The angles of a triangle form an arithmetic progression. Prove that one of them is 60 degrees.

Let us denote the angles of the triangle by α, β and γ. As you know, in any triangle, the sum of the angles of a triangle is 180 °, that is, α + β + γ = 180 °.
According to the condition of the task, the angles of the triangle α, β and γ form an arithmetic progression. Then, according to the definition of an arithmetic progression, we have: β – α = γ – β, whence α + γ = β + β = 2 * β.
Substitute the found expression α + γ = 2 * β into the equality of item 1. Then we get: 2 * β + β = 180 ° or 3 * β = 180 °, whence β = 180 °: 3 = 60 °. Q.E.D.