To find the angle at vertex A (denoted as angle A), we can use the cosine function based on the right triangle formed by points A, B, and C. From your description, we have the following:
- \( AC = 8 \) (the length adjacent to angle A)
- \( BC = 15 \) (the length opposite to angle A)
- \( AB = 17 \) (the hypotenuse)
Using the cosine definition: \[ \cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AC}{AB} = \frac{8}{17} \]
Now, we'll use the inverse cosine function to find angle A: \[ A = \cos^{-1}\left(\frac{8}{17}\right) \]
Calculating \( \frac{8}{17} \): \[ \frac{8}{17} \approx 0.4706 \]
Now using a calculator to find the inverse cosine: \[ A \approx \cos^{-1}(0.4706) \approx 62.3^\circ \]
Rounding to the nearest whole degree: \[ A \approx 62^\circ \]
Thus, the approximate interior angle measure at vertex A is \( \mathbf{62^\circ} \).