To find the angle measure at vertex A using the inverse tangent ratio, we first need to identify the correct sides of the triangle corresponding to the angle A.
In a triangle, if we denote:
- \( a \) as the length of the side opposite angle A,
- \( b \) as the length of the side adjacent to angle A,
- \( c \) as the length of the hypotenuse,
the tangent of angle A can be given by the formula:
\[ \tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b} \]
Since we need to find \( A \), the formula can be rearranged using the inverse tangent (also known as arctan):
\[ A = \tan^{-1}\left(\frac{a}{b}\right) \]
In your case, you have:
- \( b = 20 \)
- \( c = 25 \)
We need to find side \( a \) first. Since we know \( b \) and \( c \), we can find \( a \) using the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]
Substituting the values:
\[ 25^2 = a^2 + 20^2 \] \[ 625 = a^2 + 400 \] \[ a^2 = 625 - 400 \] \[ a^2 = 225 \] \[ a = \sqrt{225} = 15 \]
Now we can find angle \( A \):
\[ A = \tan^{-1}\left(\frac{a}{b}\right) = \tan^{-1}\left(\frac{15}{20}\right) = \tan^{-1}(0.75) \]
Using a calculator to find \( \tan^{-1}(0.75) \):
\[ A \approx 36.87^\circ \]
Rounding this to the nearest whole degree, we get:
\[ A \approx 37^\circ \]
Thus, the approximate angle measure at vertex \( A \) is 37 degrees.