To find the angle at vertex A using the inverse tangent ratio, we can use the sides of the triangle. Since angle C is a right angle, we can identify the sides relative to angle A.
Given:
- BC = 20 (opposite side to angle A)
- CA = 25 (adjacent side to angle A)
We can use the tangent ratio:
\[ \tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{CA} \]
Substituting the given values:
\[ \tan(A) = \frac{20}{25} = 0.8 \]
Now we will use the inverse tangent (arctan) to find the angle A:
\[ A \approx \tan^{-1}(0.8) \]
Using a calculator to find the inverse tangent, we get:
\[ A \approx 38.66^\circ \]
Rounding to the nearest whole degree, the angle at vertex A is approximately:
\[ \boxed{39^\circ} \]