Why, if in a triangle all angles are equal, then the sides are equal?

This is the equilateral triangle theorem. You just need to know this theorem. Since AB = BC, ∠A = ∠C (as angles at the base of an isosceles triangle).

Similarly, since AC = BC, ∠A = ∠B.

It follows that in an equilateral triangle all angles are equal to each other: ∠A = ∠B = ∠C

Since the sum of the angles of a triangle is 180º, then ∠A = ∠B = ∠C = 180º: 3 = 60º, that is, each angle of an equilateral triangle is 60º