The angle between the height and the bisector of a right-angled triangle drawn from the vertex

The angle between the height and the bisector of a right-angled triangle drawn from the vertex of a right angle is 12 degrees. Find the largest acute angle of this right-angled triangle.

The bisector will divide the right angle in half, and the difference in the angles between it and the height is 12, which means that the angle formed by the height, leg and hypotenuse of the original triangle (see Fig.), Which coincides with the straight line of the original, will be:

90: 2 – 12 = 45 – 12 = 33 (degrees).

In the newly formed triangle (height, leg and segment of the hypotenuse of the original triangle), one angle = 33 (defined above), the other = 90 (because the height), the third must be found. We define it by the sum of the angles in the triangle:

180 – 90 – 33 = 90 – 33 = 57 (degrees).

Answer: 57.