Special topic: application problem.

Analysis: After reading this question, we first think that isosceles triangles can be divided into acute, right and obtuse isosceles triangles. When it is an isosceles right triangle, it is impossible to have the situation mentioned in the question, so we omit it. The remaining two situations can be discussed by drawing pictures.

Solution: ① When it is an acute triangle, you can draw pictures.

The height and the right waist form an included angle of 60, which can be obtained from the sum of the internal angles of the triangle as 180, and the top angle is 30.

When it is an obtuse triangle, you can draw a picture.

At this time, the vertical foot falls outside the triangle, because the sum of the internal angles of the triangle is 180.

As can be seen from the figure, the complementary angle of the top angle of the isosceles triangle is 30,

The vertex angle of the triangle is 150,

So the answer is 30 or 150.

Comments: This topic mainly examines the nature of isosceles triangle and the interior angle and theorem of triangle. When doing the problem, we should consider the problem comprehensively and make a model to help solve it if necessary. It is the key to answer this question correctly through classified discussion with moderate difficulty.