The diagonals of the rectangle ABCD meet at point O, the angle ABO is 64 °. Find the angle AOD.

The solution to such problems is based on knowledge of the properties of the diagonals in a rectangle. Their intersection gives two pairs of equal isosceles triangles. Consider an isosceles triangle AOB, in which, by condition, the angle at the base is given.
Angle ABO = angle BAO = 64 ° (angles at the base of an isosceles triangle).
Vertex angle AOB = 180 – 2 * 64 ° = 180 ° – 128 ° = 52 °.
The angle AOD, which we need to find, is adjacent to the angle AOB and its degree measure is:
Angle AOD = 180 ° – angle AOB = 180 ° – 128 ° = 52 °.
Answer: the AOD angle is 52 °.