/(n)xx180^(@)=((9-2))/(9)xx180^(@)=140^(@)`