In rectangle ABCD, the bisector of angle A forms angles with diagonal BD

In rectangle ABCD, the bisector of angle A forms angles with diagonal BD, one of which is 105 °. Find the angle between the diagonals of the rectangle.

Let’s build a rectangle ABCD; draw diagonals AC and BD, which will intersect at point O; construct a bisector from angle A, denote AM, AM will intersect BD at point K.

Consider a triangle AKD: A = 45о (by definition of the bisector), K = 105о (by condition), D = 180о – (105о + 45о) = 30о;

Angle АDC = 90о (rectangle); angle ВDC = 60о, (90о – 30о = 60о);

Consider a triangle DОС: it is isosceles, since the diagonals in the rectangle are equal and at the point of intersection they are divided in half, and in an isosceles triangle the angles at the base are equal, therefore, D = C = 60о, the angle O also accounts for 60о (according to the theorem on the sum of angles in triangle);

The DOC is the angle between the diagonals of the rectangle.

Answer: 60o (120o).