In a rectangle, the diagonal divides the angle by a ratio of 4: 5 Find the angle between the diagonals.

Since all the corners of the rectangle are straight, the sum of the angles by which the diagonal divides the vertex angle will be 90 degrees. Let x be the coefficient of proportionality, then angle 1 = 4x, angle 2 = 5x.
4x + 5x = 90 degrees; 9x = 90 degrees; x = 90 degrees / 9; x = 10 degrees.
angle 1 = 4x = 4 * 10 degrees = 40 degrees
angle 2 = 5x = 5 * 10 degrees = 50 degrees
The diagonals of the rectangle are equal and the intersection point is halved. Thus, they form isosceles triangles.
In triangle 1, the base angles are 40 degrees. According to the theorem on the sum of the angles of a triangle, angle 1 + angle 2 + angle 3 = 180 degrees. Angle 1 and angle 2 are 40 degrees, so:
40 degrees + 40 degrees + angle 3 = 180 degrees; angle 3 = 180 degrees – 80 degrees; angle 3 = 100 degrees.
In triangle 2, the base angles are 50 degrees. According to the theorem on the sum of the angles of a triangle, angle 1 + angle 2 + angle 3 = 180 degrees. Angle 1 and Angle 2 are 50 degrees, so:
50 degrees + 50 degrees + angle 3 = 180 degrees; angle 3 = 180 degrees – 100 degrees; angle 3 = 80 degrees.
Answer: the angles between the diagonals are 100 degrees and 80 degrees.