The angle between the bisector and the height of a right triangle that is drawn from the vertex

The angle between the bisector and the height of a right triangle that is drawn from the vertex of the right angle is 12 degrees. Find the angles of the triangle ABC

In a right-angled triangle with a right angle A, the bisector AM and the height AH come out.

1) Consider the triangle AHM. Since АН – height, the angle АНМ is rectangular and equal to 90 °. By condition, we know that the angle MAH = 12 °. We know that the sum of the angles of a triangle is 180 °, which means the angle АМH = 180 ° – angle MAH – angle МНА = 180 – 90 – 12 = 78 °.

2) Since AM is a bisector, the angle BAM = the angle of the MAC, and since the angle A is rectangular, it means that the angle BAM and the angle MAC are equal to 45 ° (90 °: 2)

3) Consider the AMC triangle. We know 2 angles, we can find the angle C, it will be equal to 180 ° – angle AMC – angle MAC = 180 ° – 45 ° – 78 ° = 57 °

4) Consider the triangle ABC. Angle A is a straight line, which means it is equal to 90 °, we found angle C in action 3, and it is equal to 57 °. To find angle B from 180 ° – angle A – angle C = 180 ° – 90 ° – 57 ° = 33 °.

Answer: angle A = 90 °, angle B = 33 °, angle C = 57 °