How many sides does a convex polygon have, each angle of which is: a) 90 degrees b) 60 degrees

How many sides does a convex polygon have, each angle of which is: a) 90 degrees b) 60 degrees c) 120 degrees and d) 108 degrees

Let us prove that the sum of the interior angles of an n – gon is equal to 180 ° * (n – 2).

Let’s denote the vertices of the polygon as V1 V2 V3 … Vn.

Let’s connect the vertex V1 with the vertices V3 … V (n-1) by diagonals.

The diagonals of the polygon will divide it into n – 2 triangles, the sum of the angles of which is equal to the sum of the angles of the n – gon. Therefore, the sum of the angles of the n – gon is 180 ° * (n – 2).

If in an n-gon all the angles are equal to K, then the sum of all its angles is equal to K * n.

We have:

180 ° * (n – 2) = K * n,

180 ° * n – 360 ° = K * n,

(180 ° – K) * n = 360 °,

n = 360 ° / (180 ° – K).

a) If all angles are 90 °, then n = 360 ° / (180 ° – K) = 360 ° / (180 ° – 90 °) = 4.

b) If all angles are 60 °, then n = 360 ° / (180 ° – K) = 360 ° / (180 ° – 60 °) = 3.

c) If all angles are 120 °, then n = 360 ° / (180 ° – K) = 360 ° / (180 ° – 120 °) = 6.

d) If all angles are 108 °, then n = 360 ° / (180 ° – K) = 360 ° / (180 ° – 108 °) = 5.