In triangle ABC, angle C is 90 degrees, CH is height, and angle A is 34 degrees. Find angle BCH.

Option 1.
In a right-angled triangle AНС (CH – height by condition) we find the second acute angle AСН.
∠ АСН = 90 ° – ∠ А = 90 ° – 34 ° = 56 °.
We find the angle BCH, for this we subtract the found angle AСН from the right angle C:
90 ° – 56 ° = 34 °.
Option 2.
Consider two right-angled triangles: ABC and СВН, they are similar (acute angle B is common). Equality of angles follows from the similarity of triangles:
∠ ВСН = ∠ А = 34 °.
Answer: the degree measure of the BCH angle is 34 °.