Find the acute angles of a right-angled triangle if their values are proportional to the numbers 4 and 5.

Let ABC be a right-angled triangle, angle C = 90 degrees, angle A: angle B = 4: 5.
Let x be the coefficient of proportionality, then the angle A = 4x, the angle B = 5x.
By the theorem on the sum of the angles of a triangle:
angle A + angle B + angle C = 180 degrees;
4x + 5x + 90 = 180;
9x = 90;
x = 90/9;
x = 10.
Angle A = 4x = 4 * 10 = 40 degrees, angle B = 5x = 5 * 10 = 50 degrees.
Answer: angle A = 40 degrees, angle B = 50 degrees.