To determine the measures of the angles in the right triangle with sides [tex]\( AC = 7 \)[/tex] inches, [tex]\( BC = 24 \)[/tex] inches, and [tex]\( AB = 25 \)[/tex] inches, we need to use trigonometric functions and properties of triangles. Given that [tex]\( AB \)[/tex] is the hypotenuse, we can confirm that triangle [tex]\( ABC \)[/tex] is a right triangle because it satisfies the Pythagorean theorem:
[tex]\[
AC^2 + BC^2 = AB^2
\][/tex]
Calculating, we have:
[tex]\[
7^2 + 24^2 = 25^2
\][/tex]
[tex]\[
49 + 576 = 625
\][/tex]
[tex]\[
625 = 625
\][/tex]
This confirms that [tex]\( \angle C \)[/tex] is [tex]\( 90^\circ \)[/tex]. Now, let's determine the other two angles using trigonometric ratios.
### Angle [tex]\( \angle A \)[/tex]
We use the sine function to find [tex]\( \angle A \)[/tex]:
[tex]\[
\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{AC}{AB} = \frac{7}{25}
\][/tex]
We find the angle measure:
[tex]\[
\angle A = \sin^{-1}\left(\frac{7}{25}\right) \approx 16.26^\circ
\][/tex]
### Angle [tex]\( \angle B \)[/tex]
We use the sine function to find [tex]\( \angle B \)[/tex]:
[tex]\[
\sin(B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AB} = \frac{24}{25}
\][/tex]
We find the angle measure:
[tex]\[
\angle B = \sin^{-1}\left(\frac{24}{25}\right) \approx 73.74^\circ
\][/tex]
### Verification
Since the angle sum property of a triangle must hold:
[tex]\[
\angle A + \angle B + \angle C = 180^\circ
\][/tex]
[tex]\[
16.26^\circ + 73.74^\circ + 90^\circ = 180^\circ
\][/tex]
Both calculations for [tex]\( \angle A \)[/tex] and [tex]\( \angle B \)[/tex] add up correctly with [tex]\( \angle C \)[/tex].
Thus, we have the following measures for the angles in the triangle:
[tex]\[
\angle A \approx 16.3^\circ, \quad \angle B \approx 73.7^\circ, \quad \angle C = 90^\circ
\][/tex]
Therefore, the closest matching option for the angles is:
[tex]\[
\boxed{m \angle A \approx 73.7^\circ, m \angle B \approx 16.3^\circ, m \angle C \approx 90^\circ}
\][/tex]
