To determine the angle that correctly completes the law of cosines for the given triangle, we start with the Law of Cosines formula:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
Given the sides of the triangle:
- [tex]\( a = 7 \)[/tex]
- [tex]\( b = 25 \)[/tex]
- [tex]\( c = 24 \)[/tex]
First, we compute the squares of the given sides:
[tex]\[ a^2 = 7^2 = 49 \][/tex]
[tex]\[ b^2 = 25^2 = 625 \][/tex]
[tex]\[ c^2 = 24^2 = 576 \][/tex]
Next, we substitute these values into the Law of Cosines formula to solve for [tex]\(\cos(C)\)[/tex]:
[tex]\[ 576 = 49 + 625 - 2(7)(25) \cos(C) \][/tex]
Simplify the equation:
[tex]\[ 576 = 674 - 350 \cos(C) \][/tex]
Rearrange to solve for [tex]\(\cos(C)\)[/tex]:
[tex]\[ 674 - 576 = 350 \cos(C) \][/tex]
[tex]\[ 98 = 350 \cos(C) \][/tex]
[tex]\[ \cos(C) = \frac{98}{350} \][/tex]
[tex]\[ \cos(C) \approx 0.28 \][/tex]
Next, we find the angle [tex]\( C \)[/tex] using the arccosine function:
[tex]\[ C = \arccos(0.28) \approx 1.287 \text{ radians} \][/tex]
Convert the angle from radians to degrees:
[tex]\[ C \approx 1.287 \times \frac{180}{\pi} \approx 73.74^\circ \][/tex]
The angle closest to [tex]\( 73.74^\circ \)[/tex] among the given choices is:
- [tex]\( 90^\circ \)[/tex]
- [tex]\( 180^\circ \)[/tex]
- [tex]\( 74^\circ \)[/tex]
- [tex]\( 16^\circ \)[/tex]
Thus, the angle that correctly completes the law of cosines is:
[tex]\[ \boxed{74^\circ} \][/tex]
