The angle at the base of an isosceles triangle is 40 degrees. Find the angle at the vertex.

Let ABC be an isosceles triangle, the angle at the base of the AC is A = 40 degrees.
Since the angles at the base of an isosceles triangle are equal, then angle A = angle C, then angle C = 40 degrees.
By the theorem on the sum of the angles of a triangle:
angle A + angle B + angle C = 180 degrees;
40 + angle B + 40 = 180;
angle B = 180 – 80;
angle B = 100 degrees.
Answer: angle B = 100 degrees.