A convex polygon has 20 diagonals. Find the number of sides and the sum of the angles.

Since the polygon is convex, the number of its diagonals can be determined by the formula:

N = n * (n – 3) / 2, where n is the number of corners of the polygon, N is the number of its diagonals.

Then 20 = n * (n – 3) / 2.

n ^ 2 – 3 * n – 40 = 0.

Let’s solve the quadratic equation.

D = b2 – 4 * a * c = (-3) ^ 2 – 4 * 1 * (-40) = 9 + 160 = 169.

n1 = (3 – √169) / (2 * 1) = (3 – 13) / 2 = -10 / 2 = -5. (does not fit).

n2 = (3 + √169) / (2 * 1) = (3 + 13) / 2 = 16/2 = 8.

Answer: 8 sides.

The sum of the angles of the polygon is determined by the formula:

Sn = (n – 2) * 180 = (8 – 2) * 180 = 6 * 180 = 1080.

Answer: The sum of the angles is 1080.