A triangle with a vertex angle (angle at the top) of 40° must have two base angles of 70.° This is the only possibility for an Isosceles Triangle because of the Triangle Sum Theorem and Isosceles Triangle Theorem. In other words, an Isosceles Triangle must have a pair of congruent base angles and a pair of congruent sides opposite those base angles. Therefore, both base angles being congruent, there is only one angle measure for the two base angles to satisfy the Triangle Sum Theorem, so this is the only possibility given one angle measures 40°.
Furthermore, other triangles will meet these conditions, but just not an Isosceles Triangles. Triangles with different interior angle measures can contain a 40° angle; just not triangles with a pair of congruent angles. For example, a triangle with interior angles 60°, 40°,80° has exactly one 40° angle and sums to 180.
Constructing a triangle like the one given in my provided example is different than an Isosceles Triangle because the two base angles are not congruent, and thus the sides leading to the vertex (opposite the base angles) are also not congruent. Therefore, all 3 sides will measure different lengths and angles.