# Find the angles of a convex triangle if each of its angles is equal to: a) 144 °; b) 150 °; c) 170 °; d) 171 °

By hypothesis, all angles of a convex polygon are equal, then this polygon is regular.

The degree measure of the angle of a regular polygon is calculated by the formula:

α = (n – 2) / n * 180 °,

where α is the angle of a regular polygon, n is the number of sides of a regular polygon.

Then:

(180 ° * (n – 2)) / n = α;

(180 ° * n – 180 ° * 2) / n = α;

(180 ° * n – 360 °) / n = α;

180 ° * n – 360 ° = α * n (proportional);

180 ° * n – α * n = 360 °;

n * (180 ° – α) = 360 °;

n = 360 ° / (180 ° – α) (proportional).

a) α = 144 °.

n = 360 ° / (180 ° – 144 °) = 360 ° / 36 ° = 10.

b) α = 150 °.

n = 360 ° / (180 ° – 150 °) = 360 ° / 30 ° = 12.

c) α = 170 °.

n = 360 ° / (180 ° – 170 °) = 360 ° / 10 ° = 36.

d) α = 171 °.

n = 360 ° / (180 ° – 171 °) = 360 ° / 9 ° = 40.

Answer: a) n = 10; b) n = 12; c) n = 36; d) n = 40.