1. The measure of an interior angle of a regular polygon is 20 more than thrice the measure of its adjacent exterior angle. Find the number of sides of the polygon and its total number of diagonals.

2. Find the sum and difference between the sum of the measures of the interior angles of a convex 23-gon and 14-gon.

asked by Brianna
  1. 1. let the number of sides be n
    each interior angle = 180(n-2)/n
    exterior angle = 360/n

    180(n-2)/n = 3(360/n) + 20
    times n
    180(n-2) = 1080 + 20n
    180n - 360 = 1080 + 20n
    160n = 1440
    n = 9

    We have a 9-gon, which is called a enneagon or nonagon

    interior angle = 180(7)/9 = 140
    exterior angle = 360/9 = 40
    is 3 times 40 + 20 = 140 ? , YES

    sum interior angles of 23-gon= 180(21) = 3780
    sum of interior angles of 14-gon = ....
    then take their sum and difference

    posted by Reiny

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